2.8. Freely traveling plane wave

Sound pressure

If the rigid termination of Fig. 2.4 is replaced by a perfectly absorbing termination, a backward-traveling wave will not occur. Hence, Eq. (2.46) becomes
p˜(x)=p˜+e−jkx,
image (2.121)
where p˜+ image is the complex amplitude of the wave. This equation also applies to a plane wave traveling in free space.

Particle velocity

From Eq. (2.4a) in the steady state, we have
u˜(x)=−1jkρ0c∂∂xp˜(x).
image (2.122)
Hence,
u˜(x)=p˜+ρ0ce−jkx=p˜(x)ρ0c.
image (2.123)
The particle velocity and the sound pressure are in phase. This is mathematical proof of the statement made in connection with the qualitative discussion of the wave propagated from a vibrating wall in Chapter 1 and Fig. 1.1.

Specific acoustic impedance

The specific acoustic impedance is
Zs=p˜(x)u˜(x)=ρ0crayls.
image (2.124)
This equation says that in a plane freely traveling wave the specific acoustic impedance is purely resistive and is equal to the product of the average density of the gas and the speed of sound. This particular quantity is generally called the characteristic impedance of the gas because its magnitude depends on the properties of the gas alone. It is a quantity that is analogous to the surge impedance of an infinite electrical line. For air at 22°C and a barometric pressure of 105 Pa, its magnitude is 407 rayls.
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