■o -20 — 



B. Approximations and ranges of validity 



The wave equation for acoutic pressure p{r, t) is 



0.2 0.3 



k — cycles/km 



FIG. 3. Internal-wave spectrum as a function of mode number 

 ; and horizontal wavenumber k. Although large mode numbers 

 contribute very little to the overall spectrum, they are crucial 

 to understanding acoustic effects, since their vertical struc- 

 ture allows them to act as a scatterer of acoustic energy more 

 readily than the relatively structureless low modes. 



.(..^), 



where 



5c 

 c 



= €[e-''-(l-7,)]+5£i 



D. Numerical realization of the Internal-wave model 



M. Milder's program (ZMODE)*^ was modifed to 

 numerically generate the eigenfunctions W{j, k, z) and 

 frequencies u>(j, k). Modes with 1 sjs 24 were in- 

 cluded. Values of k ranged from - 0. 5 to 0. 5 cycle Am 

 in 254 equal steps. The 6096 different A(j, k) were 

 generated according to a Rayleigh probability distribu- 

 tion in amplitude, and variance given by the spectrum 

 described in Part B. The phase angle of each A{j, k) 

 is randomly generated in the region to 27r. Using the 

 above equations, our code can then generate the sound 

 speed at any point in space and time. 



II. PROPAGATION OF ACOUSTIC SIGNALS USING 

 THE PARABOLIC EQUATION METHOD 



A. Introduction 



The parabolic equation method was originally devel- 

 oped by Leontovich and Fok in 1946 to study long- 

 range propagation of radio waves in the troposphere.'' 

 This method was introduced into the field of underwater 

 acoustics by Tappert in 1972 and a computer program 

 based on this method was developed by Tappert and 

 Hardin to solve acoustic propagation problems of inter- 

 est to the Navy. "•'= 



V'p. 



f2 8/2 



Our knowledge that c varies from a constant only by 

 very small amounts, and that the variations are slow 

 compared with the acoustic frequency allows us to use 

 the following expression in cylindrical coordinates for 

 the pressure (far from the source): 



gUkQT-ut) 



p(r,t) = i',{y.z,0) — ^ , 



where the reduced wave function * is labeled by the 

 time t, because the 6c/c structure of the ocean is dif- 

 ferent for different times. Substituting in the full equa- 



FIG. 4. Sound-speed profiles induced by a full spectrum of 

 internal waves at a particular instant of time. Let r be the 

 range from some arbitrary point, then (a) r = 0, (b) r=14 km, 

 and (c) r = 28 km. 



204 



