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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