A bound on oscillations in an unsteady undular bore

The tidal bore is a well-known phenomenon in fluid mechanics, having been observed in many rivers around the world. Two of the better known rivers where bores regularly appear are the Severn River in England [1] and the Qiantang River in China [2]. The usual circumstances in which a bore can appear develop when a tidal swell causes a difference in surface elevation in the mouth of a river or further upstream. In this case, long waves start to propagate upstream, and if the conditions are favourable, the main front steepens, and a nearly steady transition profile develops. Through field measurements and experiments [3], [4] and [5], it has been found that bores appear in two types. If the ratio of the difference in surface elevation between the two uniform states to the undisturbed depth is greater than approximately 0.75, a so-called turbulent bore may be seen. If this ratio is smaller than 0.28, then the bore tends to exhibit undular character, in other words, the bore will feature oscillations in the downstream part. If the ratio is between 0.28 and 0.75, the bore will be turbulent, but also feature some oscillations. Quite commonly, the bore is studied in the context of shallow-water theory. In this case, an analysis using conservation of mass and momentum shows that energy must be lost at the front of the bore. In the case of an undular bore, the energy is thought to be disseminated through an increasing number of oscillations