chapter 15 wave motion - university of...

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Chapter 15 Wave Motion


If the wave enters a medium where the wave speed is different, it will be refracted—its wave fronts and rays will change direction. We can calculate the angle of refraction, which depends on both wave speeds:

15-10 Refraction


15-10 Refraction

Example:

A longitudinal earthquake wave strikes a boundary between two types of rock at an angle of 38o. As the wave crosses the boundary, the specific gravity of the rock changes from 3.6 to 2.8. Assuming the elastic modulus is the same for both types of rock, determine the angle of refraction.


When waves encounter an obstacle, they bend around it, leaving a “shadow region.” This is called diffraction.

15-11 Diffraction


The amount of diffraction depends on the size of the obstacle compared to the wavelength. If the obstacle is much smaller than the wavelength, the wave is barely affected (a). If the object is comparable to, or larger than, the wavelength, diffraction is much more significant (b, c, d).

15-11 Diffraction


•  Vibrating objects are sources of waves, which may be either pulses or continuous.

•  Wavelength: distance between successive crests

•  Frequency: number of crests that pass a given point per unit time

•  Amplitude: maximum height of crest

•  Wave velocity:

Summary of Chapter 15


•  Transverse wave: oscillations perpendicular to direction of wave motion

•  Longitudinal wave: oscillations parallel to direction of wave motion

•  Intensity: energy per unit time crossing unit area (W/m2):

•  Angle of reflection is equal to angle of incidence



•  When two waves pass through the same region of space, they interfere. Interference may be either constructive or destructive.

•  Standing waves can be produced on a string with both ends fixed. The waves that persist are at the resonant frequencies.

•  Nodes occur where there is no motion; antinodes where the amplitude is maximum.

•  Waves refract when entering a medium of different wave speed, and diffract around obstacles.



Chapter 16 Sound


• Characteristics of Sound

• Mathematical Representation of Longitudinal Waves

•  Intensity of Sound: Decibels

• Sources of Sound: Vibrating Strings and Air Columns

• Quality of Sound, and Noise; Superposition

•  Interference of Sound Waves; Beats

• Doppler Effect

Units of Chapter 16


Sound can travel through any kind of matter, but not through a vacuum.

The speed of sound is different in different materials; in general, it is slowest in gases, faster in liquids, and fastest in solids.

The speed depends somewhat on temperature, especially for gases.

16-1 Characteristics of Sound



Example: Distance from a lightning strike.

A rule of thumb that tells how close lightning has struck is, “one km for every 3 seconds before the thunder is heard.” Explain why this works, noting that the speed of light is so high (3 x 108 m/s, almost a million times faster than sound) that the time for light to travel to us is negligible compared to the time for the sound.


Loudness: related to intensity of the sound wave

Pitch: related to frequency

Audible range: about 20 Hz to 20,000 Hz; upper limit decreases with age

Ultrasound: above 20,000 Hz

Infrasound: below 20 Hz



16-2 Mathematical Representation of Longitudinal Waves

Longitudinal waves are often called pressure waves. The displacement is 90° out of phase with the pressure.



By considering a small cylinder within the fluid, we see that the change in pressure is given by (B is the bulk modulus):



If the displacement is sinusoidal, we have

where

and


The intensity of a wave is the energy transported per unit time across a unit area.

The human ear can detect sounds with an intensity as low as 10-12 W/m2 and as high as 1 W/m2.

Perceived loudness, however, is not proportional to the intensity.

16-3 Intensity of Sound: Decibels


The loudness of a sound is much more closely related to the logarithm of the intensity.

Sound level is measured in decibels (dB) and is defined as:

I0 is taken to be the threshold of hearing:



Example: Sound intensity at concert At a very loud concert, a 120-dB sound wave travels away from the loudspeaker at 343 m/s. How much sound energy is contained in each 1.0-cm3 volume of air near the this loudspeaker?


Example:

A CD player is said to have a signal-to- noise ratio of 98 dB. What is the ratio of the intensities of of the signal and the background noise for this device?


Example: Trumpet players.

A trumpeter plays at a sound level of 75 dB. Three equally loud trumpet players join in. What is the new sound level?


An increase in sound level of 3 dB, which is a doubling in intensity, is a very small change in loudness.

In open areas, the intensity of sound diminishes with distance:

However, in enclosed spaces this is complicated by reflections, and if sound travels through air, the higher frequencies get preferentially absorbed.




The intensity can be written in terms of the maximum pressure variation. With some algebraic manipulation, we find:


The ear’s sensitivity varies with frequency. These curves translate the intensity into sound level at different frequencies.



Musical instruments produce sounds in various ways—vibrating strings, vibrating membranes, vibrating metal or wood shapes, vibrating air columns.

The vibration may be started by plucking, striking, bowing, or blowing. The vibrations are transmitted to the air and then to our ears.

16-4 Sources of Sound: Vibrating Strings and Air Columns



This table gives frequencies for the octave beginning with middle C. The equally tempered scale is designed so that music sounds the same regardless of what key it is transposed into.


Tempered Chromatic Scale

f12 f0 = 524 262 = 2 = R12

! R = 21 12 = 1.0595

262"1.05951 = 277 262"1.05957 = 392262"1.05952 = 294 262"1.05958 = 415262"1.05953 = 311 262"1.05959 = 440262"1.05954 = 330 262"1.059510 = 466262"1.05955 = 349 262"1.059511 = 494262"1.05956 = 370 262"1.059512 = 524



This figure shows the first three standing waves, or harmonics, on a fixed string.


The strings on a guitar can be effectively shortened by fingering, raising the fundamental pitch.

The pitch of a string of a given length can also be altered by using a string of different density.




The sound waves from vibrating strings need to be amplified in order to be of a practical loudness; this is done in acoustical instruments by using a sounding board or box, creating a resonant chamber. The sound can also be amplified electronically.


Wind instruments create sound through standing waves in a column of air.



A tube open at both ends (most wind instruments) has pressure nodes, and therefore displacement antinodes, at the ends.



A tube closed at one end (some organ pipes) has a displacement node (and pressure antinode) at the closed end.



Example 16-10: Organ pipes.

What will be the fundamental frequency and first three overtones for a 26-cm-long organ pipe at 20°C if it is (a) open and (b) closed?


So why does a trumpet sound different from a flute? The answer lies in overtones—which ones are present, and how strong they are, makes a big difference. The sound wave is the superposition of the fundamental and all the harmonics.

16-5 Quality of Sound, and Noise; Superposition


This plot shows frequency spectra for a clarinet, a piano, and a violin. The differences in overtone strength are apparent.

16-5 Quality of Sound, and Noise; Superposition


Assignment:

Chapter 16: 8, 20, 48

chapter 15 wave motion - university of...

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