292 
= [ow | 
The Riemann function f Is a numerical multiple of c. 
f= /- dpddv.. dv = 0.3202 c. 
This relation Is extraordinarily convenient In the numerical work because it means that c can be 
found by merely multiplying (P + Q at any point by 0.1601, 
Energy wastage in shock waves, 
Shock waves, of course, are irreversible, and it is Impossible without employing perfect heat 
engines to regain In mechanical form all of the energy belng used In me'ntaining a shock wave of 
pressure D5. The water on the high pressure side has kinetic energy, already in mechanical form, 
and In virtue of Its pressure, may be made to do mechanical work by adiabatic expansion to atmospheric 
pressure against a full resistance, The water will then be hotter than it was originally (20°C) and 
the heat excess may be termed the "wastage". The wastage per gramme of water will be almost exactly 
@ calories. Avery small correction should be applied, because the temperature difference between 
D and A In Figure 1, the true wastage, Is not quite the same as the difference between C and B, called 
6 above. However, @ has not been calculated accurately enough to justify making a correction of 
this type. 
Reflexion and stagnation pressures. 
According to the theory of sound, an infinitesimal plane pressure culse p. + p,, incident 
normally on a plane rigid wall, is reflected, and the pressure on the wall Is p. + 2p, This effect 
is usually known as *doubling by reflexion". tf the plane wall is of finite area, the pressure 
Instantaneously Is doubted, but soon falls to Po * Py because of diffraction. When the incident 
pulse Is of finite intensity, variations are possible. First, the Instantaneous pressure excess 
over p, Is not doubled, but is Increased by a factor greater than 2 according to the medium and the 
precise value of Pas Second, the mass velocity of the fluid u associated with the original pressure 
pulse may be less than, equal to, or greater than the velocity of sound ¢ in the fluid carrying the 
pulse, If u<c, the situation is described as sonic; If u>e the situation Is described as 
supersonic. Details will be found In Rayleigh, Sctentif _ c Papers V, p,608; a helpful summary, 
together with numerical examples for alr (@/ = 1.4), will be found in R.C.118 (Taylor). 
Consider now a plane shock wave In water about to Impinge normally on a finite plane rigid 
walt. The first point to notice Is that conditions near the wall are sonic up to pressures wel) 
beyond 50 (say 300 tong square inch). This may be seen from Tables 1 or 2, where c Is always much 
greater than u, The pressure at the stagnation point, I.e. the polnt at which the stream has zero 
velocity, may be calculated from the pressure and velocity In the main stream by the use of 
BernoulllI's theorem, The Instantaneous pressure on the wall may be found by Introducing a 
reflected shock wave of sufficient Intensity to reduce the mass velocity to zero. The following 
values have been computed to Illustrate how the ordinary laws of sound reflexion apply reasonably 
well in water even at enormous pressures. 
TABLE 3 ceoee 
