Chapter 15, Problem 14P

(II) Dimensional analysis. Waves on the surface of the ocean do not depend significantly on the properties of water such as density or surface tension. The primary “return force” for water piled up in the wave crests is due to the gravitational attraction of the Earth. Titus the speed υ (m/s) of ocean waves depends on the acceleration due to gravity g. It is reasonable to expect that υ might also depend on water depth h and the wave’s wavelength λ. Assume the wave speed is given by the functional form υ = Cg^αh^βλ ^γ, where α, β, γ, and C are numbers without dimension. (a) In deep water, the water deep below does not affect the motion of waves at the surface. Thus υ should be independent of depth h (i.e., β = 0). Using only dimensional analysis (Section 1–7), determine the formula for the speed of surface waves in deep water. (b) In shallow water, the speed of surface waves is found experimentally to be independent of the wavelength (i.e.. γ = 0). Using only dimensional analysis, determine the formula for the speed of waves in shallow water.