Problem B.2: Shock Wave (6 Points)
This year’s qualification round featured a spaceship escaping from a shock wave (Problem B).
The crew survived and wants to study the shock wave in more detail. It can be assumed that
the shock wave travels through a stationary flow of an ideal polytropic gas which is adiabatic on
both sides of the shock. Properties in front and behind a shock are related through the three
Rankine-Hugoniot jump conditions (mass, momentum, energy conservation):
ρ1v1 = ρ2v2 ρ1v21 + p1 = ρ2v22 + p2 v212 + h1 = v222 + h2
where ρ, v, p, and h are the density, shock velocity, pressure, and specific enthalpy in front (1)
and behind (2) the shock respectively.
Shock front
v2, ρ2, p2, h2 v1, ρ1, p1, h1
(a) Explain briefly the following terms used in the text above:
(i) stationary flow
(ii) polytropic gas
(iii) specific enthalpy
(b) Show with the Rankine-Hugoniot conditions that the change in specific enthalpy is given by:
∆h = p2 p1 2 · 1ρ1 + 1ρ2
The general form of Bernoulli’s law is fulfilled on both sides of the shock separately:
v22
+ Φ + h = b
where Φ is the gravitational potential and b a constant.
(c) Assuming that the gravitational potential is the same on both sides, determine how the constant b changes at the shock front.
(d) Explain whether Bernoulli’s law can be applied across shock fronts.
Answers
Answer:
17.8 Shock Waves
LEARNING OBJECTIVES
By the end of this section, you will be able to:
Explain the mechanism behind sonic booms
Describe the difference between sonic booms and shock waves
Describe a bow wake
When discussing the Doppler effect of a moving source and a stationary observer, the only cases we considered were cases where the source was moving at speeds that were less than the speed of sound. Recall that the observed frequency for a moving source approaching a stationary observer is
f
o
=
f
s
(
v
v
−
v
s
)
.
As the source approaches the speed of sound, the observed frequency increases. According to the equation, if the source moves at the speed of sound, the denominator is equal to zero, implying the observed frequency is infinite. If the source moves at speeds greater than the speed of sound, the observed frequency is negative.
What could this mean? What happens when a source approaches the speed of sound? It was once argued by some scientists that such a large pressure wave would result from the constructive interference of the sound waves, that it would be impossible for a plane to exceed the speed of sound because the pressures would be great enough to destroy the airplane. But now planes routinely fly faster than the speed of sound. On July 28, 1976, Captain Eldon W. Joersz and Major George T. Morgan flew a Lockheed SR-71 Blackbird #61-7958 at 3529.60 km/h (2193.20 mi/h), which is Mach 2.85. The Mach number is the speed of the source divided by the speed of sound: