sound waves and sound fields acoustics of concert halls and rooms principles of sound and vibration,...

SOUND WAVES AND SOUND

FIELDS

Acoustics of Concert Halls and Rooms

•Principles of Sound and Vibration, Chapter 6

•Science of Sound, Chapter 6

THE ACOUSTIC WAVE EQUATIONThe acoustic wave equation is generally derived by considering an ideal fluid (a mathematical fiction).

Its motion is described by the Euler equation of motion.In a real fluid (with viscosity), the Euler equation is Replaced by the Navier-Stokes equation.

Two different notations are used to derive the Acoustic waveequation:1. The LaGrange description We follow a “particle” of fluid as it is compressed as well

as displaced by an acoustic wave.)2. The Euler description

(Fixed coordinates; p and c are functions of x and t.They describe different portions of the fluid as it streams

past.

PLANE SOUND WAVES

Plane Sound Waves

SPHERICAL WAVES

We can simplify matters even further by writing p = ψ/r, giving

(a one dimensional wave equation)

Spherical waves:

Particle (acoustic) velocity:

Impedance:

The solution is an outgoing plus an incoming wave

ρc at kr >> 1

Similar to:

ρ ∂2ξ/∂t2 = -∂p/∂x

outgoing incoming

SOUND PRESSURE, POWER AND LOUDNESS

Decibels

Decibel difference between two power levels:

ΔL = L2 – L1 = 10 log W2/W1

Sound Power Level: Lw = 10 log W/W0 W0 = 10-12 W (or PWL)

Sound Intensity Level: LI = 10 log I/I0 I0 = 10-12 W/m2

(or SIL)

FREE FIELDI = W/4πr2

at r = 1 m:

LI = 10 log I/10-12

= 10 log W/10-12 – 10 log 4

= LW - 11

HEMISPHERICALFIELD

I = W/2r2

at r = l m LI = LW - 8

Note that the intensity I 1/r2 for both free and

hemispherical fields; therefore, LI decreases 6 dB for each doubling of distance

SOUND PRESSURE LEVEL

Our ears respond to extremely small pressure fluctuations p

Intensity of a sound wave is proportional to the sound Pressure squared: ρc ≈ 400 I = p2 /ρc ρ = density c = speed of sound

We define sound pressure level:

Lp = 20 log p/p0 p0 = 2 x 10-5 Pa (or N/m2)(or SPL)

TYPICAL SOUND LEVELS

MULTIPLE SOURCES

Example:Two uncorrelated sources of 80 dB each will produce a sound level of 83dB (Not 160 dB)

MULTIPLE SOURCES

What we really want to add are mean-squareaverage pressures (average values of p2)This is equivalent to adding intensities

Example: 3 sources of 50 dB each

Lp = 10 log [(P12+P2

2+P32)/P0

2] = 10 log (I1 + I2 + I3)/ I0)= 10 log I1/I0 + 10 log 3 = 50 + 4.8 = 54.8 dB

SOUND PRESSURE and INTENSITY

Sound pressure level is measured with a sound level meter (SLM)Sound intensity level is more difficult to measure, and it requiresmore than one microphoneIn a free field, however, LI LP

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