Chapter 2
Waves of Finite Amplitude
2.1 Introduction
In our brief discussion of sound waves in Chap. 1 we assumed that the wave
amplitude is sufficiently small that the resulting equations are linear. As a result, it
was relatively easy to solve the equations which led to travelling waves whose
distribution of density, pressure, velocity etc. moved with constant velocity c 0 and
the profile of the wave did not change with time. This, however, is not the case when
the wave has appreciable amplitude and we will see, in due course, that the wave
profile changes its shape as it propagates.
2.2 Finite Amplitude Waves
It is necessary to return to the more exact equations of motion when the wave has
appreciable amplitude: as we have already seen, these are the continuity and
momentum equations
∂ρ
∂t
þ ρ
∂u
∂x
þ u
∂ρ
∂x
¼ 0
ð2:1Þ
and
∂u
∂t
þ u
∂u
∂x
þ
1
ρ
∂p
∂x
¼ 0,
ð2:2Þ
respectively. For isentropic flow we have the further equation, pρ
Àγ
¼ constant,
hence,
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
S. Prunty, Introduction to Simple Shock Waves in Air, Shock Wave and High
Pressure Phenomena, https://doi.org/10.1007/978-3-030-63606-7_2
43