NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET FAKULTET FOR INFORMASJONSTEKNOLOGI, MATEMATIKK

OG ELEKTROTEKNIKK

MASTEROPPGÅVE Kandidaten sitt namn: Torgeir Berge Sjølingstad Studieretning: Elektronikk Oppgåva sin tittel (norsk): Kartlegging av lydfelt i rom ved låge frekvensar Oppgåva sin tittel (engelsk): Distribution of sound fields in rooms at low

frequencies Oppgåva sin tekst: Oppgåva går ut på å analysere og kartlegge lydfeltet i eit rom ved låge frekvensar, basert på impulsresponsmålingar. Mogelegheitene for å bruke forenkla modalanalyse for verkelege, praktiske rom skal undersøkast. Modefunksjonar i rommet vert identifiserte t.d. ved målingar av lydfelt i ei rekke ulike punkt i rommet, og modefunksjonane sine parameterar blir funne. Fleire ulike rom bør analyserast og prediktert lydtrykksfordeling i rommet skal samanliknast med målt fordeling. Oppgåva gjeven: 17. januar 2005 Svaret skal leverast innan: 10. juni 2005 Svaret levert: 6. juni 2005 Utført ved: NTNU, Inst. for elektronikk og telekommunikasjon Rettleiar: Siv.ing. Lars Henrik Morset Trondheim, 17.01.2005 Peter Svensson, faglærar Institutt for elektronikk og telekommunikasjon

2

Acknowledgements I would like to thank Professor Peter Svensson at NTNU for invaluable help and guidance during the work on this project. Also I would like to thank Lars Henrik Morset for all assistance during the measuring, and tips along the way. Finally everyone at Ton Art deserve thanks for letting me do measurements in their store and giving me detailed descriptions of their listening rooms.

3

Abstract Room modes can be a problem at low frequencies and especially in smaller rooms. These modes may cause poor room acoustic, and different solutions have been suggested to deal with this problem. An even distribution of room modes along the frequency axis is desired for a room to function well acoustically. Mathematical models based on wave theory are generally accepted and used to calculate at which frequencies room modes appear, and how the sound pressure level (SPL) distributions appear at these frequencies. The validity of these models was tested by performing impulse response measurements in four different rooms. In each room, measurements were performed in several positions along the walls so that the SPL distributions could be observed. Using these measurements, the SPLs at the observed resonances, and in new positions were predicted mathematically. New measurements were performed in these positions and the measured SPLs were compared to the predicted SPLs. The measured modes that appeared most according to the theoretically calculated modes were those well separated from the nearby modes. Predicting SPL at these frequencies gave a mean error of 4 dB. The error was generally greater at the other frequencies, but the results were better when a high SPL was predicted. The error also increased with frequency, except in the room with the highest Schroeder frequency, where the error was similar at all resonances below 100 Hz. This shows that the method only gave good results at frequencies well below the Schroeder frequency. Asymmetrical properties in terms of room geometry and wall materials affected measurement results and caused SPL distributions different from the theoretical ones.

4

Contents 1 Introduction ........................................................................................................ 5 2 Theory ................................................................................................................ 7

2.1 Standing Waves............................................................................................ 7 2.2 Room modes ................................................................................................ 8 2.3 Room dimension ratios ............................................................................... 10 2.4 Predicting sound pressure level using room modes ..................................... 11

3 Experiments...................................................................................................... 12 3.1 Method of experiments ............................................................................... 12 3.2 Rooms........................................................................................................ 16

3.2.1 Basement room at Ton Art ................................................................... 16 3.2.2 Ground floor room at Ton Art .............................................................. 22 3.2.3 Room B-337 at NTNU Gløshaugen...................................................... 26 3.2.4 Room B-343 at NTNU Gløshaugen...................................................... 29

4 Measuring results and discussion ...................................................................... 33 4.1 Modes and resonant frequencies ................................................................. 33

4.1.1 Basement room at Ton Art ................................................................... 33 4.1.2 Ground floor room at Ton Art .............................................................. 40 4.1.3 Room B-337 at NTNU Gløshaugen...................................................... 48 4.1.4 Room B-343 at NTNU Gløshaugen...................................................... 52

4.2 Prediction of sound pressure levels ............................................................. 57 4.2.1 Basement room at Ton Art ................................................................... 57 4.2.2 Ground floor room at Ton Art .............................................................. 59 4.2.3 Room B-337 at NTNU Gløshaugen...................................................... 62 4.2.4 Room B-343 at NTNU Gløshaugen...................................................... 65

5 Conclusions ...................................................................................................... 68 6 Appendix .......................................................................................................... 70

6.1 Room modes .............................................................................................. 70 6.2 MATLAB................................................................................................... 74

5

1 Introduction

The dimensions in small rooms are comparable to the wavelengths of low frequency sound. This leads to reflections from the walls creating standing waves between them at certain frequencies. These frequencies are referred to as resonance frequencies, or room modes. At these frequencies, sound pressure maxima and minima will be observed in the room. Such modes are likely to create colourations in a room’s frequency response. This means that sound at some frequencies will be perceived as sounding louder than sound at other frequencies. In a rectangular room with rigid walls the appearance of a room mode will lead to a particular sound pressure level (SPL) distribution in the room. There exist simple models3 for predicting these distributions in rooms, and for calculating the frequencies at which the room modes appear. These mathematical models are based on wave theory, and use wavelengths and room dimensions in the calculations. In rooms where an even frequency response is desired, the room modes may become a serious problem. One way to deal with this problem is to find the room dimension ratios that give the most even distribution of room modes along the frequency axis. Methods for finding such ideal ratios have been developed by several researchers7,8,9, and much of this research uses the same simple models for the distribution of room modes. These models are in other words commonly accepted, and the conditions under which they are said to be valid, rectangular room shape and hard walls, are a common feature for many rooms. An interesting aspect is therefore to test under which conditions the theoretical models hold, and which conditions in the rooms might make the results deviate from the models. The validity of the models was tested in real rooms, some of which were specially designed listening rooms. By using impulse response measurements in several positions in the rooms, the sound fields and SPL distributions for the room modes were determined. The results from these measurements were compared to the expected results, based on the theoretical models. Furthermore, when having determined the SPL distributions at the resonance frequencies, these SPL distributions were used to predict the SPL in new positions in the rooms. Measurements were done in these positions and the results were compared to the predicted results. Only room modes appearing or expected to appear below 100 Hz were investigated. The results showed that a mode that is predicted to appear a minimum of Hz from the neighbouring modes is likely to display SPL distributions in accordance with the expected ones. This minimum distance to adjacent modes is dependent on the mode’s bandwidth, which is dependent on the room’s reverberation time (RT-60). Some of the rooms were almost square shaped, and this ledto an uneven distribution of room modes. The modes that are expected close to each other in frequency will interact to create a resultant room mode with a SPL distribution affected by more than one mode.3 This happened at most of the observed resonance frequncies in the rooms, resulting in unpredictable SPL distributions.

6

Other factors that affected the observed SPL distributions were the different wall materials. At hard concrete walls, SPL maximums occurred as expected most of the time, while glass doors, wooden walls, and absorptive material led to these maximums being reduced or absent. This is also in accordance with theory. However, exceptions occurred in several cases, and at a hard plaster wall there might be observed a SPL maximum at one resonant frequency, and a SPL minimum at another resonant frequency. Predicting SPL based on previously performed impulse response measurements proved not to give very good results. Some of the best results were observed at the resonant frequencies where modes uninfluenced by other modes appeared. Also, if a high SPL was predicted, the results were generally better than when a low SPL was predicted. A high predicted SPL indicated one mode dominating in that particular position. In the room with the highest Schroeder frequency, the errors were low at most resonance frequencies below 100 Hz. In the rooms with lower Schroeder frequency the error increased with frequency below 100 Hz. This indicates that the prediction method only works at frequencies well below the Schroeder frequency. The room dimensions ratios of the rooms where measurements were done resulted in very uneven distributions of room modes along the frequency axis. Therefore very few modes appeared without another mode close to it in frequency. Measurements done in a room with better ratios, would most likely give better results at more frequencies.

7

2 Theory 2.1 Standing Waves Sound waves propagating in a room with rigid walls will reflect off the walls. This results in these reflections interfering with each other, and at certiain frequencies, create sound pressure minima and maxima in different positions in the room1. This occurs because of the interfering being constructive in some places, and destructive in others. The lowest frequency where this occurs is where the wavelength is exactly twice the value of the longest room dimension, creating a standing wave condition between the two walls. At the parallel walls in each end of this dimension the particle velocity will by zero, creating pressure antinodes, i.e. maximum pressure at these walls. Halfway between the two parallel walls the pressure will reach its minimum value and have a pressure node. An example of this is displayed below in the left part of Figure 2.1. The x-axis on the plot shows the distance from one wall to the other parallel wall, and there are obvious pressure maxima at the walls, and a minimum halfway between the walls. The standing wave patterns appear like in Figure 2.1 because of them being products of cosine functions, given by6

tjnminmi

lmnezyxPp ϖ⋅= ),,( (2.1) zkykxkAzyxP ziymxnnmilmn coscoscos),,( = , (2.2)

where

xxn L

nk π= , n = 0,1,2,… (2.3)

yxm L

mk π= , m = 0,1,2,… (2.4)

zxi L

ik π= , i = 0,1,2,…, (2.5)

The mentioned lowest frequency at which a standing wave condition is set up will be one of the room’s fundamental frequencies. Subsequent standing mode patterns will occur at multiples of this frequency, making the distance between two adjacent nodes one half wavelength of the fundamental frequency. The room will have one fundamental frequency for each dimension, totally three, and they can be calculated by using n=1 in

,...)3,2,1(2

=⋅⋅= nLcnfn , (2.6)

with L being the dimension in question (e.g. length). Increasing n will give the frequencies to subsequent standing waves in the given dimension, which are called harmonics. These are the room’s resonant frequencies, which are likely to create coloration, i.e. favouring of these frequencies in the room’s frequency response. The right part of Figure 2.1 shows an example of a second order standing wave pattern,

8

with pressure maximums at the walls, and halfway between the two walls. The plots show graphs of cosine functions, generated by MATLAB.

Figure 2.1. Example of standing wave pattern, 1st order (left), and 2nd order (right). The size of a room determines to what degree standing wave patterns occur2. Such observations are likely to be done in fairly small rooms, where the dimensions are comparable to the wavelengths of the lowest audible frequency areas. This makes standing wave patterns a practical problem in most listening rooms and studios, which often have room dimensions of this size. 2.2 Room modes The resonance frequencies occuring at multiples of the fundamental frequencies may also be referred to as room modes3. The resonances described above are caused by reflections between two parallell walls, and are referred to as axial modes. Tangential modes include reflections from four surfaces and oblique modes include reflections from all six surfaces. In a three-dimensional room, resonance frequencies for normal modes can be found by using

+

+

=

222

2 zyxnmi l

ilm

lncf , (2.7)

where the integers n,m and i state the mode order for each dimension, and lx, ly, and lz are the room dimensions. In this paper the modes will be described by three numbers, e.g. (2,1,0), where these numbers denote the order of standing wave pattern in each dimension x, y and z, respectively. Consequently, the (2,0,0)-mode is an axial mode, (2,1,0) is tangential), and (2,1,1) is oblique. A (1,2,0)-mode would have the pressure distribution shown in the left plot in Figure 2.1 along the x-axis, and the distribution shown in the right plot along the y-axis. The pressure along the z-axis would in this case be constant, shown in normalized form in this report. The observed standing wave patterns for each dimension x, y and z will in this paper be referred to as pressure distributions, or standing wave patterns of the first order, second order, etc. Figure 2.1 display standing wave patterns of the first and second order. The combination of the patterns in each direction will be referred to as the mode itself, e.g. a (2,1,0)-mode.

9

With increase in frequency, oblique modes will start to dominate the room. The modes will appear with smaller frequency spacing, and eventually the number of exited modes in the room will be so high that the room is considered to contain diffuse sound fields,4. A way to estimate the frequency at which this transition happens, is to use the Schroder frequency6:

VTf 2000≥ , (2.8)

where T is the reverberation time T-60, and V is the volume of the room. The room modes contribute in creating peaks and dips in the frequency response measured in the room in question, and should according to theory appear at the frequencies calculated by equation (2.7). But not all modes are equally significant or likely to affect the perception of audio in the room. Gilford2 lists five different factors that contribute to determining the effect of room modes. These factors include:

• the bandwidth of the mode • degree of excitation of the mode • the mode’s frequency spacing from adjacent significant nodes • the position of the sound source and recording- or measuring microphone • the frequency response of the sound source and microphone.

The bandwidth of a mode is dependent on the fundamental frequency and the absorption of the walls, i.e. the reverberation time. Everest3 lists the mode bandwidth for different reverberation times, these are displayed in Table 2.1. Reverberation time [s] 0.2 0.3 0.4 0.5 0.8 1.0 Mode bandwidth [Hz] 11 7 5.5 4.4 2.7 2.2 Table 2.1. Mode bandwidth dependent on reverberation time Morse and Bolt5 state that an axial mode has twice the energy of a tangential mode, which in turn has twice the energy of an oblique mode. This is because the transmission loss increases with the number of walls involved in the reflections. In other words, axial modes should be given the largest amount of attention. Following in this vein, Gilford2 states that only axial modes can, as opposed to tangential or oblique modes, be individually significant. Using this conclusion, he says further that coloration, i.e. frequencies being boosted because of resonance, is likely to occur if an axial mode is spaced 20 Hz or more from the adjacent axial modes. A normal mode will be fully exited where it has a pressure maximum, and as noted above, this occurs at a hard wall. In a room with hard walls, a corner of the room contains three hard surfaces. So if a loudspeaker is positioned in this corner, all modes in all dimensions will be fully exited.6 Having the speaker placed where a mode has a pressure minimum would lead to this mode not being fully excited, and would lead to a lower perceived sound pressure level.

10

2.3 Room dimension ratios The distribution of room modes along the frequency axis can seriously affect a room’s acoustic properties. This becomes obvious if a room has identical room dimensions, i.e. length, width and height are equal. Using equation (2.2) to find such a room’s modes would show that the room modes appear at the same frequencies for all dimensions, leading to the room favouring these frequencies, and creating frequency areas with low or no resonance. Ideally, the room modes should be evenly spaced along the frequency axis. A fair amount of research has been done in trying to find the ideal room dimensions for this purpose. Bolt7 presented a model for determining the number of normal frequencies below a certain frequency, and the average spacing between these modes. Using these criteria he created what has become known as Bolt’s graph or Bolt’s chart. Room dimension ratios falling inside Bolt’s chart were acceptable ratios. Sepmeyer8 focuses on both frequency and angular distribution of normal modes. He concludes with three sets of room dimension ratios that give the most ideal rooms. These are listed below in Table 2.2.

