1492 
order reflections from both boundaries will follow the primary 
Wave. The case of interest here is one in which the lower 
boundary is a plane, horizontal sea bed as treated above, and 
the upper boundary is the free surface of the water. Since the 
sound velocity in air is lower than that in water, the surface 
reflection does not involve any critical angles or phase shifts 
other than the usual acoustic inversion at the free surface. 
(Finite amplitude effects introduce their own complications, 
which are not considered in this treatment.) 
The various higher order reflections will in general 
undergo more than one reflection from the sea bed. For example, 
the reflection designated by the notation M = 1, N # 2 undergoes 
the following sequence before reaching the point of observation: 
reflection from the bottom, reflection of this wave from the 
surface, and again a reflection from the bottom. The reflections 
all take place at the same angle of incidence, and the resulting 
pressure wave is given by equation (19). 
Equation (19) makes apparent the cumulative nature of 
the successive phase shifts. For example in the case of reflection 
M21, N= 2, cited above, equation (19) becomes 
Per = =f (t) cos 4€ - F(t) sin Fé (23) 
If the geometry and acoustic properties are such that 
& = 45°, equation (23) reduces to 
Paar @) = vig (¢/ 2 (24) 
= 16— 
