Question

Large waves on the deep ocean propagate at the speed v=gk--√, where g is the magnitude of the free-fall acceleration and k is the wavenumber. Seafaring mariners report that in great storms when the average peak-to-peak wave height becomes about 1/7 of the wavelength, the tops of the largest ocean waves can become separated from the rest of the wave. They claim that the wind and the wave's forward velocity cause huge "hunks" of water to tumble down the face of the wave. Some are reportedly large enough to damage or capsize small vessels. The reason these "rogue waves" appear is that the amplitude of the water waves becomes so large that the acceleration of the water in the top of the wave would have to be greater than g for the wave to stay in one piece. Because gravity is the only significant vertical force on the water, the acceleration cannot exceed g, so instead the water at the top of the wave breaks off and is blown down the side of the wave. In this problem, you will compute the ratio of amplitude to wavelength of a rogue wave. The analytic expression for the vertical displacement of the water surface when an ocean wave of amplitude a is propagating in the x direction is z(x,t)=acos(kx-Ωt). What is the ratio of amplitude to wavelength of a rogue wave?

Answer 1

The ratio of amplitude to wavelength for a rogue wave is 1/14, based on the mariners' reports that rogue waves occur when the wave height is about 1/7 of the wavelength.

Step-by-step explanation:

The ratio of amplitude to wavelength for a rogue wave can be determined by considering the point at which the wave's top would need to break in order to not exceed the acceleration due to gravity. Mariners report that rogue waves occur when the wave height is about 1/7 of the wavelength. In the analytic expression for the vertical displacement of the water surface, z(x,t) = a*cos(kx - Ωt), a is the amplitude and the wavenumber k is related to the wavelength λ by the expression k = 2π/λ. When the wave height is 1/7 the wavelength, this means the amplitude is 1/14 of the wavelength because the height is measured from peak to trough, and the amplitude is measured from the equilibrium position to the peak. Therefore, the ratio of amplitude a to wavelength λ for a rogue wave would be 1/14.