Question

Question 4. (25%) Surface gravity waves: (a) Write down the expression for phase speed c for surface gravity waves if their wavelength λ is much shorter than the water depth. Are these 'deep- water' waves or 'shallow water' waves? Explain briefly. (b) Write the expression for wavenumber k in terms of wavelength λ. (c) Evaluate the phase speeds for a wavelength of 10 meters, and for a wavelength of 500m. (d) Using these results, suppose the waves are generated by a storm 1000 km away. How long will it take for the two waves in (c) to reach you?

Answer 1

(a) when wavelength is much shorter than the water depth, the wave is deep water wave.

The phase speed is given as

c = sqrt (g/k)

where k is wave number

(b) Relation between wave number (k) and wavelength $\lambda$ is given as

k = 2 $\pi$ / $\lambda$

so,

c = sqrt (g $\lambda$ / 2 $\pi$ )

(c) Now,

for   $\lambda$ = 10 m

c = 3.949 m/s

for $\lambda$ = 500 m

c = 27.92 m/s

Therefore, we can see that longer waves can travel faster than shorter waves.

(d) time taken by 10 m wave

t = 1000,000 / 3.949 = 70.34 hours or 2.532e5 seconds

time taken by 500 m wave

t = 1000000 / 27.92 = 9.9 hours or 3.58e4 seconds