Question

The phase speed vp for deep ocean waves is given by vp = g/ω where ω = 2π/T and T is the period of the wave; g is the acceleration due to gravity. What is the group speed vg?

Answer 1

The group speed is
`(v_(p))/(2)`.

Step-by-step explanation:

Given that,

Phase speed
`v_(p)=(g)/(\omega)`

`\omega=(2\pi)/(T)`

Where, T = period of the wave

g =acceleration due to gravity

We need to calculate the group speed

Using formula of phase speed

`v_(p)=(\omega)/(k)`

Put the value of
`v_(p)`

`(g)/(\omega)=(\omega)/(k)`

`\omega=√(gk)`....(I)

We need to calculate the group speed

Using formula of group speed

`v_(g)=(d\omega)/(dk)`

`v_(g)=(d)/(dk)(√(gk))`

On differentiating

`v_(g)=(1)/(2)\sqrt{(g)/(k)}`

Put the value of k from equation (I)

`v_(g)=(1)/(2)\sqrt{(g)/((\omega^2)/(g))}`

`v_(g)=(1)/(2)*(g)/(\omega)`

`v_(g)=(v_(p))/(2)`

Hence, The group speed is
`(v_(p))/(2)`.