ABSTRACT
The changes in pressure when a sound wave travels through air are, in general, so rapid that heat cannot be exchanged between different volume elements. Consequently, the changes are adiabatic. In air, when the initial pressure and volume are P0 and V0, respectively, are changed to (P0 +p) and (V0 +V), respectively, due to sound pressure p, the following relationship is obtained
∴ 1+ p P0
= ( 1+ V
where γ is the ratio of specific heats at constant pressure and constant volume. If V/V0 is very small, then expanding the right-hand side of the above equation and approximating, it follows that
p P0
=−γ V V0
∴ p=−γP0 VV0 Putting
κ = γP0 (11.1)
p=−κ V V0
(11.2)
Therefore, sound pressure is proportional to the volume change. κ is called
B. Plane wave equation
Consider a tube of unit cross-sectional area with its axis parallel to the direction of propagation of a plane wave, as shown in Figure 11.1, where the plane at x is displaced by ξ and the plane at (x+ δx) is displaced by
ξ + δξ = ( ξ +
( ∂ξ
∂x
) δx )
This means that the original volume V = δx contained between the two planes has been increased by
V= (
∂ξ
∂x
) δx
due to the passage of the sound wave. Thus, the fractional increase in volume is
V V
= ∂ξ ∂x
which is the so-called equation of continuity. Substituting this into Equation (11.2) gives
p=−κ ∂ξ ∂x
(11.3)
When the sound pressure p acts on the plane at x, then
( p+
( ∂p ∂x
) δx )
acts on the plane at (x+ δx). The differential pressure (
∂p ∂x
) δx
sets in motion the mass of air between the two planes. If ρ is the average density of the air, then the mass between the two planes is (ρδx). Hence, using Newton’s second law, which states that force equals mass times acceleration, we have
ρδx ∂2ξ
∂t2 =−∂p
∂x · δx
∴ ρ ∂ 2ξ
∂t2 =−∂p
∂x
(11.4)
This is the equation of motion. Differentiating both sides of Equation (11.3) with respect to t and
Equation (11.4) with respect to x and eliminating ξ ,
∂2p ∂t2
= κ∂ 2p
ρ∂x2 (11.5a)
or ∂2p ∂t2
= c2 ∂ 2p
∂x2 (11.5b)
where
c= √
κ
ρ
This is the wave equation in terms of sound pressure. Similarly, eliminating p, the wave equation, in terms of the displacement ξ is obtained in the same form. Then, introducing a new function, the velocity potential ϕ defined by Equation (3.2) in Chapter 3,
∂ξ
∂t =−∂ϕ
∂x
p= ρ ∂ϕ ∂t
⎫⎪⎪⎬ ⎪⎪⎭ (11.6)
and using the fundamental Equation (11.3), the following is obtained
∂2ϕ
∂t2 = c2 ∂
∂x2 (11.7)
This has exactly the same form as Equation (11.5) for sound pressure. If the sound wave behaves as a simple harmonic vibration, then the
function may be written as ϕ ejωt where ω is the angular frequency, then Equation (11.7) may be written
( d2
) ϕ =0 (11.8)
where
k= ω c
k is called the wavelength constant or wave number.
