TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN 

 UNDERWATER ACOUSTICS 



variable-depth bottom structure are readily included in the numerical 

 calculations. Most recently, a randomly fluctuating (in depth, range, 

 and time) component of the sound speed has been implemented without 

 difficulty (Flatte and Tappert, 1974) . 



OUTLINE OF THE PARABOLIC-EQUATION METHOD 



The starting point is the reduced elliptic wave equation for the 



pressure, p, given by Equation (1) in Figure 1, where r is the 



horizontal range, z is the depth, the azimuth angle, k a reference 



o 



wavenumber, and n the index of refraction. 



Equation (2) relates k to the reference sound speed, c , and 



o o 



the angular frequency, w, and n to the variable sound speed, c. 



The basic idea behind the parabolic-equation method is expressed 



in Equation (3) . The pressure is replaced by a slowly varying 



envelope function ijj and an outgoing wave represented by the Hankel 



function of zero order, H . Two approximations are then made: 



o 



(1) tnat one is in the far field of the source (Equation 4) , and 



(2) that the angles with respect to horizontal are small (Equation 5) . 



These lead to a parabolic wave equation for the slowly varying 

 envelope function iji, shown in Equation (6) . The equation is para- 

 bolic because only the first derivative with respect to r occurs, 

 whereas two derivatives with respect to z occur. 



By further neglecting the coupling between azimuthal directions 

 (that is, the derivatives with respect to the azimuthal angle 0) , 

 the two-dimensional parabolic wave Equation (7) is obtained. This is 

 the basis for all the computer models of low-frequency acoustic 

 propagation utilizing the parabolic approximation. 



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