SECT. 4] SOUND IN THE SEA 483 



reflection, although total, occurs with a phase change 2x given by (18); a 

 reasoning similar to that used in arriving at (21) then gives 



2A;^cos^-2x = (2w-l)7r 



Ceos9=(™-i+^)^. (22) 



Since x is itself a function of d, it is not possible to obtain a simple formula 

 for the angles corresponding to the various modes, but if x/tt and (2^/A) cos d 

 be plotted against 6, the several values of d which satisfy (22) can be determined 

 graphically, and those not less than the critical angle correspond to the modes 

 which are propagated. 



Since neither the sea-bed nor the surface is perfectly smooth and horizontal, 

 there is some scattering of the energy, and the fronts of the incident and 

 reflected waves do not make equal angles with the vertical. There is some loss 

 due to this effect but multiple scattering also transfers energy from one mode 

 to another. Even without this transfer the modes are not all equally attenu- 

 ated because the small loss at the surface decreases as the angle of incidence 

 increases. 



Hitherto we have considered a single frequency ; if, however, the source of 

 sound has a continuous frequency spectrum, as, for instance, noise of ships or 

 the noise of an explosion, waves are propagated at all angles of incidence 

 greater than the critical angle. Also, as absorption of sound increases rapidly 

 with frequency, at long distances from the source, the spectrum is more and 

 more shifted towards the low-frequency end. 



To all the effects mentioned above the cylindrical spread of energy which 

 takes place in shallow water adds its own reduction of intensity with range. 



The simple treatment above applies to plane waves only and can be extended 

 to the calculation of intensity in the vertical. The problem is more complicated 

 when energy is radiated from a point transmitter at some depth zi below the 

 surface to a receiver at depth 22. Pekeris (1948) and Ide et at. (1954) show that 

 the expression for the velocity potential then contains sinusoidal factors 

 depending on zift, and 22/^, and another factor giving the relative strength of 

 the various modes. 



The expression for the phase shows that the propagation constant in the 

 horizontal direction y is, k sin 6. The phase velocity, F, or velocity with which 

 phase is propagated along y, is thus c cosec d. When instead of a single fre- 

 quency there is a continuous frequency spectrum, for instance that of a sonar 

 pulse, 6 changes with oj and so does the propagation constant. In this case an 

 important characteristic of propagation is the "group" velocity, U, which is the 

 frequency with which the spectrum is propagated; it is the value of dyjdt 

 corresponding to a stationary value of the phase and is given here approxi- 

 mately by 



jj = ^ ^^/^^ (23) 



sin d{dojldd) -f CO cos ^ 



