- #1

James Brady

- 105

- 4

## 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##