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The transformation of long waves—such as tsunamis and storm surges—evolving over a continental shelf is investigated. We approach this problem numerically using a pseudo-spectral method for a higher-order Euler formulation. Solitary waves and undular bores are considered as models for the long waves. The bathymetry possesses a periodic ridge-valley configuration in the alongshore direction which facilitates a means by which we may observe the effects of refraction, diffraction, focusing, and shoaling. In this scenario, the effects of wave focusing and shoaling enhance the wave amplitude and phase speed in the shallower regions of the domain. The combination of these effects leads to a wave pattern that is atypical of the usual behavior seen in linear shallow-water theory. A reciprocating behavior in the amplitude on the ridge and valley for the wave propagation causes wave radiation behind the leading waves, hence, the amplitude approaches a smaller asymptotic value than the equivalent case with no lateral variation. For an undular bore propagating in one dimension over a smooth step, we find that the water surface resolves into five different mean water levels. The physical mechanisms for this phenomenon are provided. 

Stability of a solitary wave disturbed by a submerged flat sill is investigated experimentally. For sills narrow compared with the solitary wave, the transmitted waves are found to be unaffected in waveform and amplitude. A wider sill disturbs the solitary wave resulting in the formation of a dispersive wavetrain following the transmitted wave. In some cases, the wave amplitude recovers, despite being perturbed, to the state of an unobstructed solitary-wave state at a certain distance beyond the sill. Wider sills cause wave breaking that occurs over the sill or, in some cases, after the wave passes through the sill. Details of waveform transformation leading toward the breaking and subsequent energy dissipation are discussed. 

Evolution of a solitary wave travelling along a submerged sill is studied. The disturbance from the sill creates a phase lag along the wave crest between the ambient water depth and the shallower depth over the sill. This phase lag causes wave diffraction between the different parts of the wave, which induces radiating waves off the edge of the sill. The radiating waves act as an outlet for wave energy, resulting in significant and continual amplitude reduction of the solitary wave. Findings from laboratory experiments are confirmed numerically by simulating a much longer propagation distance with different sill breadths. When the sill breadth is narrow, the solitary wave slowly attenuates by wave radiation, maintaining a quasi-steady wave pattern. This is not the case for a broader sill. The resulting phase lag on the sill continually changes the wave pattern and the attenuation rate is substantially greater than the rate for the case of the narrow sill. The significant energy radiation together with the continual change in the wave formation eventually leads to the complete annihilation of the solitary wave in a wave tank. We also report a wave-breaking process along the sill observed in laboratory experiments. This breaking is induced when the wave amplitude on the sill is smaller than the maximum amplitude of a solitary wave in a uniform depth. Also found is the wake-like formation of gravity–capillary waves behind the breaking crest forming on the sill. Other features associated with the breaking are presented.