handbook of modern sensors

3.10 Sound Waves 93
If we consider the propagation of a sound wave in an organ tube, each small
volume element of air oscillates about its equilibrium position. For a pure harmonic
tone, the displacement of a particle from the equilibrium position may be represented
by
y = y m cos 2π (x − vt), (3.100)
λ
where x is the equilibrium position of a particle and y is a displacement from the
equilibrium position, y m is the amplitude, and λ is the wavelength. In practice, it is
more convenient to deal with pressure variations in sound waves rather than with
displacements of the particles. It can be shown that the pressure exerted by the sound
wave is
p = (kρ 0 v 2 y m ) sin(kx − ωt), (3.101)
where k = 2π/λ is a wave number, ω is angular frequency, and the terms in the first
parentheses represent an amplitude, p m , of the sound pressure. Therefore, a sound
wave may be considered a pressure wave. It should be noted that sin and cos in Eqs.
(3.100) and (3.101) indicate that the displacement wave is 90 ◦ out of phase with the
pressure wave.
Pressure at any given point in media is not constant and changes continuously,
and the difference between the instantaneous and the average pressure is called the
acoustic pressure P . During the wave propagation, vibrating particles oscillate near a
stationary position with the instantaneous velocity ξ. The ratio of the acoustic pressure
and the instantaneous velocity (do not confuse it with a wave velocity) is called the
acoustic impedance:
Z = P ξ , (3.102)
which is a complex quantity, characterized by an amplitude and a phase. For an
idealized media (no loss), Z is real and is related to the wave velocity as
Z = ρ 0 v. (3.103)
We can define the intensity I of a sound wave as the power transferred per unit area.
Also, it can be expressed through the acoustic impedance:
I = Pξ = P 2
Z . (3.104)
It is common, however, to specify sound not by intensity but rather by a related
parameter β, called the sound level and defined with respect to a reference intensity
I 0 = 10 −12 W/m 2 β = 10 log 10
( I
I 0
)
(3.105)
The magnitude of I 0 was chosen because it represents the lowest hearing ability of a
human ear. The unit of β is a decibel (dB), named after Alexander Graham Bell. If
I = I 0 ,β= 0.