STICKLER: NORMAL MODES IN OCEAN ACOUSTICS 



representation is a cylindrical superposition of waves, and the inte- 

 gration variable corresponds to the radial wave number. 



The integration contour can be taken along the real k-axis if 



loss is introduced; if not, it can be taken just below the real axis. 



The transform P(z,z ,k) is determined by the sound-speed and density 



o 



profiles, pressure-release condition at the ocean surface, continuity 



conditions at discontinuities in the acoustic properties, and a 



radiation condition. The determination of this function and the 



evaluation of this integral constitute the central practical problem 



in a normal-mode expansion. In a liquid region (with no shear) 



P(z,z ,k) satisfies 

 o 



( — -) 



\ p(z) dz J 



p(z) ^ ( "4-, ^ ) + (k^(z) - k^)P = :ri 6(z - z ) (2) 

 dz \ o(z) dz / 2iT o 



where p (z) is the density and k(z) = a3/c(z). The presence of the 

 2Tr factor is due to the cylindrical symmetry. 



A physical interpretation of the integration contour can be 

 made in terms of incidence angles as shown in Figure 2. The polar 

 transformation shown makes it possible to describe the pressure 

 field as an integration over real and complex incidence angles or, 

 alternatively, in terms of homogeneous and nonhomogeneous "plane" 

 waves. The integration over (o, k-|^) in wavenumber space then 

 corresponds to "real" incidence angles and the integration over 

 (k-j , °°) to nonreal incidence angles. 



The discussion of normal*mode expansions is simplified by 

 transforming Equation 1 so that the integration contour is infinite 

 and the standing -wave component J (kr) is replaced by an outward- 

 going wave component H (kr) , the Hankel function. The repre- 



o 



sentation is given in Equation 3 and the integration contour is 

 shown in Figure 3. This technique is used by Brekhovskikh (1960). 



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