Height: Width: Length:Design 1: 1.00 1.14 1.39 Design 2: 1.00 1.28 1.54 Design 3: 1.00 1.60 2.33

Table 2.2. Ideal room dimension ratios, suggested by L.W.Sepmeyer. Similar ratios were suggested by M.M.Louden3:

Height: Width: Length: Best ratio 1.00 1.40 1.90 2nd best ratio 1.00 1.30 1.90 3rd best ratio 1.00 1.50 2.10

Table 2.3. Ideal room dimensions ratios, suggested by M.M.Louden. Bonello9 focuses on the energy within each one-third octave band, exploiting the limited ability of the human ear to differentiate between modes of similar frequencies. By finding the number of modes within each one-third octave band, the energy in each band is found, and Bonello’s criteria can be used to determine if the room’s dimensions are acceptable. These criteria are:

1) Each one-third octave should have more modes than, or the equal amount of modes as the preceding one.

2) No modes should appear at the same frequency, unless there are five or more modes in that particular one-third octave band.

One potential flaw with the method is that it does not take into account the difference between axial, tangential and oblique modes.

11

2.4 Predicting sound pressure level using room modes When knowing a room’s distribution of modes, it should theoretically be possible to predict the sound pressure level at any given point in the room, at any given frequency. When taking into consideration only y- and x-direction, i.e. using measuring points at the same height as the reference point, mathematically, the sound pressure in a certain point (x,y) in a room is found by10

( )( )

( )( )refref

ref

refref

refrefref yxp

yxpyxp

yxpyxpyxp

,,

,,

)(),( , ⋅⋅= . (2.9)

The point (xref,yref) can be defined as being anywhere in the room, but it is decided to use the corner opposite to the one where the measuring loudspeaker was positioned. Using the sound pressure from (2.9), a relative sound pressure level in the position (x,y) is found by

)),((log20 10 yxpSPL ⋅= (2.10) The method using equations (2.9) and (2.10) to predict sound pressure is further described in chapter 3.1. When performing measurements in a room in order to investigate its sound field, some measurements will be done closer to the measuring loudspeaker than others. Perceived sound pressure in a room consists of direct sound, pdir and sound reflected from walls and other surfaces, prefl, i.e.,

)( refldir ppp += . (2.11) Pdir is inverse proportional with r and as r 0, pdir ∞. This leads to the conclusion that if possible, measurements show not be done close to the loudspeaker. This is shown in chapter 3.1.2, under the header Comparing parallell walls, where results from measurements close to the loudspeaker are compared with results from measurements along the parallell wall that is further away from the loudspeaker.

12

3 Experiments A special method of experiments, using impulse response measurements, was used to detect room modes and their sound pressure level (SPL) distributions in four different rooms. The results from these first measurements were used in a method for predicting sound pressure levels at resonance frequencies at specific positions in the rooms. The methods were suggested by professor Peter Svensson at NTNU, who had also used the first method earlier. Both these methods are explained in the following. 3.1 Method of experiments For the following method to work in practice, the room in which measurements were done had to be more or less rectangular, so that the simple models for SPL distribution at room modes could be used. A corner in the room was defined as (0,0,0) in the room’s coordinate system, and impulse response measurements were done along the walls in all directions x, y and z, from this corner. The measurements were done along the walls because the SPL has a maximum here. In some cases measurements were also performed along the walls parallel to these. These measurements were done with a microphone placed on the floor, one cm from the wall. Each measurement was done 25 cm from the last one. For the measurements in the z-direction the microphone was hand held. The measurements were performed with a Genelec loudspeaker in one corner of the room in question, and the excitation was a sine sweep generated by WinMLS, going from 16 Hz to max (half the sampling frequency fs), with fs = 11025 Hz. The loudspeaker position in the corner was selected because the loudspeaker will then excite all normal modes of the room. A mode will be fully exited where it has a pressure antinode, and as noted above, this occurs at a hard wall. In a corner containing three hard surfaces, all modes in all dimensions will be fully exited. To move the speaker front as close to the corner as possible, the speaker was positioned so that it was facing the corner. The results from these measurements contained the measured frequency responses in all the positions along directions x, y and z, and by comparing these frequency responses the SPL distribution in the room could be plotted. An example of this is shown in Figure 3.1. The left plot shows the different frequency responses measured 25 cm apart along the axis in question, here the x-axis. By selecting a frequency in the plot of the frequency responses, the SPL distribution along this axis at the selected frequency is shown in the right part of Figure 3.1. In this example the wall was almost six meters long, and the x-axis therefore goes to six meters. As seen in the plot, at the selected frequency, approximately 40 Hz, there is a maximum at x = 0 and x = 5.96, creating a 1st order standing wave pattern. The positions of the first and last measurements will from now on be denoted by 0 and max subscripts, i.e. x0 and xmax for the x-measurements.

13

Figure 3.1. Example of frequency responses (left) and SPL distribution (right) along the x-axis. After having determined the SPL distributions in the room, nine positions in each room were selected, some of which would normally be used as listening positions. For simplicity, and because there often was no SPL distribution along the z-axis for low frequencies, only positions on the floor were used in these experiments. The SPL in these position, at certain frequencies, was then calculated using the measurements described above. This was done the following way: For a certain position (x,y), the measured SPLs at positions (x,yref) and (xref,y) were used to predict the SPLs in position (x,y), as showed in Figure 3.2 below. A corner in the room was defined as (xref,yref). In Figure 3.2 this corner is the same as (0,0,0); in some rooms the opposite corner was defined as (xref,yref), depending on the position of the loudspeaker. Finally, equation (2.9) and (2.10) was used to predict the SPL in position (x,y). The frequencies for which these calculations were done were selected among the observed resonances in each room. After having calculated, i.e. predicted, the SPL in the nine selected positions, there were done impulse response measurements in these same positions. The measured SPL in each position was then compared to the predicted SPL in each position. The predicted sound pressure level was then compared to the measured one in each point, and the errors were plotted and compared for different positions, frequencies and rooms.

Figure 3.2. Prediction of sound pressure level in position (x,y) in a room.

14

Matlab code written by Peter Svensson was modified and used for plotting the frequency responses and SPL distributions. Other equipment used during the measurement and/or analysis of the results:

• WinMLS 2000 • Free field microphone type 4190, sensitivity –25,9 dB re 1 V/pa. • MATLAB • Yamaha HA8 microphone preamplifier

Genelec loudspeaker ( • Figure 3.3) • VX Pocket 2-channel sound card

Figure 3.3. Genelec loudspeaker that was used for the measurements. Why multi-channel measuring was rejected To measure more effectively, a special microphone stand, being able to hold up to five microphones, was made. This method was not used for the measurements, but it is commented here because it explains why a single microphone was used for the measurements. The original intension was to use eight microphones, as the sound card to be used had eight outputs, but due to lack of available microphones, only five were used. This array of microphones was used for measuring alongside one wall in the room next to the anechoic chamber at NTNU, and along all three directions (x, y and z) in the audio laboratory (C 031). None of the measurements gave good results, so to be sure the microphones were equal in frequency response and other qualities, a carefully executed evaluation of calibration was done in the anechoic chamber. The microphone array with its five microphones was placed in the anechoic chamber, at a distance of 2.00 meters to a Genelec loudspeaker. The cables were laid through the door, so that the chamber wasn’t entirely closed during the meaurements, but most probably this only had a neglectable effect on the measurements. Each microphone was first calibrated using a calibrator at 1000 Hz at 93.8 dB. Then a sine sweep was generated from WinMLS, and the measurement was saved. The microphone array

15

was then moved 25 cm, so that the next microphone ended up in the exact same position as the first microphone. The same sine sweep was generated, the measurement was saved, and the procedure was repeated for all five microphones. The five measurements should then according to theory, end up being identical. And in terms of the shape of the curve, they did. But microphone four and five were extremely ustable level wise, even though all microphones had been calibrated in advance

Pos2Ch2_Pos2Ch5_Pos2Ch4_Pos2Ch3_Pos1Ch2_

Smoothed SPL Frequency Response Magnitude

Name: Pos1Ch1_Measured - 16:47:35, 03Mar2005 Plotted - 15:45:28, 11Mar 2005

Frequency [Hz]100 1 000

[SPL

], [d

B]

100

95

90

85

80

75

70

65

60

55

x=132.81 Hzy=89.882 dB

WinMLS Pro

Figure 3.4. Frequency responses from five different microphones in anechoic chamber. Naturally, microphone four and five couldn’t be used for measurements, so the alternatives were to get hold of new microphones, use a microphone array of three microphones, or use only one microphone. The last alternative was chosen for a test run of measurements in the audio laboratory. And when these results turned out much better than the ones from the last measurements in the same room, this alternative was also chosen to be used for further measurements. The most obvious advantage with multiple channel recording/measuring is the time aspect. With five microphones measuring at the same time, the measuring ought to be five times faster than with only one microphone. This turned out to be only partly true. In the rooms where the measurements were done, various objects were placed on the floor, some of them quite heavy, and fairly close to the walls. This could be amplifiers, computers, loudspeakers etc. With five cables going from the sound card to the microphone amplifier, five cables going to each respective microphone, and the microphone array itself, moving the array to the different measuring positions proved to be not as straight forward as hoped for. Especially the many cables created problems, and got easily tangled together. One alternative would be to move all these objects out of the room, but this would take some time, and the time aspect was the one in question here. If the room was empty in advance, or was needed to be empty for the measurements anyway, this would be an entirely different situation; then a multiple channel set-up probably would be the most effective. But as the microphones gave such varying results during the evaluation of the calibration, the safest way in this case, at least, was to use only one microphone. All measurements described in this paper were therefore done with only microphone.

16

3.2 Rooms Four rooms in total were measured, each room was different from the others in terms of geometry, size, acoustical treatment etc. Two of the rooms were listening rooms at the hi fi store Ton Art in Trondheim, and these two rooms were perhaps the ones most similar to each other, having the same acoustical designs in terms of absorptive materials on the walls, and having similar dimensions. The rooms are in the following described as accurate as necessary for the focus of this thesis, which is room modes at low frequencies. The geometry, room dimension ratios, wall materials, and reverberation times (RT) are given for each room. The RTs may be seen as being properties of the rooms more than results from the methods used in the experiments. They are therefore included here instead of in the results-chapter. The same applies to the calculation of room romes and the distances between them. These calculations were done independent of the resonances and SPL distributions that were actually observed. The Bonello criteria are tested on each room to get a picture of how the modes in the room are distributed along the frequency axis. As the Ton Art rooms are specially designed listening rooms, their room dimensions are also compared to some theoretically calculated ideal ratios. All dimensions in sketches of the rooms are inner dimensions, while calculations for finding room modes used outer dimensions.

3.2.1 Basement room at Ton Art The basement room at Ton Art was supposed to give a good listening experience, but the owners and employees at Ton Art were not satisfied with it*. The main problem was said to be at low frequencies. Recordings which ought to give a listening experience with a heavy bass, would lack the bass, making the room seem dead in significant low frequency areas. The room’s dimensions were as shown in Figure 3.5, where the nine measurement positions, pos.1-pos.9, for the prediction of SPL are also shown. Measurements were done with the loudspeaker in the corner between walls C and F. The position on the floor in the corner between walls A and B was defined as origo in the room’s coordinate system. Measurements were done with microphone positions along wall A, defined as the x-axis, and along wall B, defined as the y-axis as showed to the right in Figure 3.5 below. In the z-direction measurements were done vertically in the corner between walls A and B. Behind the end wall, wall C, there was a storage room 1.85 m deep, with a concrete wall at the end (wall D). As wall C was a light wooden wall which might not reflect * After the measurements for this project was finalized, Ton Art decided to move the back wall, wall C in this room, changing the dimensions. Accordint to the employees at Ton Art, this caused a major improvent in bass respons in the room.

17

all low frequencies as well as walls A and B, measurements were also done inside this room, along the wall in the y-direction. The loudspeaker was also during these measurements positioned in the main room, in the corner between walls F and C. The storage room was filled with mainly cardboard boxes, and there was a ventilation system on the concrete wall inside this room. Some of the plots that show the SPL distribution along the y-axis also include these measurements from the storage room, in order to investigate the effect of such a lightweight boundary for low frequencies.

Figure 3.5. Listening room in basement at Ton Art. The dimensions are for the inner walls. Wall E was a 10 cm thick lightwall with no insulation, and wall D was a naked concrete wall. Other wall materials are shown in Figure 3.6 below. The most important difference between the walls is walls A, B and D being concrete walls, while C and F are lighter wooden walls. Walls A and B was therefore be suspected to be more reflective than the other walls, paricularly at low frequencies, while wall D would be highly reflective at all frequencies, being a naked concrete wall. The room was almost squared in shape, but slightly umsymmetrical, as the length was 6.35 along wall B, and 6.15 m along wall F. The width was 5.97 m and the height 2.20 m. But because of the lathing and insulating creating a light inner shell compared to the concrete behind it, different room dimensions than the ones displayed in Figure 3.5 were used when calculating the room’s mode frequencies. This is because the insulation material will be transparent for low frequency waves with long wavelengths, which will be reflected by the concrete walls†. As for wall C, it was decided to include the 5 cm of insulation when setting the dimensions used in these calculations, as the insulation probably is more transparent to low frequencies than the wooden wall. Wall F had as 18 cm thick layer of wooden OSB plates facing the room, so it was decided to use the inner dimensions for this wall in the calculations.

† G. Ballou, “Handbook for Sound Engineers”, Howard W.Sams & Co, 2nd ed., p. 48 (1991)

18

The ceiling was conciderably lowered from the concrete ceiling. Perforated plaster plates of the same kind as the ones on the wall (see Figure 3.9 below) covered the surface facing the room. Above these plates there was 5 cm with absorbing material. Above this the room was divided in half, parallell with wall B, see Figure 3.5 and Figure 3.7 below. The half closest to the two doors had ca 20 cm of open space between the insulation material and the concrete, leading to a total distance of 25 cm from the inner ceiling to the concrete ceiling. The other half of the room had concrete right above the absorbing material. Using mean values for the length and height due to the rooms asymmetrical attributes, the following dimensions were used for the mode calculations:

40.605.010.02

)15.635.6( =+++=Length m

(3.1) 07.610.097.5 =+=Width m

(3.2)

35.22

45.225.2 =+=Height m

(3.3)

Figure 3.6. Wall materials, listening room in basement at Ton Art.

