Question

The phase velocity v of wave in a certain
medium is given by v=a+bX where X is
the wavelength and a and b are constants.
What is its group velocity ?

Answer 1

The phase velocity of wave

$$v=a+b\lambda$$

The wave number

$$k=\frac{2\pi}{\lambda}$$

So, the phase velocity

$$v=a+\frac{2\pi b}{k}$$

From the other side

$$v=\frac{\omega}{k}$$

Therefore the frequency

$$\omega=vk=ak+2\pi b$$

The group velocity

$$v_g=\frac{d\omega}{dk}=a$$