BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 



227 



M(2) ARBITRARY SCALE 







0.5 



M(J) ARBITRARY SCALE 

 „ 



3.01 



FIRST MODE -^ = 0.5 



M(Z) ARBITRARY SCALE 

 



MINIMUM FREQUENCY 

 FIRST MODE 

 M(Z) ARBITRARY SCALE 

 O 



2.0 



.th 



THIRD MODELS 



2.0 



/ = Frequency. 

 h = Depth of water. 

 c = Velocity of sound in water. 

 z = Distance below surface of water. 

 M(z) = Pressure amplitude at depth 2. 



Figure 28. Variation of pressure with depth along a 

 vertical line for various normal modes. Velocity of 

 sound in bottom assumed to be 1.5 X c. Density of bot- 

 tom assumed 2 times density of water. 



•'SECOND MODE," and so on, in Figures 27 and 28. 

 A noteworthy fact is that for modes of any given 

 order there is a minimum frequency below which no 

 value of X can be found which will satisfy the 

 boundary conditions. At this frequency the quantity 

 V, which is inversely proportional to the depth of 

 penetration of the disturbance into the bottom, goes 

 to zero; and the pressure distribution takes a form 

 such as that shown in Figure 28B. As the mathemati- 

 cally inclined reader can verify for himself from equa- 

 tion (24), the minimum frequency for the first mode 

 has a half-period equal to the interval which ray 

 theory would predict between the arrivals of types 

 Ig, Ilg, lllg, etc. , of Flgure 26. This half-period is 

 given by 



T l/l 



'2 = ''h^ 



Since these ray arrivals are alternately positive and 

 negative, the period of the disturbance given by ray 

 theory is the same as that for the minimum frequency. 

 It is also noteworthy that the minimmn frequency 

 for the I'th mode is (2i/- 1) times its value for the first 

 mode. This has the very important consequence that 

 any simple harmonic disturbance of low frequency 

 which is propagated over a large horizontal range can 

 be represented by a superposition of a finite number 

 of normal modes of low order. 



Let us now consider the velocity of propagation of a 

 disturbance which consists of a superposition of nor- 

 mal modes of a given order but distributed over a 

 narrow range of frequencies. It is easy to show that 

 such a disturbance, considered as a function of hori- 

 zontal distance x or of time t, will form a wave train. 

 For each component normal mode has a phase factor 

 proportional to e^-'Uxix-m, At any given time t 

 there will be some value of x for which most of the 

 component modes are approximately in phase; in the 

 neighborhood of this value of x the pressure disturb- 

 ance will therefore be large. If z differs very widely 

 from this value, on the other hand, the phases of all 

 the component normal modes will be rather randomly 

 distributed since the different modes have slightly 

 different wavelengths X; for such values of x the 

 pressure disturbance will be small. If we watch the 

 motion of the wave train in the course of time, we 

 shall find that the region of large amplitude moves 

 with a certain velocity, commonly called the "group 

 velocity." Now, if the center of the wave train is to 

 be near xi at time t\, and near x^ at time h, the phase 

 change {xi — Xi/X) — /(^ — ti) must be very nearly 

 the same for all the different normal modes contained 

 in the wave train, in order that they may continue 

 to reinforce one another. This implies, in the limit 

 where only a very narrow range of frequencies is in- 

 volved, 



dfaja — xi „ ~\ 



or, if F is the group velocity, 



(X2 - Xi) dj 



V = 



{k - k) 



<d 



(26) 



c? 



(25) 



Now the phase velocity of any single-frequency 

 component, defined as the speed of advance of a 

 point having a given constant value of the phase 

 2t{x/\ — ft), is equal to X/. If this quantity were a 

 constant independent of frequency, as is the case for 



