Shallow Water Waves in Canals of Variable Section

New theoretical solutions are derived for shallow water small amplitude waves in rectangular cross-sectional channels that have their depth and their width varying in the form of a polynomial. These solutions can closely approximate the long period seiches in shallow water canals, harbors, or lakes that are either open or closed at either end. The solutions are obtained by using the well-known linearized shallow water theory, shown to be best suited for studying the usual seiche in a lake or a harbor because of the fact that the fundamental standing wave that can be formed has a relatively small amplitude in comparison with its long wave length. The fundamental seiche usually has a long period and corresponds to the free oscillation of all of the contained water following a sudden impulse, such as that produced by an abrupt change in the atmospheric pressure or by an earthquake, or by resonance with tidal waves. These solutions may also be used to study the amplification of sea waves as they enter a shallow water canal or harbor. The new solutions that are presented allow a greater variation in the shape of the shallow body of water because they are applicable to any canals, harbors, or lakes that can have their depth and width approximated by polynomials. It is shown that all of the previously available theoretical solutions for varying rectangular cross sections are special cases of this new family of solutions.