Another technique to compute bottom attenuation, which works for nonisovelocity cases is to assume a thin isovelocity layer just above the bottom. In this layer, normal modes are represented by up- and downgoing waves with a reflection coefficient which includes the bottom attenuation term as in the problem above. The field and its derivative must be continuous in the water column. Take the limit of zero layer thickness to obtain the ratio of the normal mode to its derivative in terms of the reflection coefficient. Assume the modes and wavenumbers are complex and write down the eigenvalue equation and its complex conjugate.Multiply these equations by their complex conjugate mode function, respectively. Taking the difference of these two equations and integrating by parts will yield a relation connecting the
imaginary part of the wavenumber with the normal mode and its derivative. Use this
method to derive an expression for the modal attenuation coefficient.
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