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WAVES Loads
Published in Gerrit J. Schiereck, Introduction to Bed, Bank and Shore Protection, 2017
The shape of waves with increasing steepness in deep water (upper right corner in Figure 7-4) has to be described with more sine components, leading to more complex solutions of the equation of motion. Waves with considerable wave height in shallow water, can be described with the cnoidal wave theory and, on the verge of breaking, with the solitary wave theory. Here only the linear or first-order wave theory will be used, in which the shape of the wave is a simple sine. In appendix 7.7.1, an overview of the formulae in this theory is given. According to Figure 7-4, this theory may only be applied with relatively small waves in deep water (another name for this theory is the small-amplitude wave theory). The approximation of waves by a simple sine function is a crude simplification. For an adequate understanding, however, the linear theory is very useful and attractive, since it gives a simple, but complete description of the pressure and velocity field. Calculated values outside the range of validity can serve as a first indication.
Waves and offshore engineering
Published in P. Novak, A.I.B. Moffat, C. Nalluri, R. Narayanan, Hydraulic Structures, 2017
P. Novak, A.I.B. Moffat, C. Nalluri, R. Narayanan
The steepness of the wave is H/L. If the height of the wave is extremely small compared with the wavelength and the depth of water, the governing equations are linear and the waveform is usually referred to as a linear or Airy wave. Some other waves that a coastal engineer may find as a better approximation of waves on shorelines are shown in Fig. 14.2. These are non-linear waves which occur for large wave heights. In non-linear theory it is usual to classify waves in terms of the wavelength relative to the water depth. In deep water or for short waves, a finite-height wave known as a Stokes wave is applicable (Fig. 14.2(a)). In shallow water or for long waves, the cnoidal wave theory is applied as an approximation. Both Stokes and cnoidal waves (Fig. 14.2(b)) are asymmetrical with respect to the still-water level and have sharp crests and elongated troughs. A solitary wave characterized by a single hump above still water, moving in shallow water, is shown in Fig. 14.2(c). Linear wave theory is widely used in engineering applications because of its simplicity, but, for cases where better evaluation of wave properties is required, complex non-linear wave theories have to be applied. However, if the waves are not large in relation to the depth, or steep enough to break, linear theory is sufficiently accurate.
The KdV Soliton
Published in David S. Ricketts, Donhee Ham, Electrical Solitons, 2011
This periodic waveform is known as a cnoidal wave. Its amplitude, pulse width and velocity are written in terms of the zeros of the cubic function in (2.13). The solution, (2.20), can be further simplified by assuming a zero offset (z2 = 0) as follows: () u(x,t)=z3⋅cn2[z3−z12ξ;m].
Miracles, misconceptions and scotomas in the theory of solitary waves
Published in Geophysical & Astrophysical Fluid Dynamics, 2019
A cnoidal wave is a nonlinear generalisation of a cosine wave. Both cnoidal waves and cosine waves depend on a single phase variable of the form where c is the phase speed. The phase variable defines a “moving coordinate” such that a wave steadily translating at the speed c is stationary in the moving frame; that is to say, in the coordinates , the wave is a function of X only. The infinitesimal amplitude cosine wave is a single Fourier component of the form for some constants A and k.The cnoidal wave is a finite amplitude spatially-periodic solution that is a Fourier series in the phase variable: A fuller discussion is given in section 5.7, Boyd (1982, 1998c), and Remoissent (1999).
Allowable span length of submarine pipeline in shallow water
Published in Marine Georesources & Geotechnology, 2018
Changjing Fu, Pingyi Wang, Tianlong Zhao, Guoying Li
At present, the linear wave theory is frequently used for calculating the span length of submarine pipelines. A wide array of prior research has shown that the calculation error of the linear wave theory is within allowable ranges in deep water (the relationship between wave height H and wavelength L is H/L→0), but it causes large errors in shallow water (Fu, Li, and Zhao 2015). In summary, it is necessary to take full account of the characteristics of wave motion when studying pipelines that span shallow waters. For practical applications of wave theory, Le Mehaute (1976) stated that the linear wave theory was applicable for deep water, but the cnoidal wave theory is more suitable for shallow water conditions.