21 
The interesting point arises, however, that each component in the 
incident pulse experiences a change of phase, the value of which is 
independent of the frequency. If all the components of a Fourier transform 
suffer the same change of phase, the resultant shape of pulse is changed, 
especially inthe region of a steep front. 
This simple but important conclusion agrees with experiment but 
recent work at Woods Hole has shown that the reflected wave can only be 
reproduced by the mathematical analysis if an appreciable absorption of 
energy is assumed in the reflecting medium. Particularly interesting 
is the work, both theoretical and experimental, on the multiple wave system 
in shallow water (12b). 
The considerable algebraic and computational difficulties of the 
problem of the reflection and refraction of finite pressure pulses meeting 
an interface are demonstrated in an article by TauKk12c). A step pulse, 
lead by a shock wave, incident obliquely at an interface with a "denser" 
medium, always gives a shock in the denser medium, and a reflected Mach 
or a regularly reflected wave, depending on the angle of incidence and the 
mechanical parameters, in the lighter medium. A general solution has not 
been obtained. 
61. To sum up, for soft sea-beds the presence of the sea-bed may in 
general be neglected and the explosion assumed to take place in an infinite 
depth of water so far as the pressure pulse is concerned. For hard sea= 
beds the effects differ appreciably according as the charge is well away 
fran the sea=-bed or not. If the charge is not near the sea=bed the latter 
acts as a partially reflecting surface and the maximum effect of the sea=bed 
occurs for points on the sea-bed. On the other hand, a charge of weight W 
on a hard sea=bed can be equivalent to a charge of weight 2 W in mid-water 
so far as the pulse arriving at points well away from the sea-bed is concerned 
At the same time, it can hehave as of weight less than W for the pulse at 
points on the sea-bed. This phenomenon has yet to be satisfactorily 
explained. 
The gas bubble 
Oscillation and rise of bubble in mid-water 
62. After the pressure pulse has been propagated well away from the 
explosion the motion of the gas bubble and the surrounding water take 
Place relatively slowly in comparison with the initiel events producing 
the pulse. Hence, whilst the compressibility of the water is an all 
important factor for the pressure pulse, it is quite a good approximation 
to neglect this compressibility during most of the bubble motion. The 
mathematical treatment is, therefore, based primarily on the ordinary laws 
for incompressible flow as given in standard text books of hydrodynamics. 
