Short-Term Barotropic Exchange

The barotropic flow in a channel of small width W and shallow depth H is treated as propagation of long waves in several text books of fluid dynamics, for example. Gill (1982). The pressure is assumed to satisfy the hydrostatic equation 

In case of a channel getting narrower at x = 0, that is, (cW)- (cW)+, the sea level will be higher in the narrowed section after the wave has passed the jump at x = 0. 

Natural channels have gradually varying cross sections. If the change in the cross section is small along a wavelength and every cross section in the channel can cany the transport associated with the incoming wave, the problem can be treated as infinitesimal continuous reflections along the channel. 

In this case, there is virtually no reflection of the incoming wave but the transmitted wave 

The complete description of the reflected and transmitted parts of the incoming wave is 

---