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
≈
