# Finding the Group Velocity for Shallow Water Wave

In summary, the group velocity for a shallow water wave is given by v_g = \frac{d\omega}{dk} where k is the wave number.

## Homework Statement

Find the group velocity for a shallow water wave: ##\nu = \sqrt{\frac{2\pi\gamma}{\rho\lambda^3}}##

## Homework Equations

Phase velocity: ##v_p = \nu\lambda##
group velocity: ##v_g = \frac{d\omega}{dk}##

##k=\frac{2\pi}{\lambda}##
##\omega = 2\pi \nu##

## The Attempt at a Solution

Get frequency in terms of wave number:

##\nu(k)=\sqrt{\frac{\gamma k}{\rho\lambda^2}}##
##\omega(k) = 2\pi\sqrt{\frac{\gamma k}{\rho\lambda^2}}##

##\frac{d\omega}{dk}=2\pi(\frac{1}{2}(\frac{\gamma k}{\rho \lambda^2})^{-1/2} \frac{\gamma}{\rho \lambda^2})##

We can rewrite the function, getting rid of k. Also did some canceling and moved the pi:##v_g=(\frac{\gamma 2\pi}{\rho \lambda^3})^{-1/2} \frac{\gamma\pi}{\rho \lambda^2}## ##v_g = \frac{\frac{\gamma\pi}{\rho \lambda^2}}{(\frac{\gamma 2\pi}{\rho \lambda^3})^{1/2}}##

Note that given the definition for phase velocity, we can write it as:

##v_p = \sqrt{\frac{2\pi\gamma}{\rho\lambda}}##

##v_g = \frac{\frac{\gamma\pi}{\rho \lambda^2}}{v_p \lambda} = \frac{\frac{\gamma\pi}{\rho \lambda^3}}{v_p}##

So this is where I'm stuck... The correct answer from the book is ##\frac{3}{2}v_p##

Get frequency in terms of wave number:

##\nu(k)=\sqrt{\frac{\gamma k}{\rho\lambda^2}}##
Try expressing ##\nu(k)## completely in terms of ##k##, rather than a mixture of ##\lambda## and ##k##.

##\frac{d\nu}{dk}=\frac{1}{2}(\frac{\gamma k^3}{4 \pi^2 \rho})^{-1/2}\frac{3k^2\gamma}{4\pi^2\rho}##

So I see we have that nice 3/2 term in there, but if my phase velocity term is correct, ##v_p = \nu\lambda = \sqrt{\frac{2\pi \gamma}{\rho \lambda}} = \sqrt{\frac{\gamma k}{\rho}}##, then every thing other than the 3/2 coefficient must reduce to this...

So maybe this is just some algebra problem right now...?

##\frac{\frac{\gamma k^2}{4\pi^2 \rho}}{\frac{\gamma k^3}{4 \pi^2 \rho}^{1/2}} = \sqrt{\frac{\gamma k}{\rho}}##

Was writing out this question when I figured out that this does reduce to 3/2v_p. Just as long as we do ##\frac{d \omega}{dk}## (include the 2pi), instead of ##\frac{d \nu}{dk}##

I figured I'd post my thoughts anyway. Thanks for tip @TSny.

## 1. What is group velocity?

Group velocity is the velocity at which the energy of a wave propagates in a medium. It is the speed at which the envelope of a wave packet (group of waves) moves through space.

## 2. Why is finding the group velocity important for shallow water waves?

Finding the group velocity for shallow water waves is important because it allows us to understand how the energy of the waves is transferred through the water. This information is crucial for predicting the behavior and potential impacts of these waves on structures and coastlines.

## 3. How is group velocity calculated for shallow water waves?

The group velocity for shallow water waves can be calculated using the following formula: c = √(gd), where c is the group velocity, g is the acceleration due to gravity, and d is the depth of the water.

## 4. Can group velocity change for shallow water waves?

Yes, the group velocity for shallow water waves can change depending on the depth of the water. As the water depth decreases, the group velocity increases. This means that shallow water waves will travel faster in shallower water.

## 5. How does group velocity affect the behavior of shallow water waves?

The group velocity affects the behavior of shallow water waves by determining how quickly the energy of the waves moves through the water. A higher group velocity means that the waves will have more energy and can potentially cause more damage to coastlines and structures. It also affects the spacing and frequency of the waves, which can impact their interference patterns.

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