Topic

# Wave shoaling

About: Wave shoaling is a research topic. Over the lifetime, 4276 publications have been published within this topic receiving 103039 citations.

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29 Jan 1978TL;DR: In this article, a model was developed for the prediction of the dissipation of energy in random waves breaking on a beach and the probability of occurrence of breaking waves was estimated on the basis of a wave height distribution with an upper cut-off which in shallow water is determined mainly by the local depth.

Abstract: A description is given of a model developed for the prediction of the dissipation of energy in random waves breaking on a beach The dissipation rate per breaking wave is estimated from that in a bore of corresponding height, while the probability of occurrence of breaking waves is estimated on the basis of a wave height distribution with an upper cut-off which in shallow water is determined mainly by the local depth A comparison with measurements of wave height decay and set-up, on a plane beach and on a beach with a bar-trough profile, indicates that the model is capable of predicting qualitatively and quantitatively all the main features of the data

1,463 citations

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TL;DR: In this paper, an analytical theory is presented to describe the combined motion of waves and currents in the vicinity of a rough bottom and the associated boundary shear stress, and the resulting linearized governing equations are solved for the wave and current kinematics both inside and outside the wave boundary layer region.

Abstract: An analytical theory is presented to describe the combined motion of waves and currents in the vicinity of a rough bottom and the associated boundary shear stress. Characteristic shear velocities are defined for the respective wave and current boundary layer regions by using a combined wave-current friction factor, and turbulent closure is accomplished by employing a time invariant turbulent eddy viscosity model which increases linearly with height above the seabed. The resulting linearized governing equations are solved for the wave and current kinematics both inside and outside the wave boundary layer region. For the current velocity profile above the wave boundary layer, the concept of an apparent bottom roughness is introduced, which depends on the physical bottom roughness as well as the wave characteristics. The net result is that the current above the wave boundary layer feels a larger resistance due to the presence of the wave. The wave-current friction factor and the apparent roughness are found as a function of the velocity of the current relative to the wave orbital velocity, the relative bottom roughness, and the angle between the currents and the waves. In the limiting case of a pure wave motion the predictions of the velocity profile and wave friction factor from the theory have been shown to give good agreement with experimental results. The reasonable nature of the concept of the apparent bottom roughness is demonstrated by comparison with field observations of very large bottom roughnesses by previous investigators. The implications of the behavior predicted by the model on sediment transport and shelf circulation models are discussed.

1,412 citations

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TL;DR: In this article, the authors show that Rademacher's theorem for functions with values in Banach spaces implies that the function m is almost everywhere differentiable on (0, T) with dm dt (t)=vt~(t, x) -m( t ) <<.

Abstract: WAVE BREAKING FOR NONLINEAR NONLOCAL SHALLOW WATER EQUATIONS 233 THEOREM 2.1. Let T>O and vE C 1 ([0, T); H 2(R)). Then for every t~ [0, T) there exists at least one point ~(t)ER with ,~(t) := in~ Ivy(t, x)] = ~ ( t , ~(t)), and the function m is almost everywhere differentiable on (0, T) with dm dt (t)=vt~(t,~(t)) a.e. on (O,T). Proof. Let c>0 stand for a generic constant. Fix te[0, T) and define m(t):=infxcR[v~(t,x)]. If m(t))O we have that v(t , . ) is nondecreasing on R and therefore v(t,. ) 0 (recall v(t,. )cL2(R)) , so that we may assume re(t)<0. Since vx(t,. ) c H I ( R ) we see that limlxl~ ~ Vx(t, x)=0 so that there exists at least a ~(t) e R with re(t) =v~(t, ~(t)). Let now s, tC [0, T) be fixed. If re(t) <~rn(s) we have 0 < re(s) .~(t) = i n f [~x (~, x ) ] ~ ( t , ~(t)) <. ~=(~, ~(t)) -~x(t, ~(t)), and by the Sobolev embedding HI(R) C L ~ ( R ) we conclude that Im(8)-.~(t)l ~< Ivx(t)-v~(s)lL~(~) < c Iv~(t)-v~(8)l.l(R). Hence the mean-value theorem for functions with values in Banach spaces-Hi(R) in the present case--yields (see [12]) jm(t)-m(s)l<~clt-s j m a x [IVt~(T)JHI(R)], t, se[O,T). O~T~max{s,t} Since vt~cC([O,T), Hi(R)) , we see that m is locally Lipschitz on [0, T) and therefore Rademacher's theorem (cf. [14]) implies that m is almost everywhere differentiable on (0,T). Fix tC(0, T). We have that v~(t+h)-vx(t)h vt~(t) Hl(R) ---~0 as h--*O, and therefore vx( t+h ,y ) -vx ( t , y ) sup vtx(t,y) --~0 as h---~O, (2.1) ycl~ h in view of the continuous embedding H 1 ( R ) c L ~ (R). 234 A. C O N S T A N T I N AND J. E S C H E R By the definition of m, m(t+h) = v~(t+h, ((t+h)) <. v~(t+h, ((t)). Consequently, given h>0 , we obtain m( t+h) -m( t ) <<. h Letting h--~O + and using (2.1), we find lim sup m(t+h) -m( t ) h~_~0 + h

1,361 citations