SOUND WAVES AND SOUND FIELDS Acoustics of Concert Halls and RoomsPrinciples 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:The LaGrange description We follow a particle of fluid as it is compressed as well as displaced by an acoustic wave.)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 WAVESWe can simplify matters even further by writing p = /r, giving(a one dimensional wave equation)

Spherical waves:Particle (acoustic) velocity:Impedance:The solution is an outgoing plus an incoming wavec at kr >> 1 Similar to: 2/t2 = -p/x outgoing incoming

SOUND PRESSURE, POWER AND LOUDNESSDecibelsDecibel difference between two power levels:

L = 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/4r2at r = 1 m:

LI = 10 log I/10-12 = 10 log W/10-12 10 log 4p = LW - 11

HEMISPHERICALFIELDI = W/2pr2

at r = l m LI = LW - 8Note that the intensity I 1/r2 for both free and hemispherical fields; therefore, LI decreases 6 dB for each doubling of distance

SOUND PRESSURE LEVELOur ears respond to extremely small pressure fluctuations p

Intensity of a sound wave is proportional to the sound Pressure squared: c 400 I = p2 /c = 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 SOURCESExample:Two uncorrelated sources of 80 dB each will produce a sound level of 83dB (Not 160 dB)

MULTIPLE SOURCESWhat we really want to add are mean-squareaverage pressures (average values of p2)This is equivalent to adding intensitiesExample: 3 sources of 50 dB eachLp = 10 log [(P12+P22+P32)/P02] = 10 log (I1 + I2 + I3)/ I0)= 10 log I1/I0 + 10 log 3 = 50 + 4.8 = 54.8 dB

SOUND PRESSURE and INTENSITYSound 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