Question

For wavelengths longer than about a meter the dispersion relation for gravity waves (waves for which the restoring force is gravity)is approximately given by : ω = k g Tanh(k d) where g is the gravitational acceleration, and d is the fluid depth, and you know what k and ω are. An undersea earthquake suddenly thrusts a 100km chunk of undersea land upwards, generating waves of wavelength of many scales, from λ=1 meter to to λ = 105 meter: a tsunami! Numerically calculate the phase velocity and group velocity of waves of: a) λ=1 meter b) λ= 103 meter b) λ = 105 meter (assume ocean depth is 3000 m)

Answer 1

Dispersion relation for gravity wave is,

.

Using expansion,

dispersion relation becomes

ignoring higher order terms,

.--------------- (1)

Phase velocity and group velocity is defined as,

From equation (1) and (2),

(a)