19

Figure 3.7. Material in the ceiling, basement room at Ton Art.

The floor was hard, but with a soft carpet, sized 1.66*2.25 = 3.74 m2 , and 1 cm thick, placed in the middle of the room. Moveable drapes of the type displayed in the middle in Figure 3.8 below were placed on walls A (two drapes sized 2.60 m2), B (one sized 2.60 m2 and C (two sized 1.50 m2). Wall F had three thinner drapes (total size 2.75 m2), see left part of Figure 3.8. Behind wall F there was open space, i.e. a hallway. A probable weak spot, as will be discussed later, was the glass door on wall F. One set of measurements were done with this door open, and these measurements are compared to the ones with door closed. The door can be seen in Figure 3.8 below. As expected, the door being open made the measuring results less in accordance with theory than when the door is closed. The most useful results in terms of a typical listening situation are thus the ones from the measurements with the door closed and these results are given main focus.

Figure 3.8. Drapes on the walls in the basement room at Ton Art. Wall F (left), and wall B (middle). Walls A and C had the same type of drapes as wall B. Open glass door on wall F (right). Walls A, B and C were covered with perforated plaster plates and OSB distributed as showed in Figure 3.9 below. Wall F was covered with OSB all over.

20

Figure 3.9. Distribution of OSB and plaster plates, walls A, B and C, basement room at Ton Art (left and right), listening positions when loudspeakers are placed along wall A (right). A natural position for the loudspeakers in a listening situation would be along wall A, while the listening position would be 1.20 m from the back wall C, and halfway from wall F to B (see Figure 3.5). In carthesian coordiantes, a typical listening position would be the point (3.00, 5.00, 1.20), although there were placed several chairs in the room, so the listening position might vary, especially in x-direction. This is shown in Figure 3.9 above. Pos.7, pos.8 and pos.9 in Figure 3.5 above are typical positions for the chairs in Figure 3.9. Another possible loudspeaker position sometimes used was along wall B, making the point (4.25, 4.75, 1.20) a natural listening position. This position is defined as pos.4 in Figure 3.5. The measured reverberation times for each octave band are displayed in Table 3.1. The results are from two different measurement sets. The first set was done with thin carpet samples placed in the corners and along the walls in the room, while the second set was performed with a new carpet placed all over the floor. From this point on, all references to RT in this room will refer to the values from “RT without carpet” in Table 3.1. The RT-values are mean values, using the nine positions pos.1 – pos. 9 in Figure 3.5. The RT-60 values in the room were measured to be low, particularly for high frequencies. In the 250 Hz octave band and higher frequency bands, the RT varied between 0.20 s and 0.25 s depending on measuring position. Octave bands 63 Hz 125 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz RT without carpet [s] 0.71 0.38 0.24 0.21 0.21 0.22 RT with carpet [s] 0.64 0.48 0.23 0.20 0.21 0.22 Table 3.1. RT-60 for each octave band for the basement room at Ton Art. The Schroeder frequency might be expected to be in the 125 Hz-band, so using this band’s RT, 0.38 s in equation (2.8), the schroder frequency is calculated to be f = 128 Hz. The selected frequency area between 20 and 100 Hz should therefore be well inside the area where room modes are most likely to be observed.

21

As mentioned earlier‡, having almost equal length and width in a room makes the room modes appear with an uneven spacing along the frequency axis. This is not a desired situation when wanting to create best possible listening conditions, and as shown later, this also affects the observed standing wave patterns. When room modes appear close to each other, these modes affect each other and together create resonance, displayed as a peak in the frequency response. Unless one of these modes dominate over the others, the observed SPL distributions are products of several modes, and these distributions often look less like theoretically expected room modes than modes not appearing in the vicinity of other modes. The dimensions for the hard outer shell of the room were width = 6.12, length = 6.40 and height = 2.35. Using these dimension the ratios between the room dimensions could be found. These were:

Height: Width: Length: Ratios, basement room at Ton Art 1.00 2.60 2.72

Table 3.2. Room dimension ratios, basement room at Ton Art.

Compared to the room dimension ratios suggested by Sepmeyer and Louden‡, these ratios were very different. In addition to the width and length being almost equal, another reason for this was the low ceiling. A height of 2.35 meters is a very small number compared to six meters. These ratios suggest that the room had a very uneven distribution of modes. The theoretically calculated modes for the room are listed in Table 7.1 in the appendix, and according to that list, no modes should appear between 28 Hz and 39 Hz, between 39 Hz and 54 Hz, and between 62 Hz and 73 Hz. This fact appeared to match the measured results fairly well, with no resonances appearing in the mentioned intervals (with the glass door closed). A way of analyzing the distribution of modes along the frequency axis is to use Bonello’s criterion. The use of Bonello’s criteria‡ on the room’s distribution of modes gave the result displayed in Figure 3.10 below. The room fulfilled the criteria of more modes or equal amount of modes for bands of higher frequency. But a better distribution would not make the curve in Figure 3.10 so suddenly steep already at 80 Hz. The 63 Hz band has two modes, while the 80 Hz band has eight modes.

‡ See chapter 2.3 on ideal room dimensions.

22

Figure 3.10. Calculation of resonance modes in each 1/3-octave band for the basement room at Ton Art. Focusing only on the axial modes, which have the most energy§, the distance between

each axial mode appear below 100 Hz is listed in Table 3.3. This table shows that no mode below 100 Hz has a big spacing in frequency in both ends. The (0,0,1)-mode at 73.2 Hz is the only mode that has more than 5 Hz distance to another axial mode both downwards and upwards in frequency.

n m i Frequency Distance to next axial mode

0 1 0 26.9 Hz 1.2 Hz 1 0 0 28.1 Hz 25.7 Hz 0 2 0 53.8 Hz 2.4 Hz 2 0 0 56.2 Hz 17.0 Hz 0 0 1 73.2 Hz 7.4 Hz 0 3 0 80.6 Hz 3.7 Hz 3 0 0 84.3 Hz

Table 3.3. Distance between axial modes, basement room at Ton Art.

3.2.2 Ground floor room at Ton Art This room was according to the employees of Ton Art their best listening room. An interesting aspect is thus to compare the results from the measurements in this room with the ones done in the listening room in the basement, which they were not satisfied with. A sketch of the room is shown in Figure 3.11, with the inner dimensions and the nine measurement positions where the SPL was predicted. As seen in Figure 3.11, the position defined as (0,0,0) in the room’s coordinate system is the position where the measuring loudspeaker was positioned, in the corner between walls A and B. Measurements were done along all four walls, resulting in measurement sets x, y, x2 § See chapter 2.2 on room modes.

23

and y2 as seen on the right in Figure 3.11. But because of the closeness to the measuring loudspeaker, measurement sets x and y could not be used in the analysis. So unless otherwise stated, all references to the x-axis or y-axis refers to measurement sets x2 and y2 in Figure 3.11, along walls C and D, respectively. Measurements were also done along the z-axis in all corners of the room. Because of the position of the measuring loudspeaker, set z2 was useless, but sets z, z3 and z4 gave very similar results. One of them, set z4, is therefore used when plotting the results.

Figure 3.11: Listening room, ground floor at Ton Art. The dimensions are for the inner walls. This room was smaller than the basement room, with length 5.14 m, length 4.28 m, and height 2.38 m, as shown in Figure 3.11. The floor was reasonably hard, with a carpet covering one layer of tiles, with concrete below the tiles. Wall A (see Figure 3.11) was a hard wall with 5 cm insulation material next to a 30-40 cm thick concrete layer. Wall B was a thinner, 18 cm thick wooden wall. Walls B and D were divided in three parts horisontally, as shown left in Figure 3.12, and in

Figure 3.13.

24

Figure 3.12. Distribution of OSB and plaster plates on walls B and D, ground floor room at Ton Art (left), material in wall D, ground floor room at Ton Art (right) Wall D and C had the materials as shown right in Figure 3.12 and in Figure 3.14, respectively. Wall C is also depicted in Figure 3.15.

Figure 3.13. Picture of wall D, ground floor room at Ton Art.

Figure 3.14. Wall materials, wall C, ground floor room at Ton Art.

25

Figure 3.15. Picture of wall C, with coach in listening position, ground floor room at Ton Art. The ceiling was covered with wood boards, 18 mm apart, as seen in the upper part of Figure 3.13 above. Above these there was 5 cm with absorptive plates, and 5 cm of open space. The loudspeakers in a listening situation would be placed along wall A, while the listening position would be ca 1.25 m from the back wall C, and halfway from wall F to B, i.e. in the coach in Figure 3.15 above. The positions pos.7, pos. 8 and pos. 9 are partly typical listening positions, particularly pos.8, having the coordiantes (2.38, 3.75). A natural listening position would of course be 1.20 m over the ground, while pos. 8 has z=0. As this was a smaller room than the basement room, and with less hard walls, the reverberation time was even shorter here than in the basement room, as displayed in Table 3.4 below. For the higher octave bands, the RT was 0.20 s in the 250 Hz band, and gradually increased with frequency, being 0.27 s in the 2 kHz band. Octave bands 63 Hz 125 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz RT [seconds] 0.56 0.32 0.20 0.23 0.25 0.27 Table 3.4. RT-60 for each octave band for the first floor room at Ton Art. Again expecting the Schroeder frequency in the 125 Hz, band, using 0.32 s as the RT in (2.8), the schroder frequency is calculated to be 150 Hz. This room being smaller than the basement room made the Schroeder frequency larger for this room. Using the dimensions the ratios between the room dimensions can be found. These are

given in Table 3.5 below, along with Sepmeyer’s design 3‡, which of the ratios mentioned in chapter 2, are the ones closest to the ratios on the ground floor room. Compared to the basement room they were much better.

Height: Width: Length: Ratios, ground floor room at Ton Art 1.00 1.77 2.11 Sepmeyer’s design 3: 1.00 1.60 2.33

26

Table 3.5. Room dimension ratios, ground floor room at Ton Art. The above ratios should imply that the room had a more even distribution of room modes along the frequency axis than the basement room. The use of Bonello’s criteria on the room’s distribution of modes gave the results displayed in Figure 3.10 below. Compared to the results in the basement room in Figure 3.10, this plot is surprisingly similar, considering the assumingly better ratios ( Table 3.5), and the satifaction with the room from the owners of Ton Art compared to the basement room. The room was approved in accordance with the criteria, and the slope is a bit less steep than the one observed for the basement room, which implies a more even distribution.

Figure 3.16. Calculation of resonance modes in each 1/3-octave band for the ground floor room at Ton Art.

Table 3.6 shows the distance between each axial mode below 100 Hz in the ground floor room, and it is worth noticing that only five axial modes appear below 100 Hz, and the distance in frequency between them is very uneven.

n m i Frequency Distance to next axial mode

0 1 0 32.8 Hz 6.5 Hz 1 0 0 39.3 Hz 30.1 Hz 0 0 1 69.4 Hz 9.1 Hz 2 0 0 78.5 Hz 20.0 Hz 0 3 0 98.5 Hz

Table 3.6. Distance between axial modes, ground floor room at Ton Art.

3.2.3 Room B-337 at NTNU Gløshaugen

27

Rooms B-337 and B-343 were meeting rooms located at NTNU Gløshaugen. In general, they had not been given the acoustical treatment that had been given to the listening rooms at Ton Art. With the exception of curtains in front of the windows, none of the walls had any sort of absorbing material. The exception was the ceiling, which was partly covered with perforated plates plates. These are displayed in Figure 3.19 below. But where there were insulation material behind the similar plates in the basement room at Ton Art, no such material was placed behind these plates, naturally making the absorptive effects of the plates suffer. Room B-337 had the dimensions as in Figure 3.17 below. Wall D was a hard concrete wall, and so was wall B, but only up to a height 1,03 m, up from which the wall mainly consisted of windows, as seen in Figure 3.18 below. Walls A and C were 20 cm thick plaster walls with insulation, and the wall was hard and without any sort of carpet. As all the inner walls were made of either concrete or plaster, and should therefore be highly reflective at low frequencies, these inner dimensions were also used when calculating the expected mode frequencies. The positions used when predicting sound pressure levels are also showed in Figure 3.17. As this was not a listening room as such, these positions are chosen somewhat randomly.

Figure 3.17. Room B-337 at NTNU Gløshaugen, department of acoustics.

28

Figure 3.18. Distribution of wall materials and windows, wall B, room B-337 and B-343 at NTNU Gløshaugen.

Figure 3.19. Perforated acoustic plates used in the ceiling in rooms B-337 and B-343 at NTNU Gløshaugen. As the basement room at Ton Art, this room had the problem of being practically squared in shape, leading to large frequency areas with no resonance frequencies. The measured reverberation time was longer in this room than in the Ton Art rooms, due to the lack of absorbing materials on the walls and floor. Some curtains hung in front of the windows, partly lifted aside. Apart from these curtains, most surfaces, i.e. walls, chairs, tables etc, were hard and reflective. The RT-values are displayed in Table 3.7 below. Octave bands: 63 Hz 125 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz RT [seconds] 1.02 0.85 0.61 0.46 0.43 0.43 Table 3.7. RT-60 for each octave band for room B-337 at NTNU Gløshaugen. Using the RT 0.85 s from the 125 Hz band where the Schroeder frequency might be expected, results in this frequency being 181 Hz.

29

As this room was even more squared in shape than the Ton Art-rooms, an uneven distribution of modes along the frequency axis would come as no surprise. Using Bonello’s criteria on the room confirms this suspicion, as displayed in Figure 3.20 below. The room did not match the criteria of the test, as the 40 Hz, 50 Hz and 80 Hz bands all included fewer modes than the previous band.

Figure 3.20. Calculation of resonance modes in each 1/3-octave band for room B-343 at NTNU Gløshaugen. Table 3.8 shows the distance between the axial modes appearing below 100 Hz in this room. The table shows that below 100 Hz, two frequency areas almost 30 Hz in size have no axial modes. The axial modes that do appear come in clusters of two or three modes that are destined to affect each other unless one of the would tend dominate over the other.

n m i Frequency Distance to next axial mode

1 0 0 28.9 Hz 0.8 0 1 0 29.7 Hz 27.6 0 0 1 57.3 Hz 0.5 2 0 0 57.8 Hz 1.5 0 2 0 59.3 Hz 27.4 3 0 0 86.7 Hz 2.3 0 3 0 89.0 Hz

Table 3.8. Distance between axial modes, room B-337 at NTNU Gløshaugen.

