Shallow water wave

In most solids, the sound speed is an increasing function of pressure, and it is that property that causes a compression wave to steepen into a shock. The situation is similar to a shallow water wave, whose velocity increases with depth. As the wave approaches shore, a small wavelet on the trailing, deeper part of the wave moves faster, and eventually overtakes similar disturbances on the front part of the wave. Eventually, the water wave becomes gravitationally unstable and overturns. 

For a shock wave in a solid, the analogous picture is shown schematically in Fig. 2.6(a). Consider a compression wave on which there are two small compressional disturbances, one ahead of the other. The first wavelet moves with respect to its surroundings at the local sound speed of Aj, which depends on the pressure at that point. Since the medium through which it is propagating is moving with respect to stationary coordinates at a particle velocity Uj, the actual speed of the disturbance in the laboratory reference frame is Aj - -Ui- Similarly, the second disturbance advances at fl2 + 2- Thus the second wavelet overtakes the first, since both sound speed and particle velocity increase with pressure. Just as a shallow water wave steepens, so does the shock. Unlike the surf, a shock wave is not subject to gravitational instabilities, so there is no way for it to overturn. 

Shallow water waves have large wavelengths compared to the water depth such that kH 1. Then the dispersion relation (2.26) is approximated by... 

That means that shallow water waves are nondispersive, that is, an initial disturbance of the sea surface consisting of an ensemble of shallow water waves keeps its form along its path of propagation. 

The particle velocity associated with the shallow water wave is elliptic ally with the major axis aligned parallel to the bottom and the minor axis perpendicular to it. The amplimde of the particle velocity decays with depth but remains finitely by close to the bottom where the minor axis of the particle motion approaches to zero. This implies that particle motions of shallow water waves are quite capable of resuspending sediments as well as to develop a turbulent bottom boundary layer. 

A solvent dissolution, a vapor adsorption, any kind of surface-active substance exchange between the surface and the adjacent subphase, or heating makes the surface tension locally vary, thus generating Marangoni stresses and convection. Then, gravitocapillary waves (wavelength X and amplitude q) excited and sustained by the Marangoni effect in the shallow water waves approximation can be described by the equation ... 

The time required for a shallow-water wave to travel from Newport to New Haven is about 2.5 hr, from New London to New Haven, about 1.3 hr. Since the greatest Ah occurs earlier at New Haven than at New London, the rise in water level in LIS is not a surge advancing as a progressive wave from the sea. Evidence of a storm surge in LIS is found only for very intense, rapidly moving storm systems, such as the 1938 hurricane (Redfield and Miller, 1957). 

From the selected wind field, the deep water, transition water and shallow water waves should be evaluated. In simplified methods for such an evaluation it is assumed that wind is unidirectional these methods are based on semi-empirical relationships and use as input the fetch, wind speed and wind duration... 

The waves generated directly by the action of the wind on transition water and shallow water are also evaluated independently from deep water waves. After deep water waves have travelled into shallow water, they dissipate part of their energy and they may be reduced to such a height as not to represent the critical wave at the site. On the basis of an appropriate aUgnment of the critical fetch to the nuclear power plant site both deep water and shallow water waves should therefore be evaluated. 

The near shore waves critical for the design of the plant should be identified by comparing the histories of various heights of incident deep water, transition water and shallow water waves and hmiting breaking waves, with account taken... 

With the seaward boundary condition now established, for the case of shallow water wave-breaking and the consideration of depth limited breaking across the surf zone, the wave setup is... 

J. W. van der Meer, Extreme shallow water wave conditions. Report H198. Delft Hydraulics Laboratory, The Netherlands (1990). 

The surface slopes MEM method (i.e., A-r MEM) performs about as well as the Tf—u—v MEM over a wide range of wave conditions and water depths. Tests conducted with bimodal seas, nonUnear shallow water waves, and even breaking waves, indicate that both methods perform reasonably well, even under these challenging conditions. The surface slopes MEM offers a number of operational advantages, such as calibration compared with alternative methods, and is a viable alternative for accurate and reliable directional wave analysis in the laboratory. However, the optimal array radius being dependent on the dominant wavelength could be perceived as a disadvantage of the surface slopes method. 

J. S. Readshaw, W. F. Baird and E. P. D. Mansard, Shallow water wave generation An engineering perspective, Proc. Sem. Wave Anal. Generation, 22nd lAHR Conf, Lausanne, Switzerland (1987), pp. 397-410. 

Vol. 15 I. Kinnmark The Shallow Water Wave Equation Formulation,... 

Craig measured the wave amplitude as a function of time for the first few seconds at a distance of 4 meters from a 2.54 cm diameter PBX-9404 explosive sphere initiated at its center in 3 meters of water. He included mass markers in the water. Mass markers located 1 meter below the water surface and markers located 0.5 meter below the surface and 1 meter from the explosive showed no appreciable movement compared with those located nearer the surface or explosive charge. These results showed that the wave formed was not a shallow water wave. 

This result is identical to that given by Basset (page 253. Vol 2. 1888) thus we know that it was available to chemical engineers in 1888. Although Basset took the time to derive this form of the macroscopic mechanical energy balance. I am not able to find the macroscopic momentum balance In his two volume treatise. However, the elements of the macroscopic momentum balance are evident in Basset s treatment of shallow water waves, which is essentially identical to that given by Lamb (1879). 

The measured swirl period could be adequately approximated by (5.1) and divided into two categories, the limit being at Hi /D 0.3. The swirl period agrees with the period of fundamental harmonics of the rotary sloshing (t = 1) for H] /D > 0.3 within a scatter of -20 to 0%, whereas agreement with the period of the second harmonics i = 2) for H, /D < 0.3 is within a scatter of 0 to +25%. This limit seems to be associated with the limit between the shallow water wave and the deep water wave in a cylindrical vessel [16,17] given by Hi /D = 0.3. In this case, a wave affected significantly by the bottom wall of the vessel is termed shallow... 

Waves are normally recorded by wave buoys in offshore waters (Figure B.32). In shallow water, waves can also be recorded by an ADCP (by translating wave orbital velocities to wave parameters). 

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