200 
4l | APPENDIX I 
2. REFLECTION OF WAVES 
6° being the angle of refraction so that 
c2 Sin 6 = ¢, sin 0’ {10} 
As before, the coefficient of reflection is R= (p"/p)?. 
If, however, as in all actual cases, sliding between the mediums does not 
occur, the boundary conditions cannot be satisfied merely by superposing upon the in- 
cident wave a reflected and a refracted one. A local disturbance must then occur near 
the interface, which involves shearing motion in both mediums. There should, however, 
be no appreciable effect upon the waves, so long as the amplitude remains small. If 
the waves are not spherical, or if c, sin@>c,, so that total reflection occurs, the 
phenomena at the interface are more complicated. Special effects due to this cause 
are utilized in geophysical sound-ranging. 
APPENDIX I 
3. FINITE AMPLITUDES 
III. WAVES OF FINITE AMPLITUDE 
Waves of appreciable amplitude should possess none of the properties listed 
for waves of indefinitely small amplitude, except in approximate degree as the ampli- 
tude becomes rather small. In water, effects of finite amplitude should be appreci- 
able at wave pressures exceeding 2000 pounds per square inch. 
The various parts of a wave of finite amplitude travel at different speeds 
for two reasons. In the first place, the wave is carried along by the medium in its 
motion; and in the second place, the wave velocity itself usually increases with in- 
creasing density of the medium. Hence regions of higher pressure are propagated 
through space faster than regions of lower pressure. Consequently, a compression, as 
it advances should become pro- 
gressively steeper at the rear, 
as suggested in Figure 16. There 
is some experimental evidence in Ep 
support of this conclusion from 
theory, at least in the case of 
+ 
sound waves in air. 
The final result of 
such a process would obviously p<0 
be the production of infinite 
gradients, i.e., discontinuities Figure 16 
of pressure and of particle ve- 
locity. When a discontinuity comes into existence, however, the ordinary laws of 
hydrodynamics fail. A special theory for the further propagation of such discontinui- 
ties has been given by Riemann (10) and Hugoniot (15). This theory will next be de- 
scribed. 
