482 VIGOUKEUX AND HERSEY [CHAP. 12 



Corresponding to (17) there is a "bound wave" 



ptip = 2 cos ;(;e*^i[(ci/c)- sin^ e-ljViz (I9) 



of phase ojt — ky sin d + x- Owing to the exponential term in (19) this wave is 

 rapidly attenuated with depth below the sea-bed, along which it propagates 

 with a velocity c cosec 6 equal to Ci at the critical angle and less than ci for all 

 greater angles of incidence. There is no transmission of energy into the lower 

 medium but a diffraction across the interface associated with and bound to 

 the reflected wave (Officer, 1958). 



For all angles of incidence less than sin~i (c/ci), however, (15) is real and 

 gives for the ratio of reflected to incident energy 



Apicot di j [pi cot 01^ 2 



/ y p cot 6 I \p cot 6 



(20) 



On the provisional assumption made above that the bottom of the sea is 

 homogeneous and infinite, the proportion of energy lost on reflection is the 

 second term of (20) taken with the positive sign. 



F. Propagation in Shallow Water 



In shallow water the sound wave is continuously reflected from the surface 

 and the bottom, and there is rapid attenuation for angles of incidence less than 

 the critical angle even if the second term of (20) is small, so that propagation is 

 impossible even at moderate range. But for angles of incidence greater than the 

 critical angle, reflection is total and sound is propagated to great ranges. 

 Propagation at any one frequency is, however, restricted to a few discrete 

 angles of incidence, as can be seen by considering the phases of the two waves. 

 If, for simplicity, we flrst assume a perfectly rigid sea-bed (pc= oo) and a per- 

 fectly "soft" air above the surface {pc = 0), the wave p incident on the sea-bed 

 (Fig. 1) has phase ojt -\- kz cos d — ky sin 6 and the wave pr reflected from it has, 

 since x of (18) vanishes, phase cut — kz cos d — ky sin 6. Since at the surface 

 the pressure must vanish, these two waves differ in phase by n or more generally 

 (271— 1)77- if w is a positive integer ; thus, if the depth be ^, 



2A;^cos 6 = {2n-l)TT 



^cos^ = (n-i)A/2. (21) 



This equation shows that n, called the "order" of the "mode" of propagation , 

 cannot have all integral values, since it cannot exceed | -1- 2^/A. In fact if the 

 depth t, is less than A/4 the waves are not propagated. 



In practice the air above the surface is a sufficiently good approximation to 

 a perfectly soft medium, although there is attenuation due to the small amount 

 of energy which finds its way into it, but the sea-bed is not infinitely rigid and 



