1429 
I. INTRODUCTION 
The reflection phenomena are complicated problems in 
acoustics or more general in the theory of waves. The usual 
solution for the linearized equations and infinitely extended 
plane waves being reflected by an infinite baffle is simple and 
well known. However, the exact solution of the reflection of 
spherical acoustical waves by the plane boundary of two mater- 
ials requires a complicated mathematical investigation. 
The situation:is more complicated for waves of finite 
amplitudes and shockwaves owing to the fact that the hydrody- 
namic equation can no longer be linearized. Exact solutions 
can be derived for the free propagation of plane shockwaves. 
The waves of practical interest, however, are spherical 
shockwaves; they cannot be treated without simplifying as- 
sumptions (Bethe, Kirkwood, Brinkley). 
The exact treatment of reflection phenomena of plane 
shockwaves present, considerable difficulities. The case of 
plane shockwaves, reflected by a rigid surface, has been inves- 
tigated by J. von Neumann (Ref. 1) and some calculations concern- 
ing the reflection of plane shockwaves from a free water surface 
have been carried out by Dr. Penney. 
A partial but interesting solution for the reflection of 
the spherical shockwave at a free water surface is given in the 
following. Only shockwaves produced by an underwater explosion 