3.2.4 Room B-343 at NTNU Gløshaugen This room was similar to B-337 in terms of wall material. Walls D, C, E and F were hard concrete walls, wall B was hard up to the height 1,03 m, up from which there were windows, as seen in Figure 3.18 above. Wall A was a 20 cm thick plaster wall with insulation, and there was a small kitchen room behind the wall. And the ceiling was covered with similar

30

plates as in room B-337, but with no insulation material behind them. The main difference between this room and B-337 is the size of the rooms, with this room being almost twice as long as it is wide. The room’s geometry is displayed in Figure 3.21, and as seen there, the room is not entirely rectangular. This asymmetry is also showed in Figure 3.22

Figure 3.21. Room B-343 at NTNU Gløshaugen, department of acoustics.

Figure 3.22. Unsymmetrical properties of room B-343 at NTNU Gløshaugen.

The loudspeaker was placed in the corner between walls C and D because both these were hard concrete walls, and would thus help excite as many modes as possible. One thing that might have affected the measurement results is that the loudspeaker in that corner had to be placed below the cabinet that is depicted in Figure 3.23. This cabinet may have affected particularly the excitation of modes along the z-axis.

31

Figure 3.23. Cabinet in the corner between walls A and B in room 343, under which the loudspeaker was positioned. Measurements were done along side walls A and B, but as wall A was shorter than the whole width of the room, four measurements were done along wall E to complete the measurement set in the x-direction. But these measurements do not naturally belong together with the measurements done along wall A, because any standing wave conditions along wall A would be caused by reflections from walls B and F, and would create SPL maximums at these walls. The measurements along wall E are nevertheless included in some plots, to illustrate how this geometrical irregularity in the room may create a more complex sound field in the room. As this room is twice as big as B-337 it might be expected that the reverberation time is longer. This is not the case, which can be seen by comparing Table 3.7 and Table 3.9 below, where the RT-values are displayed for rooms B-337 and B-343, respectively. Octave bands: 63 Hz 125 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz RT [seconds] 1.02 0.89 0.66 0.48 0.44 0.47 Table 3.9. RT-60 for each octave band for room B-343 at NTNU Gløshaugen. Using RT = 0.89 s in equation (2.8) results a Schroeder frequency of 139 Hz. When calculating mode frequencies, the dimensions x = 5.80, y = 10.04 and z = 3.00 are used. Here, the x- and y-values are mean values calculated because of the unsymmetrical appearance of the room. The Bonello criteria are used on the room in Figure 3.24, and the room failed the test miserably, with both the 40 Hz- and 80 Hz- 1/3 octave bands having more modes than the previous band. The geometry of the room was the cause of this. The mode appearing in the 16 Hz band may be ignored, as it is not in the spectrum audible for the human ear.

32

Figure 3.24. Calculation of resonance modes in each 1/3-octave band for room B-343 at NTNU Gløshaugen. Table 3.10 shows the distance between the axial modes that appear below 100 Hz in this room. As in the other three rooms, the modes are rather unequally distributed along the frequency axis. But as many as eight (not counting the 17.1 Hz mode) axial modes appear below 100 Hz, making the biggest gaps with no axial modes 17 Hz wide, which may be to narrow to be noticable to the human ear.

n m i Frequency Distance to next axial mode

0 1 0 17.1 Hz 12.6 Hz 1 0 0 29.7 Hz 4.6 Hz 0 2 0 34.3 Hz 17.1 Hz 0 3 0 51.4 Hz 5.9 Hz 0 0 1 57.3 Hz 2.0 Hz 2 0 0 59.3 Hz 9.2 Hz 0 4 0 68.5 Hz 17.2 Hz 0 5 0 85.7 Hz 3.3 Hz 3 0 0 89.0 Hz

Table 3.10. Distance between axial modes, room B-343 at NTNU Gløshaugen.

33

4 Measuring results and discussion All measuring results from the four different rooms are in this chapter analysed and compared to theory. The observed room modes are first compared to the theoretically expected ones. Secondly the observed SPL distributions at these room modes are used to predict the sound pressure levels in selected positions in the room, and for selected resonance frequencies. All values of sound pressure and sound pressure levels used in tables and plots are uncalibrated and relative values. Unless otherwise stated the frequency responses in x-, y- and z-direction in the same measuring set (same figure) are comparable, because of the same settings and calibration being used for the measurements. But sound pressure levels from a set of frequency responses are not comparable to the sound pressure levels from another set of frequency responses when these are done in another room, or with changes being done to the room. None of the sound pressure levels are given relative to Pa61020 −⋅ , which is the lower limit for what is audible to the human ear. Measurements in positions where the SPLs were predicted were of course performed with the same settings and calibrations as the measurements along the walls. 4.1 Modes and resonant frequencies The observed resonant frequencies and SPL distributions in the four rooms are in this chapter analysed and compared to theoretically calculated room modes.

4.1.1 Basement room at Ton Art Measurements in this room were performed on three occasions, with some changes being done to the room compared to the last measurements. These changes are summarized in Table 4.1. As the measurements for the pressure prediction were performed on April 12 along with set 2, this set of measurements is given main focus below. Measurement sets 2 and 3 displayed almost identical results in terms of resonant frequencies and SPL distributions. The measurements along the z-axis from set 3 is therefore in some figures plotted along side the measurements from set 2, as this set lacks measurements along the z-axis. Measurement set 1, March 4

Measurement set 2, April 12

Measurement set 3, May 3

Glass door open Glass door closed Glass door closed Thick carpet 3.74 m2, middle of floor.

Thick carpet, and thin carpet samples along the walls.

Thin carpet all over.

Drapes partly lifted out of the way

Drapes hanging freely Drapes hanging freely

x, y and z-axis x and y-axis x, y and z-axis Table 4.1. Differences in the basement room at Ton Art between each time measuring was done. Major changes are in italic.

34

Where sets 2 and 3 gave similar results, measurement set 1 displayed results differing from the other two sets. This is mainly because of the open glass door. The effect of this door is showed below. The carpet placed on the floor during measurement set 3 was of the same kind as the samples placed along the walls, mainly in the corners, for set 2. These were of the size 0.25 m2, and there were 40 of these in the room during set 2. According to employees at Ton Art, these carpet samples had bettered the listening experience in the room, in terms of clarity and avoiding noise. The theoretically calculated modes for the room are listed in Table 7.1 in the Appendix. A comparison between these modes and the observed resonances is summarized in Table 4.2 after the plotting of and discussion on the observed modes and SPL distributions. Measurements were done along the x-axis in the room, but as will be shown later, no SPL Distributions appeared below 80 Hz in this direction. Therefore, most often only distributions along the x-and y-directions are shown. At 37.8 Hz the first real resonance appeared, as seen in Figure 4.1. A (1,1,0)-mode is expected at 39.1 Hz, and the observed standing wave patterns were almost exactly like expected. This resonance also appeared in the results from set 1, where the glass door was open, but this resonance was a weak one with a small peak, and appeared at 38.7 Hz (Figure 4.2). For comparative reasons Figure 4.2 shows the distribution at 37.8 Hz and not 38.7 Hz, but the SPL distributions at these frequencies were almost equal. The effect of opening and closing the glass door is obvious. The pressure at xmax, i.e. by the door, fails to reach a maximum in Figure 4.2, but succeeds in doing so in Figure 4.1, creating the expected (1,1,0)-mode at this frequency. At other frequencies displaying resonance with the door closed a similar phenomenon was observed, with the open door letting out sound waves, denying standing wave conditions to be set up. A comparison between the frequency responses in Figure 4.1 and Figure 4.2 showed that there were resonances at different frequencies when door is open than when it was closed. According to theory, no modes should appear between the (1,1,0)-mode at 39.1 Hz and the (0,2,0)-mode at 53.8 Hz. Figure 4.2 shows that this is not the case when the door is open, displaying peaks at 43.2 Hz in x- and y-direction, and at 52.2 Hz in y-direction. However, in Figure 4.2 there are no peaks between 37.8 Hz and 53.6 Hz, which matches well with the theoretically calculated mode frequencies. For sound field to appear as calculated theoretically it is in other words essential that doors and windows in the room are closed. The (1,1,0)-mode at 37.8 Hz was the mode in the room with the largest distance in frequency to adjacent modes. And most likely because of this, it was the only mode that appeared with SPL distributions that were almost equal to the expected distributions.

35

Figure 4.1. Frequency responses in x- and y-direction (left) and normalized distribution at 37.8 Hz (right), main room in basement. Measurement set 2.

Figure 4.2. Frequency responses in x- and y-direction (left) and normalized distribution at 37.8 Hz (right), main room in basement room at Ton Art. Measurement set 1 (open glass door). The highest peak in the frequency responses appeared at 53.6 Hz A (0,2,0)-mode was expected at 53.8 Hz, and along the y-axis there were maximums at the walls and halfway across the room, although two of the maximums were low. In the x-direction there seemed to be an unwanted minimum in the middle of the room, as the pressure ideally should be constant across the room. This decrease in SPL may have occured because measurements done along side this wall were done behind a large shelf with cd-players, receivers etc. The sound waves may have resonated between the wall and the objects in the shelf, and be absorbed by the fairly heavy drapes on the wall. Also, the (2,0,0)-mode due at 56.7 Hz might have affected the distribution at this resonance.

36

Figure 4.3. Frequency responses in x- and y-direction (left) and normalized distribution at 53.6 Hz (right), main room in basement room at Ton Art. Measurement set 2. The SPL distributions at 53.6 Hz also demonstrates another repeating pattern for the modes in this room. The sound pressure maximum always occured at positions x0 and y0, which are the same position – the corner between walls A and B. These walls were both hard concrete walls, which explained the maximum. But at xmax and ymax the maximums were low at 53.6 Hz, and at other resonance frequencies, there were minimums in these positions. This was evident at the peak at 63.9 Hz (Figure 4.4). The glass door in the corner between walls A and F, at xmax, and the wooden wall C at ymax seemed to be much less able to reflect sound waves at these frequencies than the concrete walls. Wooden walls and thin glass windows like the ones used in the glass door do generally have higher apsorptivity in the 125 Hz octave band than concrete and concrete blocks6. But such extreme minimums as those at xmax and ymax at 63.9 Hz are surprising. These minimums also occured in the SPL distributions at the smaller peaks at 28.5 Hz and 96.5 Hz. The SPL distributions observed at 63.9 Hz are clearly results of two modes appearing close to each other in frequency. A (1,2,0)-mode was expected at 60.8 Hz, and a (2,1,0)-mode at 62.7 Hz. The resultant resonant frequency displayed SPL distributions that make out a (2,2,0)-mode with missing maximums at two of the walls. Also at other of the smaller resonance frequencies in the room several modes appeared close to each other, resulting in unpredictable SPL distributions. This occured at the peaks at 28.5 Hz, 86.0 Hz and 96.5 Hz.

37

Figure 4.4. Frequency responses in x- and y-direction (left) and normalized distribution at 63.9 Hz (right), main room in basement room at Ton Art. Measurement set 2. The (2,2,0)-mode was expected at 77.8 Hz, and a small resonance occured at 75.4 Hz. The SPL distributions made out the standing wave patterns of a (2,2,0)-mode, but where the highest peaks were halfway across the room both in x- and y-direction. The peaks at the walls were present, but lower. These distributions were unusual, and did not occur at any other frequencies in any of the rooms. The measurements with the new carpet, measurement set 3, also showed a small peak at 96.5 Hz. This peak was not clearly defined in the frequency responses resulting from measurement set 2, as shown e.g. in the plots of the frequency responses along the x- and y-axes to the left in Figure 4.4. The SPL distributions looked similar for both sets of measurements though, and the SPL distributions along the x- and y-axes again, like at 63.9 Hz, resembled 2nd order standing wave patterns with missing maximimums at one end. The (2,2,1)-mode due to appear at 106 Hz might have been the dominating mode at this frequency, but 10 Hz is a big difference from the observed resonance frequency. Another and more likely possibility was that the observed patterns are resultants of different modes affected each other, i.e. the (1,2,1), (2,1,1) and (2,3,0)-modes due at 95.1 Hz, 96.4 Hz and 98.6 Hz, respectively. By comparing the frequency responses in Figure 4.5 with those in Figure 4.4, the effect of covering the whole floor with a carpet can be observed. With the carpet (Figure 4.5) the high peak at 53.6 Hz looked smoother than without the carpet (Figure 4.4), but the distortion of this peak seem only to count for some measurements, and may have been caused by other things than the carpet itself. Also, two peaks appeared close to each other around 37 Hz with the carpet; only one peak appeared without it. There should only be one mode appearing at this frequency, so this is another unpredictable effect of doing changes to a room. At both the resonances at 86 Hz and 96.5 Hz, the SPL was lower at the ceiling than at the floor. This is likely to be caused by the insulation material that was placed in the ceiling, above the perforated plaster plates. The perforations let through sound waves, which resonated behind the plates, and were absorbed by the insulation material.

38

Figure 4.5. Frequency responses in x- and y-direction (left) and normalized distribution at 96.5 Hz (right), main room in basement. Measurement set 3. Table 4.2 below summarizes the above discussion on the observed modes and the theoretically expected mode frequencies. Observed resonance

Theoretical calculated room modes

Comments on the observed pressure distributions

28.5 Hz 26.9 Hz (0,1,0), 28.3 Hz (1,0,0) Low peak, maximum at x0 and y0, but minima at other ends.

37.8 Hz 39.1 Hz (1,1,0) Perfect (1,1,0)-mode 53.6 Hz 53.8 Hz (0,2,0), 56.7 Hz (2,0,0) Resembles a (1,2,0)-mode 63.9 Hz 60.8 Hz (1,2,0), 62.7 Hz (2,1,0)

(2,2,0)-mode with missing maxima at end of x-axis and y-axis

75.4 Hz 77.8 Hz (2,2,0), 78.0 Hz (0,1,1), 78.5 Hz (1,0,1)

The (2,2,0)-mode require control, but low maxima.

