KUTSCHALE: LOW-FREQUENCY PROPAGATION IN THE ICE-COVERED ARCTIC OCEAN 



Figure 17 shows the form of the integral solution of the wave 

 equation derived from a harmonic point source in a multilayered, 

 interbedded liquid-solid half space (Kutschale, 1970) . Solid layers 

 represent the ice and they may represent bottom layering. To take 

 account of reflection loss of waves boioncing off the ice, the surface 

 reflection coefficient is modified to yield a specific dB loss per 

 bounce at each ice contact. For simplicity, the loss is taken as 

 independent of the grazing angle, but an angle dependence could be 

 included. The loss per bounce (in dB) appears to increase with fre- 

 quency in a regular manner. Below about 10 Hz it appears to be 

 essentially zero, except possibly in the areas of roughest ice. Two 

 convenient ways to evaluate the integral solution are: 



• Contour integration in the complex k-plane yielding a 

 branch-line integral plus a stun of normal modes. 



The normal modes predominate at long ranges. 



• Direct integration of the integral solution by the 

 Fast Field Program (FFP) technique introduced by Marsh 

 and DiNapoli (see, for example, DiNapoli, 1971). 



In the FFP technique the integral solution of the wave equa- 

 tion is evaluated rapidly as a function of range by direct numerical 

 integration employing the Fast Fourier Transform (FFT) algorithm. 

 The prescription is shown on the bottom line of Figure 17. Singu- 

 larities in the integrand corresponding to the normal-mode poles are 

 removed from the axis of integration by including attenuation co- 

 efficients for compressional and shear waves in each layer. In a 

 liquid layer, of course, both the shear velocity and corresponding 

 attenuation coefficient are zero. The effects of this attenuation 

 can be removed at the final stage of the loss computations. A con- 

 venient starting point for the FFP is the integral solution of the 

 wave equation derived by the Thomson-Haskell matrix method (Thomson, 

 1950; Haskell, 1953) for propagation from point harmonic sources in 

 a multilayered, liquid-solid half space (Kutschale, 1973). 



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