Why do waves increase in amplitude when approaching a beach?

[hanson]

Hi all.
I am wondering why when a wave approach a beach, its amplitude will increase and then break?
From my study of nonlinear wave, I know that the velocity of upper paricles of the pulse will travel faster than the lower particles and hence steeping of wave occur. But this does not explain why the AMPLITUDE become large, actually why? Also, the before-said steeping of wave is significant only when approaching a beach? why?

 

[arildno]

The amplitude increase is present in the linear case as well, and is called Green's law.
It follows from energy conservation considerations.

Non-linearity effects really only becomes significant when the non-linearity paremter (say amplitude/depth, (or more proplerly, the Ursell parameter)) is large.
This will typically occur when the depth is "small", i.e, in approaching the beach, rather than at deep waters.

 

[hanson]

The amplitude increase is present in the linear case as well, and is called Green's law.
It follows from energy conservation considerations.

Non-linearity effects really only becomes significant when the non-linearity paremter (say amplitude/depth, (or more proplerly, the Ursell parameter)) is large.
This will typically occur when the depth is "small", i.e, in approaching the beach, rather than at deep waters.

Thanks for the reply. So, you mean the amplitude increase has nothing to do the nonlinearity or dispersion effect etc. It is just a consequence of energy conservation?

In other words, the steepening of wave and the increase in wave amplitude act independently in a wave approaching a beach?

By the way, will a wave break if there is no steepening by nonlinearity but just amplitude increase due to decrease in wave depth?

 

[arildno]

Thanks for the reply. So, you mean the amplitude increase has nothing to do the nonlinearity or dispersion effect etc. It is just a consequence of energy conservation?

No, nonlinear effects will perturb the actual increase, but the dominant contribution for small non-linearities is predictable by energy conservation arguments for the linearized model.

In other words, the steepening of wave and the increase in wave amplitude act independently in a wave approaching a beach?

No, linear and non-linear effects are not strictly "independent", but the effects for the linearized model will dominate the actual picture when non-linearity is small.

By the way, will a wave break if there is no steepening by nonlinearity but just amplitude increase due to decrease in wave depth?

Indeed, a linearized model fails in that it cannot predict wave-breaking.

 

[russ_watters]

The amplitude does not actually increase, it's just that since the depth of the water decreases, in order to keep the amplitude the same, the wave has to rise.