86.0 Hz 83.0 Hz (1,1,1), 85.0 Hz (3,0,0), 85.5 Hz (1,3,0), 89.2 Hz (3,1,0), 90.8 Hz (0,2,1)

Resembles a (2,2,0)-mode. Many close modes probably affect each other.

96.5 Hz 95.1 Hz (1,2,1), 96.4 Hz (2,1,1), 98.6 Hz (2,3,0)

(2,2,1)-mode with missing maxima at xmax, ymax and zmax. The two oblique modes at 95 and 98 Hz expected to require control, but also 1st order pattern with missing maximum in z-direction.

Table 4.2. Observed resonant frequencies in the basement room at Ton Art compared to the theoretically calculated modes. Modes assumed to dominate based on observations are highlighted (bold type).

39

When adding the measurements from the adjacent storage room, the total length along which measurements were done in the y-direction is 8.35 m. In Figure 4.6 to Figure 4.8 below these last measurements are added to the ones from the main room, and the SPL distribution is plotted at some selected resonance frequencies. Because the measurements in the storage room were performed along with measurement set 1 these are resonance frequencies appearing for these measurements and may differ from the ones from sets 2 and 3. The glass door was open during the measuring, but most probably this did not affect the relative difference between the SPL in the main room and the storage room. Only the frequency- and pressure distributions in y-direction are plotted here, as these are the only ones that became altered by including the measurements from the storage room. The last measurement in the main room was done at y=6.35, and the first one in the storage room at y = 6.75. Figure 4.6 below shows the SPL distribution at 39.4 Hz. Figure 4.6 shows that the pressure dropped drastically behind the wall to the storage room. Wall C was a light wooden wall with insulation, and what probably happened is that the sound waves entered through the perforations on the wall facing the main room, resonanted inside the room, and was absorbed by the insulation material. The same phenomena was observed at the peaks at 43.4 Hz, 52.0 Hz and 75.6 Hz, with the pressure decreasing abruptly behind the wall.

Figure 4.6. Frequency responses in the three directions (left) and normalized distribution at 39.4 Hz (right), basement room with inclusion of storage room. Measurement set 1. At 60.7 Hz and 61.4 Hz, where there were peaks in x- and y-direction, respectively, a minumum in SPL was observed by wall C. The pressure in the storage room then remained at this level.

40

Figure 4.7: Frequency responses in the three directions (left) and normalized distribution at 61.4 Hz (right), basement room with inclusion of storage room. Measurement set 1.

For most resonant frequencies, the SPL behaved like displayed in Figure 4.6, with the SPL being lower inside the storage room. The exception occured at the very small peak in the y-direction appearing at 25.8 Hz. At this frequency the pressure increased drastically inside the storage room, displaying the pressure distribution in Figure 4.8 below.

Figure 4.8: Frequency responses in the three directions (left) and normalized distribution at 25.8 Hz (right), basement room with inclusion of storage room. Measurement set 1. The measurements discussed above showed that for most resonance frequencies pressure maxima were observed at wall C, i.e. the peaks at 39 Hz, 43 Hz, 52 Hz, 76 Hz, 78 Hz, and 83 Hz. Only for the peaks at 26 Hz and 61 Hz the wall failed to create expected pressure maximums, leading to SPL minimums at the wall. As the wall between the main room and the adjacent storage room was only an 18 cm thick wooden wall, one possibility was to consider the storage room as part of the main room in the calculation of room mode frequencies, i.e. using y = 8.30 instead of y=6.35. These calculations resulted in mode frequencies that did not match the measurement results at all. And the results from the measurements in the storage room also showed that the wall definitely was hard enough to reflect sound waves, creating the standing wave patterns observed in the main room, and leading to the tendency of the pressure dropping abruptly behind wall C. Therefore, this theory was rejected.

4.1.2 Ground floor room at Ton Art Also for this room, the theoretically calculated modes due to appear in the room are listed in the Appendix, in Table 7.2. A comparison between these modes and the observed resonances is summarized in

41

Table 4.2 after the plotting of and discussion on the observed modes and standing wave patterns. Measuring was done on two occasions, resulting in measurement set 1 and 2. Set 2 was done the same day as the pressure-prediction measurements, and is therefore used in the plots below. Although the frequency responses looked slightly different, resonance occured at the same frequencies for both measurement sets, displaying the same pressure distributions. Measurements along the z-axis from set 1 are therefore used in the plots below, as set 2 lacks measurements in z-direction. The highest peak in the frequency resonses occored at 49.3 Hz (Figure 4.9). A (1,1,0)-mode was expected at 51.2 Hz, and the SPL distributions observed at 49.3 Hz looked almost like expected, as the maximums at x0 and y0 were lower than those at xmax and ymax. Interestingly, there was also a peak appearing at 53.5 Hz, displaying similar pressure distributions, but with clearer defined maximums at x = 0 and y = 0. This was a peak that should not appear according to theory. As Figure 4.9 shows there was no pressure distribution along the z-axis for frequencies below 65 Hz. Therefore, for peaks appearing below that frequency, only the distribution along the x- and y-axis is shown.

Figure 4.9: Frequency responses in the three directions (left) and normalized distribution at 49.3 Hz (right), room at ground floor at Ton Art. Measurement set 2 (x- and y-axis), set 1 (z-axis). Between 30 Hz and 40 Hz there were four or five small peaks in the frequency response. At one of these, at 30.4 Hz, the SPL distributions made out the patterns that are expected for a (0,1,0)-mode, which was expected at 32.8 Hz. But also here there was a low maximum at y0. At the peak of similar size at 34.4 Hz the distribution looked similar to the one in at 30.4 Hz, but there the SPL distribution along the y-axis was reversed compared to the one at 30.4 Hz, i.e. the clearest defined maximum being at y0,.with a low maximum at ymax.

42

Also at 40.9 Hz and 63.6 Hz (Figure 4.10), the SPL distributions looked almost like expected, i.e. for a (1,0,0) and (0,2,0)-mode, respectively. The distribution along the y-axis at 63.6 might not have looked ideal, with both the first maximum and minimums not clearly defined, but as is shown below, doing measurements along the y-axis along the parallel wall, the SPL distribution looked more like the expected one. As seen in Figure 4.10, the peak appeared at 62.6 Hz for the measurements along the x-axis, but at 63.6 Hz for the measurements along the y-axis. At 62.6 the maximum was clearer defined at y0, and with a decreasing SPL at xmax. The common factor for the four lowest modes, appearing at 30.4 Hz, 40.9 Hz, 49.3 Hz and 63.6 Hz, was that the spacing to neighbouring modes was big, almost 10 Hz in all cases. The same phenomena that was observed in the basement was thus observed in this room – a big frequency spacing is necessary for SPL distributions to look like predicted by the models.

Figure 4.10: Frequency responses in the three directions (left) and normalized distribution at 63.6 Hz (right), room at ground floor at Ton Art. Measurement set 2 (x- and y-axis), set 1 (z-axis). At all resonant frequencies the SPL was lower at y0 than at ymax, and at all resonant frequencies exept at 81.3 Hz, the SPL was lower at x0 than xmax. In this room xmzx and ymax denoted the same positon, which was the corner between walls C and D, which were concrete walls closest to this corner. The first measurements along the x-axis, along wall C, were done along side a glass door. And the other wall in the corner at x0, wall B, was a lighter, wooden wall. Along the y-axis close to y0, part of the wall consisted of a type of thicker glass. A heavy glass plate is less absorptive than thin glass6, but still more absorptive than concrete, and next to this thick glass in the corner there was a wooden door. All these factors may have contributed to the missing maxima and low maxima at x0 and y0. The modes appearing above 70 Hz appeared according to the models much more close in frequency than those below 70 Hz. The SPL distributions at the observed resonant frequencies were therefore products of several modes. An example of this appears at 74.1 Hz. Several modes were expected close to each other in this frequency area, a (1,2,0)-mode, a (0,1,1)-mode, a (2,0,0)-mode and a (1,0,1)-mode were due at 76.5 Hz, 76.7 Hz, 78.5 Hz and 79.7 Hz, respectively. The observed pattern along the

43

z-axis could be a 1st order standing wave pattern with the maximum at the ceiling missing. The absorbing material in the ceiling probably absorbed the sound waves at this frequency, so that no maximum was created at zmax. The problem with missing maximums at x0 and y0 was also obvious here.

Figure 4.11: Frequency responses in the three directions (left) and normalized distribution at 74.1 Hz room at ground floor at Ton Art. Measurement set 2 (x- and y-axis), set 1 (z-axis). Also at the peak at 81.3 Hz two modes appearing close to each other most probably affected each other. There should be a (2,1,0) – mode at 85.1 Hz, and a (1,1,1)-mode at 86.2 Hz.. The observed SPL distributions created a something looking like a (1,1,1)-mode with a missing maximum at y = 0, but the distributions were most probably influenced by the (2,1,0)-mode aswell. The observed resonance frequencies are listed below in Table 4.3, along with the modes that are likely to create the observed pressure distributions. Observed resonance

Theoretically calculated room modes

Comments to observed SPL distributions

30.4 Hz 32.8 Hz (0,1,0) Clear (0,1,0)-mode 40.9 Hz 39.3 Hz (1,0,0) (1,1,0)-mode missing maxima at x0 49.3 Hz 51.2 Hz (1,1,0) Perfect (1,1,0)-mode 63.6 Hz 65.6 (0,2,0) Resembles a (0,2,0)-mode with a

missing minimum at x = 1. 74.1 Hz 76.5 Hz (1,2,0), 76.7 Hz (0,1,1),

78.5 Hz (2,0,0), 79,7 Hz (1,0,1) (1,2,0) or (1,2,1)-mode missing maxima at two or three walls.

81.3 Hz 85.1 Hz (2,1,0), 86.2 Hz (1,1,1) (1,1,1)-mode with low maxima at y0. Table 4.3. Observed resonant frequencies in the ground floor room at Ton Art compared to the theoretically calculated modes. Modes assumed to dominate based on observations are highlighted (bold type).

44

Comparing parallel walls The measurements along walls A and B could not be used due to their being adjacent to the corner where the speaker was placed. But from these measurements it could still be observed that the sound fields along these walls appeared to be different than the ones along their parallel walls. Examples of this are given in the figures below, where the measurements along walls A and B are discussed and compared to the ones along walls C and D. By observing the plots of the measurements, it is clear that the five measurements closest to the loudspeaker gave results that were so affected be the speaker that they could not be used in further discussion. Because of this, measurements x_0 to x_4, and y_0 to y_4 used in the plots below were in fact copies of measurements x2_0 to x2_4, and y2_0 to y2_4. This also contributes to illustrate the differences in measurements done along two parallel walls. Figure 4.12 and Figure 4.13 display very different distributions of SPL, even though they should in principle show identical patterns. These differences have more than one cause. First, there are the properties of the room, specifically the walls. The minimum at the end of the x-direction in Figure 4.12 could be caused by the glass on the wall in this corner of the room, and the maximum at the end of wall C in Figure 4.13 is probably due to both walls C and D consisting of hard concrete in this corner. It is also apparent that the loudspeaker did affect the measurements along the walls adjacent to it, as the observed pressure increased abruptly for the fifth measurement in both x- and y-direction (Figure 4.12). The four first measurements were, as mentioned, from the parallel walls.

Figure 4.12. Frequency responses in the three directions (left) and normalized distribution at 73.9 Hz (right), room at ground floor. Measurements done along walls A and B.

45

Figure 4.13. Frequency responses in the three directions (left) and normalized distribution at 73.9 Hz (right), room at ground floor. Measurements done along walls C and D. Figure 4.14 and Figure 4.15 compare the pressure distributions at the resonant frequency that appeared at 62,2 Hz. Remembering that the scales on the distribution plots are relative, the small maximum in x-direction in Figure 4.14 could be explained by closeness to the loudspeaker. The minimum at 1,5 meters along the y-axis in Figure 4.14 was interesting, because it was more obvious along wall B than wall D. Especially was this interesting because this is the wall closest to the loudspeaker, and the distribution still displayed an obvious minimum only 1,5 meters from the speaker. A (0,2,0)-mode is expected at 65.6 Hz, so in this case the measurements along wall B (close to the loudspeaker) gave the distribution closest to theory.

Figure 4.14. Frequency responses in the three directions (left) and normalized distribution at 62.2 Hz (right), room at ground floor. Measurements done along walls A and B.

46

Figure 4.15. Frequency responses in the three directions (left) and normalized distribution at 62,2 Hz (right), room at ground floor. Measurements done along walls C and D. In general it is observable in the plots from measurements along walls A and B that closeness to the loudspeaker affected the distribution patterns more as the frequency increases. Figure 4.12 and Figure 4.14 show that from approximately 65 Hz, the measurements at distance 1.25 m-2.00 m from the speaker corner differed significantly in shape and strength from those further away. Comparing measurement sets As mentioned, two sets of measurements were performed in the ground floor room at Ton Art. When the measurements for set 1 were done, this room had a large number of speakers along the walls. These were removed for set 2, leaving only one pair of speakers. This was the only change done to the room between the two measurement sets. Figure 4.16 and Figure 4.17 show the frequency responses and pressure distributions at 49.3 Hz for measurement set 1 and 2, respectively. It is obvious that the pressure distributions were practically identical for the different measurement sets. The frequency responses displayed small differences, but with a margin of +/- 1 Hz, the resonances appeared at the same frequencies.

47

Figure 4.16: Frequency responses in the three directions (left) and normalized distribution at 49.3 Hz (right), room at ground floor at Ton Art. Measurement set 2.

Figure 4.17. Frequency responses in the three directions (left) and normalized distribution at 49.3 Hz (right), room at ground floor at Ton Art. Measurement set 1. Figure 4.18 compares the frequency response in one particular position for the two measurement sets. And plotting the results this way, the difference between the measurement sets seem to be bigger than first assumed. This can be seen for instance at 42 Hz, where the old measurement is 1 dB stronger, and 79 Hz, where the new measurement is 5 dB stronger. This shows that it is difficult to predict the affect these kinds of changes have on a room’s acoustic properties. Moving of furniture etc might cause unexpected and in some cases unwanted changes in a room’s sound field. But as the pressure distributions looked alike at the resonance frequencies, and these resonances appeared at the same frequencies, the plotting of the distribution along the z-axis from set 1 along side the x- and y-axis distributions from set 2 can be defended. Using results from set 1 to predict SPL and compare with measurements from set 2 is not recommended though.

