In this problem, we will consider the breaking of giant waves. 1 Seafarers report that in major storms, when the average wave height measured from peak to bottom is about ? of the wavelength: the peaks of the largest ocean waves detach from the rest of the wave. They say that the wind and the propagation of the wave cause large lumps of water to fall down along the surface of the wave. Some lumps are large enough to damage or crash small ships. The reason for the rupture of such rogue waves is that the amplitude of the waves increases so large that the acceleration of the wave peak should be greater than the fall acceleration g in order for the wave to remain in size. Since gravity is the only signicant vertical force acting on the water, the acceleration cannot exceed the acceleration of the fall. Thus, the peak of the wave detaches from the rest of the wave and rolls down along the edge of the wave. a.) Consider water waves propagating in the deep sea with a wave height (amplitude) much smaller than their wavelength. Use dimensional analysis to determine the relation between the wave propagation velocity '1) and its wavenumber k 21s The wavenumber is dened as k = T , where A is the wavelength. The dimensional analysis does not give a value for the constant C , but it turns out that this prefactor is C = 1 ! Hint: Assume the amplitude of the wave is 'small' (what does 'small' mean?) The result should be 1,- 02) = C E k b.) The simplest model for a wave is a harmonic vertical deection on the water surface z (3';? t) = A 005(ka.' wt) Consider the surface of the water at different points in this wave. At what point in the wave is the downward acceleration greatest and what is its mangtude? c.) Solve the critical ratio for the amplitude A of the wave and the wavelength A at which the wave begins to rupture. d.)At the beginrn'ng, we assumed that the amplitude of the wave is small. This assumption was needed for a simple dimensional analytical calculation of the wave propagation velocity. Check that the critical amplitude you calculate satises this condition. ('We would probably like this amplitude to be even smaller: but the result obtained is suicient for our simple backoftheenvelope calculation). Finally, at least near the breaking point; the assumption of a harmonic wave must be A broken. Thus, the critical ratio _ can only be understood as an estimate. A