6.—Smooth Fronted Waves in the Shallow Water Approximation

Abstract

SynopsisThis paper examines the mathematical problem of the propagation of a smooth fronted wavein the context of shallow water theory. Here, a smooth fronted wave will be taken to be one in which the surface slope is continuous across a line in the free-surface, while the second derivative of the surface slope is discontinuous across that same line. This discontinuity line in the surface then plays the role of the wavefront. After establishing that such wavefronts propagate along the characteristics, and deriving the appropriate transport equations, the explicit form is found for the acceleration with respect to distance of the horizontal component of the water velocity of the surface immediately behind the wavefront as a function of position and seabed profile when the wave propagates into still water. The result is then used to prove that in this approximation such a wave cannever break immediately behind the wavefront before the shore line is reached.