48

20 40 60 80 1000

0.5

1

1.5

2

2.5x 10

4 Frequency responses along the y-axis

Pre

ssur

e am

plitu

de

[P

a un

cal]

Frequency [Hz]30 40 50 60 70 80 90 100

70

72

74

76

78

80

82

84

86

88

Frequency responses along the y-axis

Pre

ssur

e am

plitu

de

[dB

]

Frequency [Hz] Figure 4.18. Frequency response measured in position y_0, measurement 16/3 (stippled), and 12/4 (line). Left: pressure amplitude in Pascal. Right: Pressure amplitude in dB.

4.1.3 Room B-337 at NTNU Gløshaugen In this room the measurements along the z-axis were done on a different occasion, denoted measurement set 2, than the ones done along the x- and y-axis (measurement set 1). Therefore the SPLs in the z-direction-plots were not comparable to the pressures on the x- and y-direction plots. But no changes had been done to the room between these measuring occasions, and the resonances appeared at the exact same frequencies. The calculated room modes based on the room dimensions are listed in Table 7.3 in the Appendix. As this room was even more squared in shape than the basement room at Ton Art, some modes appeared very close to each other in frequency, leading to frequency areas with no modes. To recapitulate the description of the room, walls A and C consisted of plaster, and should be able to reflect sound waves at low frequencies, in order to create the expected maximums at y0 and ymax. Wall D was a hard concrete wall, and wall B was part windows, part concrete, leading one to expect pressure maximums at xmax and maybe x0, dependent on the affect of the windows. Some absorptive material was placed on the ceiling, which might affect the standing waves patterns along the z-axis, and maybe corrupt the maximums that should appear at the ceiling, zmax. As in the Ton Art-rooms, the first oblique mode, appearing at 40.7 Hz, showed SPL distributions looking exactly like expected (Figure 4.19). A look at the list of modes expected below 100 Hz, revealed that the (1,1,0), (1,1,1) and (2,2,1)- modes expected at 41.4 Hz, 70.7 Hz and 100.7 Hz stand out because of their distance in frequency from other modes. The modes appeared almost exactly at the frequency they were expected, and the SPL distributions appeared according to the theoretical models. The (1,1,1)-mode, which appeared at exactly 70.7 displayed SPL distributions almost

49

perfect according to theory, while the (2,2,1)-mode had distributions deviating slightly from the expected (Figure 4.20).

Figure 4.19. Frequency responses in all three directions (left), normalized distribution at 40.7 Hz (right), room B-337 at NTNU Gløshaugen. Measurements set 1 (x- and y-axis), set 2 (z-axis).

Figure 4.20. Frequency responses in all three directions (left), normalized distribution at 100.0 Hz (right), room B-337 at NTNU Gløshaugen. Measurements set 1 (x- and y-axis), set 2 (z-axis). Apart from these three modes, the square shape of the room ledto all other modes appearing in the close vicinity of other modes. The SPL distributions at the observed resonances were therefore affected by often three or even four different modes.

50

One example of this occured at the resonance at 56.6 Hz (Figure 4.21), which was caused by an interaction of the (0,0,1)-, (2,0,0)- and (0,2,0)-modes expected at 57.3 Hz, 57.8 Hz and 59.3 Hz respectively. The (0,0,1)-mode created the observed pattern along the z-axis, and the absorbing material in the ceiling seemed to absorb sound waves at this frequency, so that the expected maximum at zmax did not appear. The 1st order standing wave pattern observed along the x-axis was hard to explain, though, as a 2nd order pattern was expected. One thing that was obvious is that the (0,0,1)- and (2,0,0)-modes dominated over the (0,2,0)-mode, so that no pressure distribution was observed along the y-axis.

Figure 4.21. Frequency responses in all three directions (left), normalized distribution at 56.6 Hz (right), room B-337 at NTNU Gløshaugen. Measurements set 1 (x- and y-axis), set 2 (z-axis). Four modes were predicted to appear between 86 and 89 Hz, and these affected each other in a similar way in creating the peak at 86.9 Hz and the pressure distributions shown in Figure 4.22 below. The expected modes were a (2,1,1)-mode and a (3,0,0)-mode at 86.7 Hz, a (1,2,1)-mode at 87.4 Hz, and a (0,3,0)-mode at 89.0 Hz. And the SPL ditributions did not make out any of the expected distributions. What was even more surprising is the sound pressure minimum at the floor, something that only happened at this resonance frequency.

51

Figure 4.22. Frequency responses in all three directions (left), normalized distribution at 86.9 Hz Hz (right), room B-337 at NTNU Gløshaugen. Measurements set 1 (x- and y-axis), set 2 (z-axis). At 82.6 frequency it seemed like the (2,2,0)-mode expected at 82.8 Hz dominated over the modes expected at 81.4 Hz and 82.5 Hz. And the pressure distribution appearing at the peak at 92.5 Hz were most probably a mixture of the (3,1,0) and (1,3,0)-modes at 91.7 Hz and 93.5 Hz. But the (1,3,0)-mode seemed to dominate, as the pattern along the y-axis displayed four maximums, and three dips (two large and one small) in SPL. At most resonance frequencies there were sound pressure maximums at the walls, indicating that there was no major difference between these walls in terms of absorptivity at low frequencies. Observed resonance

Theoretically calculated room modes

Comments on the observed SPL distributions

28.7 Hz 28.9 Hz (1,0,0), 29.7 Hz (0,1,0) Both modes contribute, but low maximum at xmax, and minimum at ymax

40.8 Hz 41.4 Hz (1,1,0) Perfect (1,1,0)-mode 56.6 Hz 57.3 Hz (0,0,1), 57.8 Hz (2,0,0),

59.3 Hz (0,2,0) Resembles (1,0,0)-mode or (1,0,1) with missing maximum at zmax

64.6 Hz 64.2 Hz (1,0,1), 64.5 Hz (0,1,1), 65.0 Hz (2,1,0), 66.0 Hz (1,2,0)

(1,1,1) or (2,1,1)-mode with low maxima at ymax and zmax

70.7 Hz 70.7 Hz (1,1,1) Low peak, but almost perfect (1,1,1)-mode

82.6 Hz 81.4 Hz (2,0,1), 82.5 Hz (0,2,1), 82.8 Hz (2,2,0)

All modes interact. Resembles a (2,1,1) mode with minimum at zmax.

86.9 Hz 86.7 Hz (2,1,1), 86.7 Hz (3,0,0), 87.4 Hz (1,2,1), 89.0 Hz (0,3,0)

All modes interact. Minimum at z = 0.

92.5 Hz 91.7 Hz (3,1,0), 93.5 Hz (1,3,0) The modes interact, but the pattern along

52

the y-axis strongest resembles 3rd order pattern.

100.0 Hz 100.7 Hz (2,2,1) Clear (2,2,1)-mode. Table 4.4. Observed resonant frequencies in room B-337 at NTNU Gløshaugen compared to the theoretically calculated modes. Modes assumed to dominate based on observations are highlighted (bold type).

4.1.4 Room B-343 at NTNU Gløshaugen The room modes that according to theory should appear in this room below 100 Hz are listed in Table 7.4 in the appendix. Because of the big size of this room compared to the three others, the number of modes predicted to appear below 100 Hz was also much bigger. 33 modes have frequencies below 100 Hz in this room, as opposed to only 13 for the ground floor room at Ton Art. Also the observed number of resonance frequencies was big, as there can be observed approximately 19 peaks in the frequency response in one or more dimensions. Some of these resonances are listed in Table 4.5, after the plots of frequency responses and SPL distributions. Walls D, C, E and F were hard concrete walls, wall B consisted of concrete and windows, and wall A was a 20 cm thick plaster wall. With such hard walls, pressure maximums should be expected at all walls, with the possible exception of wall B, i.e. at x0, because of the windows, and maybe wall A, at y0, because of the insulated plaster wall. As showed in chapter 3, the last four measurements along the x-axis were done along wall E, and not along the same wall as the other measurements along the x-axis, wall A. As stated earlier, they did not naturally belong together with the measurements along wall A. The measurement along wall A show the SPL distribution for standing waves appearing between walls B and F, creating SPL maximums at these walls. The measurements along wall E show part of the SPL distribution caused by standing waves between walls D and B. These last four measurements are included in the plots for illustrative purposes, demonstrating how such irregularities in the geometry of the room complicate the room’s sound field. Among the calculated room modes in Table 7.4 the (1,0,0)-, (1,2,0)-, and (0,3,0)- modes expected at 29.7 Hz, 45.3 Hz and 51.4 Hz all have a space of 5 Hz or more to neighbouring modes. Out of these three modes, only the (1,2,0)-mode appeared with expected SPL distributions, at 44.0 Hz (Figure 4.23). It is obvious that there was a SPL maximum at x = 5.00, which was at wall F, and the SPL was lower along wall E. This can be explained by looking at the SPL distribution along the y-axis, which had a minimum around y = 1.50, where wall E was positioned compared to wall A.

53

At most resonant frequencies the SPLs measured along wall E were lower than those measured along wall A if the SPL distribution along the y-axis indicated that the SPL was lower 1.50 m from wall A than at wall A.

Figure 4.23. Frequency responses in the three directions (left) and normalized distribution at 44.0 Hz (right), room B-343 at NTNU Gløshaugen. At 58.1 Hz a peaks appeared in the frequency responses. Four modes were expected close to each other in frequency, i.e. a (0,0,1)-mode at 57.3 Hz, a (2,0,0)-mode and a (1,3,0)-mode at 59.3 Hz, and a (0,1,1)-mode at 59.8 Hz. But here the axial (0,0,1)-mode seemed to dominate, which is logical as an axial mode is expected to have more energy than tangential modes5. The only factor that made the SPL distributions look less like expected for a (0,0,1)-mode was the low SPL at measurements close to y0 compared to measurements close to ymax. The (0,0,1)-mode is also spaced 6 Hz and 2 Hz to neighbouring modes. The (1,0,1)-mode expected at 64.5 Hz has spaces 3 Hz and 2 Hz to neighbouring modes, and did also display recognizable SPL distributions. But apart from these two modes, for frequencies higher than 57 Hz, all modes were expected within 2 Hz of at least one other mode. SPL distributions did therefore not look like the expected distributions for particular modes at any of the other resonant frequencies. This goes e.g. for the smaller peaks that appeared at 68.6 Hz, 72.4 Hz, 76.7 Hz and 90.8 Hz. The SPL distribution along the y-axis at 58.1 Hz also demonstrated another repeating phenomenon. The SPL level was at several resonant frequencies lower at y0 than at ymax. This also happened at 29.7 Hz and 63.6 Hz, which shows that the plaster wall A at y0 was not as reflective at low frequencies as the concrete wall C at ymax.

54

Figure 4.24. Frequency responses in the three directions (left) and normalized distribution at 58.1 Hz (right), room B-343 at NTNU Gløshaugen. As an exception to the reccuring low SPL at y0, at 32.5 Hz there appeared a SPL maximum at y0 and a minimum at ymax. This resonance was probably a product of the (1,1,0) and (0,2,0)-modes expected at 34.2 and 34.3 Hz, respectively. The minimum at the end of the y-axis is hard to explain, as this was next to a hard concrete wall where a maximum should be expected. This just underlines the unpredictability this room inhabited concerning room modes and SPL distributions.

Figure 4.25. Frequency responses in the three directions (left) and normalized distribution at 32.5 Hz (right), room B-343 at NTNU Gløshaugen. As the room was over 10 meters long, several third order, and even fourth order standing wave patterns were predicted to occur at frequencies below 100 Hz. And at several of the resonance frequencies, SPL distributions with several peaks and dips did occur, although they mostly did not look entirely as predicted by the models.

55

An example of this is shown in Figure 4.26, where the SPL distributions at 72.4 Hz are displayed. A (1,2,1)-mode should appear at 73.1 Hz, a (1,4,0)-mode at 74.7 Hz, and a (0,3,1)-mode at 77.0 Hz. At 72.3 Hz there was a high peak in the x-direction and a smaller one in the y-direction, where there was a slightly higher peak at 71.7 Hz. Along the y-axis the SPL distribution clearly resembled a 3rd order standing wave pattern, with maximums at the walls and two smaller maximums between the walls. In the x-direction, the four last measurements, done along wall E, clearly altered what would otherwise closely resemble a first order standing wave pattern.

Figure 4.26. Frequency responses in the three directions (left) and normalized distribution at 72.4 Hz (right), room B-343 at NTNU Gløshaugen. Table 4.5 lists some of the resonant frequencies that appeared in the room, along side the modes that are likely to have caused the resonances, and comments on the SPL distributions. Observed resonance

Theoretical calculated room modes

Comments on the observed SPL distributions

29.7 Hz 29.7 Hz (1,0,0) (1,0,0)-mode with low SPL at y0 and xmax

32.5 Hz 34.2 Hz (1,1,0), 34.3 Hz (0,2,0) Resembles (1,2,0)-mode with minimum at xmax and ymax.

35.9 Hz 34.2 Hz (1,1,0), 34.3 Hz (0,2,0) (1,1,0) mode without clear minimum halfway between y0 and ymax.

44.0 Hz 45.3 Hz (1,2,0) Clearly defined (1,2,0)-mode. 51.0 Hz 51.4 Hz (0,3,0) Minimum at xmax, and SPL distribution

along y-axis deviating close to ymax. 58.1 Hz 57.3 Hz (0,0,1), 59.3 Hz (2,0,0)

59.3 Hz (1,3,0), 59.8 Hz (0,1,1) (0,0,1)-mode with low SPL at y0.

63.6 Hz 61.7 Hz (2,1,0), 64.5 Hz (1,0,1) Minimum at y0 and z0. 64.5 Hz 61.7 Hz (2,1,0), 64.5 Hz (1,0,1) No distribution along x-axis until

halfway to xmax.

56

68.6 Hz 66.8 Hz (1,1,1), 66.8 Hz (0,2,1) 68.5 Hz (2,2,0), 68.5 Hz (0,4,0)

Looks like (1,3,1)-mode. All modes interact.

72.4 Hz 73.1 Hz (1,2,1), 74.7 Hz (1,4,0) Also (1,3,1)-mode, but lower maximum at zmax. All modes interact.

76.7 Hz 77.0 Hz (0,2,1), 78.5 Hz (2,3,0) Undefinable, all modes interact. 90.8 Hz 89.0 Hz (3,0,0), 89.3 Hz (0,4,1)

89.3 Hz (0,4,1), 90.6 Hz (3,1,0) 90.6 Hz (2,4,0), 90.6 Hz (1,5,0)

Undefinable, all modes interact.

