Wavepackets and growth of water waves as they approach a beach

[nrivera1]

Homework Statement

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

Homework 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)

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!

 

[mark726]

You are correct in your understanding that waves get larger as they approach a beach due to the decrease in dispersion. To further explain this phenomenon in terms of wave packets and group velocity, let's first define what these terms mean.

A wave packet is a group of waves with similar frequencies and wavelengths that travel together as a unit. In other words, it is a localized disturbance in a medium that is made up of multiple waves. The group velocity of a wave packet is the velocity at which the peak or envelope of the wave packet travels. It is different from the phase velocity, which is the velocity at which individual waves within the packet travel.

Now, let's consider a wave packet approaching a beach. As the wave packet gets closer to the beach, the depth of the water decreases. This decrease in depth causes the phase velocity and group velocity to decrease, as given by the dispersion relation √gh. As the group velocity decreases, the envelope of the wave packet slows down and becomes more compact, while the individual waves within the packet continue to travel at their own phase velocities.

At the same time, the decrease in depth also causes the waves to refract, which means their direction of travel changes. This change in direction causes the individual waves to overlap and interfere constructively, resulting in a larger wave. This is similar to how a magnifying glass focuses sunlight to create a larger, more intense spot of light.

In conclusion, as a wave packet approaches a beach, the decrease in depth causes the group velocity to decrease, resulting in a more compact and slower-moving envelope of the wave packet. This, combined with the refraction of the waves, leads to constructive interference and a larger wave at the shore. I hope this explanation helps clarify the role of wave packets and group velocity in the increase of wave size at a beach.