The waves on the ocean are surface waves: They occur at the interface of water and air, extending down into the water and up into the air at the expense of becoming exponentially reduced in amplitude. They are neither transverse nor longitudinal. The water both at and below the surface travels in vertical circles, with exponentially smaller radius as a function of depth.
Both empirical measurements and calculations beyond the scope of introductory physics give the propagation speed of water waves as:
v= √(g/k)
where g=9.8m/s^2 is the magnitude of the acceleration due to gravity and k is the wavenumber.
This relationship applies only when the following three conditions hold:
1. The water is several times deeper than the wavelength.
2. The wavelength is large enough that the surface tension of the waves can be neglected.
3. The ratio of wave height to wavelength is small.
4. The restoring force (analogous to the tension in a string) that restores the water surface to flatness is due to gravity, which explains why these waves are often called "gravity waves."
Determine:
(A) Find the speed v of water waves in terms of the wavelength λ.
Express the speed in terms of g, λ, and π.
(B) Find the speed v of a wave of wavelength λ=8.0m.
Give your answer in meters per second to a precision of two significant figures.
(C) Find the period T for a wave of wavelength. Express the period in terms of λ, π, and g.