Table 4.5. Observed resonant frequencies in room B-343 at NTNU Gløshaugen compared to the theoretically calculated modes. Modes assumed to dominate based on observations are highlighted (bold type). In general this room displayed the worst results in terms of SPL distributions looking like predicted by the theoretical models. This is because of the room being large, leading to many modes appearing close also at low frequencies, and because of the assymetrical geometry of the room. The plots of the frequency responses showed that the frequency responses measured along the y-axis seemed to be more coherent than those done along the x-axis, particularly for frequencies higher than 60 Hz. This is most probably due to the room’s unsymmetrical geometry near wall A where the x-axis measurements were done.

57

4.2 Prediction of sound pressure levels Nine positions in each room were selected, and impulse response measurements were done in these positions. Based on the measurements done along the walls, the sound pressure levels at selected frequencies were predicted at each of these positions. These frequencies were chosen among the observed resonances in each room. The errors between predicted and measured SPL were compared for all positions and selected frequencies in order to find variations based on frequency, position, predicted SPL and/or room. If a high pressure was predicted, this would mean that one mode is expected to dominate at this frequency and in this position, leading one to expect a small error between predicted and measured values. Because only small variations in pressure distribution along the z-axis was observed in the Ton Art – rooms, all measuring positions were chosen to be on the floor. Although pressure distribution along the z-axis appeared at lower frequencies in the NTNU-rooms, the same practice was used there, with z = 0 for all measuring positions.

4.2.1 Basement room at Ton Art The listening room in the basement at Ton Art is displayed in Figure 4.27 below. The nine measuring positions are marked with pos.1 through to pos.9. All nine positions were on the floor, i.e. z = 0, so that only measurements in x- and y-direction were used to predict the SPLs. Positions 7-9 were chosen to be in positions that normally would be used as listening positions. This was done to be able to observe if minimums or maximums occurred in these positions, thus affecting the listening experience. The microphones were for the mentioned positions 7-9 placed under the chairs on which a listener would be seated. This might have affected the measurements in these positions, but most probably only for mid- and high frequencies, as the chairs’ sizes were not comparable to the wavelengths of low frequencies.

Figure 4.27. Listening room in basement at Ton Art, with nine measuring positions

58

In total, six or seven resonances were observed below 100 Hz in this room. The five most obvious of these were at 37.8 Hz, 53.7 Hz, 63.7 Hz, 75.3 Hz, 85.9 Hz, and the measured sound pressure levels at these frequencies were selected to be compared to the predicted ones. Figure 4.28 shows the predicted sound pressure levels plotted against the measured sound pressure levels. In total 45 values were compared, nine positions and five frequencies, leading to the 45 points in Figure 4.28. Each frequency is represented by a different colour and symbol in the plot. In this plot, the dashed sloped line indicates perfect correlation between predicted and measured SPL, i.e. points on this line indicate an error of 0 dB. If a higher predicted pressure should lead to a lower error, the points in Figure 4.28 ought to be closer to the dashed slope line for higher values of predicted SPL. This seems to be the case to some degree, as many of the points with a predicted relative SPL of ca 10 dB are fairly close to the line. A division of the points into three groups, based on the predicted SPLs, is shown in Table 4.6. There are not an equal number of points in each group, but the results from this division shows that the error did decrease with higher predicted SPL.

Figure 4.28. Predicted and measured sound pressure level in all combinations of nine different positions and frequencies 37,8 Hz, 53,7 Hz, 63,7 Hz, 75,3 Hz, 85,9 Hz in the basement room at Ton Art. Class: Predicted SPL (uncal.) Points: Mean error: Class 1: Lp > 10 dB 12 5,2 dB Class 2: 5 dB < Lp < 10 dB 24 6,5 dB Class 3 Lp < 5 dB 9 8.8 dB Table 4.6. Mean difference between predicted and measured SPL dependent on predicted SPL, basement room at Ton Art. Table 4.7 shows the mean error for all positions, for each frequency. And although the difference in error is not big, it seems like the results were best at lower frequencies. These were also the resonances with the highest peaks in the frequency response,

59

which matches the results from Figure 4.28 and Table 4.6 well, where the results get better when a high SPL was expected. The resonance at 37.8 Hz-mode was also the only resonance that displayed SPL distributions that closely resembled the predicted ones. Frequency [Hz] 37.8 Hz 53.7 Hz 63.7 Hz 75.3 Hz 85.9 Hz Lp∆

Lp∆ [dB] 6.2 6.1 6.3 8.0 7.4 6.8

Table 4.7. Mean values for the difference between predicted and measured SPL, [dB], at the resonant frequencies 37.8 Hz, 53.7 Hz, 63.7 Hz, 75.3 Hz, 85.9 Hz. (vertical), basement room at Ton Art. In Table 4.8 the mean errors at each frequency is listed based on the measuring positions. As seen in Figure 4.27 positions 1,2 and 3 have the same y-coordinate, as do positions 4,5 and 6, and 7,8 and 9. The same symmetry does not apply along the x-axis, but positions 1,4 and 7 have the highest x-coordinates, positions 2,5 and 8 have the same x-coordinate, and positions 3,6 and 9 are closest to x0. The results in Table 4.8 say that along the x-axis, the results are worst in the middle of the room, while along the y-axis, the results are best in the middle of the room.

Positions: 1 and 4 (x = 3.75)

7 (x = 4.00)

2,5 and 8 (x = 3.00)

9 (x = 2.00)

3 and 6 (x = 1.25)

Lp∆ , dB]

1, 2 and 3 (y = 3.00)

5.9 11.1 4.3 7.1

3,4 and 5 (y = 4.00)

5.5 8.2 5.4 6.4

7,8 and 9 (y = 5.00)

6.1 7.8 6.7 6.9

Lp∆ [dB] 5.7 6.1 9.0 6.7 4.9 6.8

Table 4.8. Mean values for the difference between predicted and measured SPL, [dB], at each measuring position, basement room at Ton Art. This room had the lowest Schroeder frequency value, 128 Hz. This value is so close to 100 Hz that it is likely to have affected the measurement results. The method with prediction of SPL can only work at frequencies where the room modes create the sound field. If the transition to a diffuse sound field occurrs around 128 Hz, 100 Hz may be too high a frequency for using the method.

4.2.2 Ground floor room at Ton Art For the room on the ground floor at Ton Art, the points were chosen as in the figure below. Positions 7-9 were chosen to be in positions that normally would be used as listening positions. The microphones were for these positions placed under the couch on which a listener would be seated. This might have affected the measurements in these positions.

60

Figure 4.29. Listening room at ground floor at Ton Art, with nine measuring positions. As in the basement room, five different frequencies were selected because of them being observed as resonant frequencies in the room. These were 30.4 Hz, 49.3 Hz, 62.5 Hz, 74.1 Hz and 81.1 Hz. Figure 4.30 shows the predicted sound pressure levels plotted against the measured sound pressure levels in the same way as Figure 4.28. Likewise does Table 4.9 show the mean error for three classes of points, after having divided the points according to predicted SPL. Figure 4.30 show that in this room, the correlation between the predicted and measured SPL was overall better than in the basement room. And the theory stating that this correlation should get better with increase in predicted SPL is definitely in accordance with Figure 4.30. Another apparent pattern observable in Figure 4.30 is that the measured SPLs often were higher than the predicted values, and in the few cases that the predicted values are higher, the differences were small.

61

Figure 4.30. Predicted and measured sound pressure level in all combinations of nine different positions and frequencies 30.4 Hz, 49.3 Hz, 62.5 Hz, 74.1 Hz, 81.1 Hz in the first floor room at Ton Art. Dividing the points in Figure 4.30 into three groups based on the predicted further proved that the error definitely decreased with increase in predicted SPL. The 10 points where the predicted SPLs was highest only ledto a mean error of 3 dB, compared to a mean error of 8,6 dB for the eight points where the predicted SPL was lowest. Class: Description: Points: Mean error: Class 1: Lp > 76 dB 10 3,0 dB Class 2: 65 dB < Lp < 76 dB 27 4,8 dB Class 3 Lp < 65dB 8 8,6 dB Table 4.9. Mean difference between predicted and measured SPL dependent on predicted SPL, 1st floor room at Ton Art. Predicted and measured sound pressure levels at the selected resonance frequencies are compared in Table 4.10, where the ∆Lp values are mean values for all nine positions. The table shows that for this listening room it is even more obvious that the error increased with frequency. The three lowest resonances (30.4 Hz, 49.3 Hz and 62.5 Hz) were the three resonances that display SPL distributions predicted by the theoretical models. And here these three resonances display the smallest errors when predicting SPLs. Frequency [Hz] 30.4 Hz 49.3 Hz 62.5 Hz 74.1 Hz 81.1 Hz Lp∆

Lp∆ [dB] 1.5 4.2 4.0 6.8 8.8 5.1

62

Table 4.10. Mean values for the difference between predicted and measured SPL, [dB], at the resonant frequencies 30.4 Hz, 49.3 Hz, 62.5 Hz, 74.1 Hz, 81.1 Hz, ground floor room at Ton Art. In Table 4.11 below, the mean error for each measuring position is displayed. A close look at these values reveals that the positions closer to wall D than the parallel wall B, gave better results. Positions 1, 4 and 7 had a mean error of 2.9 dB, positions 2, 5 and 8 had 4,4 dB, and positions 3, 6 and 9 had 7.9 dB mean error. The same pattern can be observed when comparing the rows in Table 4.11. Each rows marks an increase of the y-value along the y-axis. The measurements closest to wall A gave better results than the one closest to wall C. As discussed earlier, the pressure distributions at parallel walls were not necessarily equal, which is something that probably affected these results. Measurements were done along sidewall D, not wall B, which probably explains why the positions closest to wall D gave the best results. The difference in wall materials for the four walls is probably an important reason for the difference in error for the various positions. The same explanation can not be used when comparing in the y-direction though, as measurements were done along side wall C, which is closest to measurement positions 7,8 and 9, where the results are worst. An explanation for this might be that these positions were under the coach placed in listening position, and this piece of furniture might have affected the measurements, e.g. because of its absorbing properties.

Positions: 1,4 and 7 (x = 2.88)

2,5 and 8 (x = 2.38)

3,6 and 9 (x = 1.88)

Lp∆ , [dB]

1, 2 and 3 (y = 2.00)

1.8 4.2 7.4 4.5

3,4 and 5 (y = 2.75)

3.7 2.8 8.2 4.9

7,8 and 9 (y = 3.75)

3.3 6.1 8.2 5.9

Lp∆ [dB] 2.9 4.4 7.9 5.1

Table 4.11. Mean values for the difference between predicted and measured SPL [dB], at each measuring position, ground floor room at Ton Art. Overall, this listening room displayed several obvious patterns. The error decreased with decreasing predicted SPL and decreasing frequency. In addition, parallel walls with different characteristics affected the ability to predict SPL using measurements done along only two of the walls. The Schroeder frequency was calculated to be 150 Hz for this room. As with the basement room, this frequency is notmuch bigger than 100 Hz, and contributes to explain the increase in error at the high frequencies.

4.2.3 Room B-337 at NTNU Gløshaugen Measurements were in this room done in positions 1-9 in Figure 4.31, and six frequencies were selected among the observed resonance frequencies in the room.

63

Figure 4.31. Room B-337 at NTNU Gløshaugen, department of acoustics. Figure 4.32 shows the predicted SPL plotted against the measured SPL for all positions and frequencies, and in Table 4.12 the points in Figure 4.32 are divided into groups based on how high the predicted SPL was. For most frequencies the error seemed to decrease with increase in predicted SPL, particularly at the resonances occuring at 82.6 Hz and 87.2 Hz a high SPL was predicted, and the measurements gave good results. The exceptions were the results at 56.6 Hz, that seemed to be bad no matter what SPL was predicted. Apart from these results, and those at 28.7 Hz, the errors were in general small for this room.

Figure 4.32. Predicted and measured sound pressure level in all combinations of nine different positions and frequencies 28,7 Hz, 40,6 Hz, 56,6 Hz, 64,8 Hz, 70.7 Hz, 82,6 Hz and 87,2 Hz for room B-337 at NTNU.

64

Calculating the mean error in the groups in Table 4.12 showed that no particular pattern could be observed in terms of e.g. decreasing error for increasing level for the predicted SPL. But if the results at 56.6 Hz had not been included, the error in class 1 would be reduced. Class: Description (relative): Points: Mean error: Class 1: Lp > 0 dB 21 6.4 dB Class 2: 0 dB < Lp < -10 dB 20 4.7 dB Class 3 Lp < -10 dB 13 7.0 dB Table 4.12. Mean difference between predicted and measured SPL dependent on predicted SPL, room B-337 at NTNU. Table 4.13 contains the mean errors for each frequency. These results show that the error did not vary much depending on frequency. Apart from the two resonant frequencies at 29 Hz and 57 Hz, the mean error for each frequency was approximately 4 dB. The SPL distributions that looked best according to the expected ones appeared at 40.6 Hz, and 70.7 Hz. And as seen in Table 4.13, the errors at these resonances are fairly low. The low errors at 87.2 Hz and 82.6 Hz can be explained from Figure 4.32. A high SPL was predicted in all positions at these frequencies, and the measured SPLs were correspondingly high. At both these frequencies two or three modes interfered with each other to create the observed SPL distributions, but the resultant resonance frequencies display large peaks in the frequency responses. Frequency [Hz] 28.7 40.6 56.6 64.8 70.7 82.6 87.2 Lp∆

Lp∆ [dB] 8.8 4.3 10.5 4.3 5.0 4.4 3.9 5.9

Table 4.13. Mean values for the difference between predicted and measured SPL, [dB], at the resonant frequencies 28,7 Hz, 40,6 Hz, 56,6 Hz, 64,8 Hz, 82,6 Hz and 87,2 Hz, room B-337 at NTNU. Table 4.14 shows the errors based on position, and it is clear that no patterns at all are observable, as the mean error was approximately 5-6 dB for most positions.

Positions: 1,4 and 7 (x = 3.70)

2,5 and 8 (x = 2.70)

3,6 and 9 (x = 1.70)

Lp∆ , dB]

1, 2 and 3 ( y = 1.50)

5.0 4.9 4.3 4.7

3,4 and 5 (y = 2.50)

5.1 9.1 6.1 6.8

7,8 and 9 (y = 3.50)

4.9 6.1 7.5 6.2

Lp∆ [dB] 5.0 6.7 6.0 5.9

Table 4.14. Mean values for the difference between predicted and measured SPL, [dB], at each measuring position, room B-337 at NTNU.

