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term exp / to / vy) - (One can easily demonstrate this by 
showing that the Cauchy - Riemann conditions are not satisfied.) 
For this reason, it is not possible to evaluate an expression fox 
fF by means of the usual methods of contour integration, 
which are only applicable to analytic functions. It is felt that 
whatever additional information might be gained would probably 
not be sufficiently valuable to justify further laborious 
manipulation of the complex integrals involved. 
Reflection of spherical waves. 
Sel The preceding discussions (section 3 in particular) have 
been based on certain simplifying assumptions which may be re- 
stated briefly as follows: 
1. The ray approximation was used to calculate the 
paths along which energy passed from the source to the receiver. 
These rays included direct transmission through the water and 
various orders of reflection from the surface and bottom. 
2 The shape of the individual reflections was 
obtained by using the plane wave theory of Section 2. 
3e The effect of the point source was allowed for 
by assuming that the amplitude of the wave was subject to normal 
spherical attenuation ( ~V VR ). No other effects due to prop- 
agation from a point source were taken into account. 
As will be seen later, these approximations represent 
the first order term in an asymptotic expansion of the spherical 
wave solution. Before proceeding to this asymptotic expansion, 
