SOUND WAVES AND SOUND FIELDS
Principles of Sound and Vibration, Chapter 6
Science of Sound, Chapter 6
The 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 wave
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
We can simplify matters even further by writing p = ψ/r, giving
(a one dimensional wave equation)
Particle (acoustic) velocity:
The solution is an outgoing plus an incoming wave
ρc at kr >> 1
ρ ∂2ξ/∂t2 = -∂p/∂x
SOUND PRESSURE, POWER AND LOUDNESS
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
Sound Intensity Level: LI = 10 log I/I0 I0 = 10-12 W/m2
I = W/4πr2
at r = 1 m:
LI = 10 log I/10-12
= 10 log W/10-12 – 10 log 4p
= LW - 11
I = W/2pr2
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
Our ears respond to extremely small pressure
Intensity of a sound wave is proportional to the sound