65

The two observed patterns for the two Ton Art-rooms, i.e. better results for low frequencies, and better results when a high SPL was predicted, were not present in these measurements, but it is worth noticing that the results at 28.7 Hz and 56.6 Hz are much worse than at the other frequencies. Not including the results at these frequencies would lead to a mean error of 4.4 dB in this room, which is very low compared to the other rooms. The Schroeder frequency had the highest value for this room, 181 Hz. This may explain why the error not increased at the higher frequencies below 100 Hz, as it did in the Ton Art rooms.

4.2.4 Room B-343 at NTNU Gløshaugen The measurement positions in room B-343 are showed in Figure 4.33 below.

Figure 4.33. Room B-343 at NTNU Gløshaugen, department of acoustics. Figure 4.34 and Table 4.15 show that the error was big for several measurements where a high SPL was predicted. So like for room B-337, there seems to be no guarantee for a small error if the predicted SPL was high.

66

Figure 4.34. Predicted and measured SPL level in all combinations of nine different positions and frequencies 32,4 Hz, 44,3 Hz, 58,1 Hz, 63,7 Hz, 72,3 Hz, 83,1 Hz and 90,6 Hz for room B-343 at NTNU.

Class: Description (relative): Points: Mean error: Class 1: Lp > 25 dB 25 9.2 dB Class 2: 15 dB < Lp < 25 dB 18 4.9 dB Class 3 Lp < 15 dB 20 6.5 dB Table 4.15. Mean difference between predicted and measured SPL dependent on predicted SPL, room B-343 at NTNU. To some degree it might be stated that the results were better for low frequencies than for high frequencies (Table 4.16). Focusing only on the mean error for frequencies 44.3 Hz, 58.1 Hz, 63.7 Hz and 83.1 Hz, the error gradually increased with frequency. The remaining three selected frequencies did not follow this pattern though. The only resonance frequency that displayed SPL distributions that looked like expected, was the one at 44.3 Hz. And at this frequency the mean error was by far the lowest. Frequency [Hz] 32.4 44.3 58.1 63.7 72.3 83.1 90.6 Lp∆

Lp∆ [dB] 7.3 3.8 5.1 8.3 6.4 12.5 6.0 7.1

Table 4.16. Mean values for the difference between predicted and measured SPL, [dB, at the, resonant frequencies 32,4 Hz, 44,3 Hz, 58,1 Hz, 63,7 Hz, 72,3 Hz, 83,1 Hz and 90,6 Hz, room B-343 at NTNU. Table 4.17 displays the mean errors distributed on the measuring positions. It seem like the results got worse along the y-axis, with measurements 1,2 and 3 together resulting in a mean error higher than the next two groups with higher y-values. In

67

other words, measurements close to wall A gave slightly better results than measurements close to wall C.

Positions: 1,4 and 7 (x = 4.50)

2,5 and 8 (x = 3.00)

3,6 and 9 (x = 1.50)

Lp∆ ,[dB]

1, 2 and 3 (y = 2.25)

8.7 7.3 8.9 8.3

3,4 and 5 (y = 4.25)

8.3 6.8 5.6 6.9

7,8 and 9 (y = 6.25)

7.8 5.0 5.0 5.9

Lp∆ [dB] 8.3 6.4 6.5 7.1

Table 4.17. Mean values for the difference between predicted and measured SPL, [dB], at each measuring position, room B-343 at NTNU. The Schroeder frequency was calculated to be 139 Hz in this room, but based on the observed frequency responses, this value might be even lower. The room modes expected above 65 Hz were expected to appear with 2-3 Hz between them, and the peaks in the frequency response appeared very close for frequencies higher than 70 Hz

68

5 Conclusions Several points and patterns were observed from the measurement results. The rooms in which measurements were performed were either almost squared in shape, or almost twice as long as wide. This led to the modes being predicted to appear either far from other modes or near other modes. Using the Bonella criteria revealed this to some degree, with the room failing the criteria in the worst manner having the least even distribution of modes along the frequency axis. But a room fulfilling the criteria may still have an unwanted distribution of modes, as the Ton Art basement room did. The modes predicted to be spaced far from other modes displayed SPL distributions that closely resembled the theoretical ones. When two or more modes were expected at almost the same frequencies, the results were resonance frequencies with unpredictable SPL distributions. Whether the expected modes were axial, tangential or oblique, they all seemed to interfere with each other, and most often one particular mode did not dominate over others. The bandwidth of a mode increases with RT, and with the measured RTs in these rooms, bandwidths sized 2-4 Hz might be expected3. But based on the performed measurements, a space of 4-5 Hz to the next mode in both directions seemed to be necessary in all rooms for the mode to appear according to theory. Prediction of SPL at the frequencies where modes appeared unifluenced by other modes also produced some of the lowest errors. The mean error from all the predictions, was 5.6 dB. The mean error for the resonances that were caused by modes spaced far from other modes and produced the predicted SPL distributions, was 4.1 dB. Wall materials also affected the measurement results. SPL maximums mostly occurred as expected at hard concrete walls. At wooden walls and walls with glass windows, lower maximums and sometimes even SPL minimums occurred at some frequencies. Some materials did not act like expected, and in some cases surprisingly low SPLs were observed at hard plaster walls. Some of the rooms had different materials on the different walls, and in some cases different materials different places on the same wall. These factors contributed in creating unpredictable SPL distributions. The SPL distribution in one direction was in the ground floor room at Ton Art observed to be different when measurements were done along the parallel wall. Ideally the distribution should not change by doing so. There is no doubt about the fact that such unsymmetrical properties in terms of wall material affected the results when the SPLs were predicted. As expected11, unsymmetrical geometry also complicates the sound field in a room, here in room B-343. The errors when predicting SPLs were generally smaller in the ground floor room at Ton Art. But this is mainly because three of the five frequencies where comparisons were done were room modes appearing far from other modes. As these are low frequency room modes, this is also the reason why the results were better at lowest frequencies in this room and partly in the basement room.

69

For three of the four rooms, the prediction of a high SPL resulted to some degree in better results than when a low SPL was predicted, indicating that one mode dominated at that position. This effect was most apparent in the smallest room, the ground floor room at Ton Art. In the largest room, such a pattern was not observed. This was most likely caused by several resonances close to each other in the frequency response, instead of fewer, larger peaks like in the other rooms. Because of the modes appearing so close, the 184 m2 was too big to gain good results in the experiments. The highest Schroeder frequency, 181 Hz, was calculated for room B-337, and in this room the mean prediction error did not increase with frequency below 100 Hz, like it did for the Ton Art rooms. This implies that the prediction method only works at frequencies well below the Schroeder frequency. All in all, SPL distributions appear as predicted when the room modes have a minimum distance in Hz to neighbouring modes. Using these SPL distributions to predict SPL in selected positions seem to be possible, within an error margin of +/- 4 dB, at these frequencies and to some degree, it is also possible at other frequencies where large resonances are observed. For the best results, the room should have symmetrical geometry, and hard and reflective walls. The experiments would be interesting to perform in a small room with room dimension ratios that create a more even distribution of room modes.** More modes would then appear uninfluenced by other modes, and the measurement results would be expected to be better at more modes than was the case in the four rooms discussed here. The ratio between RT and room volume would also have to be such that they lead to a high Schroeder frequency.

** See chapter 2.3 on room dimension ratios.

70

6 Appendix 6.1 Room modes Room modes, basement room at Ton Art:

Mode frequencies, fnmi Mode no. n m i Frequency axial tangential oblique 1. 0 1 0 26.9 Hz x 2. 1 0 0 28.3 Hz x 3. 1 1 0 39.1 Hz x 4. 0 2 0 53.8 Hz x 5. 2 0 0 56.7 Hz x 6. 1 2 0 60.8 Hz x 7. 2 1 0 62.7 Hz x 8. 0 0 1 73.2 Hz x 9. 2 2 0 77.8 Hz x 10. 0 1 1 78.0 Hz x 11. 1 0 1 78.5 Hz x 12. 0 3 0 80.6 Hz x 13. 1 1 1 83.0 Hz x 14. 3 0 0 85.0 Hz x 15. 1 3 0 85.5 Hz x 16. 3 1 0 89.2 Hz x 17. 0 2 1 90.8 Hz x 18. 2 0 1 92.6 Hz x 19. 1 2 1 95.1 Hz x 20. 2 1 1 96.4 Hz 21. 2 3 0 98.6 Hz 22. 3 2 0 100.6 Hz x

Table 6.1. Calculated room mode frequencies, main room in basement. Dimensions used: Width(x) = 6.07, Length(y) = 6.40, Height(z) = 2.35.

71

Room modes, ground floor room at Ton Art

Mode frequencies, fnmi Mode no. n m i Frequency Axial Tangential oblique

1. 0 1 0 32.8 Hz x 2. 1 0 0 39.3 Hz x 3. 1 1 0 51.2 Hz x 4. 0 2 0 65.6 Hz x 5. 0 0 1 69.4 Hz x 6. 1 2 0 76.5 Hz x 7. 0 1 1 76.7 Hz x 8. 2 0 0 78.5 Hz x 9. 1 0 1 79.7 Hz x 10. 2 1 0 85.1 Hz x 11. 1 1 1 86.2 Hz x 12. 0 2 1 95.5 Hz x 13. 0 3 0 98.5 Hz x

Table 6.2. Calculated room mode frequencies, ground floor listening room at Ton Art. Dimensions: Width(x) = 4.38, Length(y) = 5.24, , Height(z) = 2.48.

72

Room modes, rooom B-337 at NTNU Gløshaugen

Mode frequencies, fnmi Mode no. n m i Frequency Axial Tangential oblique 1. 1 0 0 28.9 Hz x 2. 0 1 0 29.7 Hz x 3. 1 1 0 41.4 Hz x 4. 0 0 1 57.3 Hz x 5. 2 0 0 57.8 Hz x 6. 0 2 0 59.3 Hz x 7. 1 0 1 64.2 Hz x 8. 0 1 1 64.5 Hz x 9. 2 1 0 65.0 Hz x 10. 1 2 0 66.0 Hz x 11. 1 1 1 70.7 Hz x 12. 2 0 1 81.4 Hz x 13. 0 2 1 82.5 Hz x 14. 2 2 0 82.8 Hz x 15. 2 1 1 86.7 Hz x 16. 3 0 0 86.7 Hz x 17. 1 2 1 87.4 Hz x 18. 0 3 0 89.0 Hz x 19. 3 1 0 91.7 Hz x 20. 1 3 0 93.5 Hz x

Table 6.3. Calculated room mode frequencies, room B-337 at NTNU Gløshaugen. Dimensions: Width(x) = 5.95, Length(y) = 5.80, , Height(z) = 3.00.

73

Room modes room B-343

Mode frequencies, fnmi Mode no. n m i Frequency axial tangential oblique 1. 0 1 0 17.1 Hz x 2. 1 0 0 29.7 Hz x 3. 1 1 0 34.2 Hz x 4. 0 2 0 34.3 Hz x 5. 1 2 0 45.3 Hz x 6. 0 3 0 51.4 Hz x 7. 0 0 1 57.3 Hz x 8. 2 0 0 59.3 Hz x 9. 1 3 0 59.3 Hz x 10. 0 1 1 59.8 Hz x 11. 2 1 0 61.7 Hz x 12. 1 0 1 64.5 Hz x 13. 1 1 1 66.8 Hz x 14. 0 2 1 66.8 Hz x 15. 2 2 0 68.5 Hz x 16. 0 4 0 68.5 Hz x 17. 1 2 1 73.1 Hz x 18. 1 4 0 74.7 Hz x 19. 0 3 1 77.0 Hz x 20. 2 3 0 78.5 Hz x 21. 2 0 1 82.5 Hz x 22. 1 3 1 82.5 Hz x 23. 2 1 1 84.3 Hz x 24. 0 5 0 85.7 Hz x 25. 3 0 0 89.0 Hz x 26. 2 2 1 89.3 Hz x 27. 0 4 1 89.3 Hz x 28. 3 1 0 90.6 Hz x 29. 2 4 0 90.6 Hz x 30. 1 5 0 90.6 Hz x 31. 1 4 1 94.1 Hz x 32. 3 2 0 95.3 Hz x 33. 2 3 1 97.2 Hz x Table 6.4. Calculated room mode frequencies, room B-343 at NTNU Gløshaugen. Dimensions: Width(x) = 5.96, Length(y) = 10.27, Height(z) = 3.00.

74

6.2 MATLAB See enclosed CD for MATLAB code that was used for the project.

75

References 1 R. E. Berg, David G. Stork, “The Physics of Sound”, 2nd ed., Prentice-Hall, inc. (1995), p. 66-90. 2 C. L. S. Gilford, “The Acoustic Design of Talks Studios and Listening Rooms”, J. Audio Eng. Soc., vol. 27 (1979), p. 17-30. 3 F. A. Everest, “Master Handbook of Acoustics”, 4th ed.,McGraw-Hill (2001), p.317-352. 4 A. Krokstad, ”Akustikk for ingeniører”, NTNU (1999), p. 148. 5 P. M. Morse and R. H. Bolt, ”Sound Waves in Rooms”, Reviews of Modern Physics, vol. 16, no. 2 (April 1944), p. 85. 6 Kinsler, Frey, Coppens, Sanders, “Fundamentals of Acoustics”, John Wiley & sons, inc (2000), p. 246-248, p.340-341, p.348-355. 7 R. H. Bolt, “Note on Normal Frequency Statistics for Rectangular Rooms”, J.Acoust. Soc. Am., vol. 18 (1946), p. 130-133. 8 L. W. Sepmeyer, “Computed Frequency and Angular Distribuion of the Normal Modes of Vibration in Rectangular Rooms”, J. Aciyst, Soc. Am., vol. 37 (1965), p. 413-423. 9 O. J. Bonello, “A New Criterio for the Distribution of Normal Room Modes”, J. Audio Eng. Soc., vol. 29 (1981), p. 597-605. 10 P. Svensson, conversations January 2005 – June 2005. 11 G. Ballou (ed.), “Handbook for Sound Engineers”, Howard W.Sams & Company, 2nd ed. (1991), p. 43-66.