Wavepackets and growth of water waves as they approach a beach

1. Feb 24, 2013

nrivera1

1. The problem statement, all variables and given/known data

Why do waves get larger as they approach a beach? Discuss in terms of wave packets and group velocity.

2. Relevant equations

Let h be the height of the water. We can write h(x,t) as:

h(x,t) = ∫dk*A(k)*cos(kx-wt), where w = w(k)

3. The attempt at a solution

What I thought was this: In shallow water, the dispersion relation for water waves is phase velocity = group velocity = √gh.

Therefore, as a wavepacket that contains a narrow group of wave-vectors approaches the beach, the dispersion effect lessens and lessens.

As a result of this, if we look at the fourier transform in part 2 at a particular place in space, x = a, we see that the phases of the waves are given approximately by (k'a-w(k)t), where k' is the central wave vector of the packet, and furthermore that the phase difference between component waves lessens because essentially the only thing causing the phase difference is the w(k)t term but as the dispersion goes away, the angular frequencies of the waves all become the same and the phases become nearly identical, allowing for more complete constructive interference.

Is this accurate?

If so, can someone then explain what happens in terms of group velocity and the envelope of the wave? I said group velocity was relevant because it's equal to dw/dk and is a measure of how much the phases change for adjacent component waves in k-space. Help would be appreciated as I've been thinking about this for a while. Thanks!