We derive analytic solutions for the forced linear shallow water equation of the following form: <?TeX ∂^2 Y ∂ t^2}-b{∂ ∂ x}(x {∂ Y ∂ x})={∂^2 f ∂ t^2 ?> for x >0, in which Y ( x,t ) denotes an unknown variable, f ( x,t ) a prescribed forcing function and b a positive constant. This equation has been used to describe landslide-generated tsunamis and also long waves induced by moving atmospheric pressure distributions. We discuss particular and general solutions. We then compare our results with numerical solutions of the same equation and with the corresponding solutions of the nonlinear depth-integrated equations and discuss them in terms of landslide-generated tsunamis.
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Liu et al. (2003) studied this question.
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