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196 Lecture 11 
Using a value sin@=1 for convenience, we have plotted F against b for fixed H 
in Fig. 11.1, and F against f for fixed H in Fig. 11.2. Figure 11.1 permits a 
comparison of back scattering with specular reflection. 
-4, CONCLUSIONS 
Figure 11.2 shows the comparative insensitivity of reverberation to wave 
height and frequency, for the large value of sin@ which corresponds to grazing 
incidence of the incident radiation. Both of these phenomena have been well 
known experimentally for some years. We are entitled, therefore, to have some 
confidence in the theory of sea surface scattering. In addition, the essential 
features of the sea surface spectrum are contained in I. We, therefore, have 
some additional confirmation that the Neumann—Pierson spectrum is correct. 
REFERENCES 
1. H.W. Marsh, "Exact Solution of Wave Scattering by Irregular Surfaces,” J. Acoust. Soc. Am., Vol. 33, 
330-333 (1961). 
2. H.W. Marsh, M. Schulkin, and S.G. Kneale, "Scattering of Underwater Sound by the Sea Surface,” J. 
Acoust. Soc. Am., Vol. 33, pp. 334-340 (1961). 
DISCUSSION 
MR. A.G.D. WATSON thought the work most important and valuable and he 
was sorry the lecture was so condensed. In his talk, Dr. Marsh had raised many 
problems in mathematics, physics, oceanography, and underwater sound, but he 
was only going to comment on the mathematical aspect. He doubted the validity 
of the lecturer's use of Wiener's theory in the resolution of the reflected field 
into plane waves, but the final results are unaffected as Dr. Marsh does not 
employ Wiener's formula in his calculations. The justification of the resolution 
into plane waves could be, perhaps, carried out by considering a Fourier analysis 
of the field over a plane on the positive side of the scattering surface. It is not, 
however, necessary to use Wiener's formula for this purpose, if we follow Dr. 
Marsh and employ delta functions. Mr. Watson, then, spoke of the second step 
in the development of the coefficients of the power series in sigma which was 
also followed, without justification, by Rayleigh and others. Thirdly, there is the 
evaluation of the coefficients in terms of the correlation function of the surface 
and this is where, in Mr. Watson's opinion, Dr. Marsh has made a great con- 
tribution. He had, himself, made a similar approach and obtained the same 
result. 
Mr. Watson said that it was difficult to reconcile Dr. Marsh's formula for 
his function A with results of other workers. Using Dr. Marsh's notation and 
following a crude argument, assuming a Gaussian distribution of heights on the 
rough surface, one obtains a specular reflection coefficient of exp (-4y a2) and 
this agrees, to terms in o”, with Rayleigh's result, for a sinusoidal surface, in 
terms of Bessel functions. Finally, Mr. Watson asked Dr. Marsh if he would 
indicate how his formula is reconciled with the conservation of energy, when 
the film is integrated over all angles. 
DR. MARSH: Wiener's method is formally applicable to any functions which 
are invariant under time translation. It is formally applicable to the resolution 
