During a storm

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Waves, acoustics and compressible flow · Waves, acoustics and compressible flow

Signal Speeds in Fluids, Ch Waves of finite height: Stokes waves, shoaling and breaking

Problem

During a storm, breakers of height Hb=3.5 mH_{\rm b} = 3.5\,\mathrm{m} form on a gently sloping beach. Outside the break the mean level is lowered by the set-down, which at the break point is ηˉb=−Hb2/(16hb)\bar\eta_{\rm b} = -H_{\rm b}^2/(16h_{\rm b}), where hb=Hb/0.78h_{\rm b}=H_{\rm b}/0.78 is the mean depth there. Inside the surf zone the broken waves keep H=0.78(h+ηˉ)H=0.78(h+\bar\eta), and the mean level rises by K=38(0.78)2/[1+38(0.78)2]K=\tfrac38(0.78)^2/\bigl[1+\tfrac38(0.78)^2\bigr] for every unit fall in the still-water depth hh. ((a)) Find the mean depth hbh_{\rm b} at the break. ((b)) Find the set-down ηˉb\bar\eta_{\rm b} at the break point, measured from the far-offshore still-water level. ((c)) Find the set-up at the still-water shoreline, where h=0h=0, measured from the same level. ((d)) Using the mean momentum balance ρg(h+ηˉ) dηˉ/dx=−dSxx/dx\rho g(h+\bar\eta)\,\mathrm{d}\bar\eta/\mathrm{d}x=-\mathrm{d}S_{xx}/\mathrm{d}x, with xx increasing shoreward, explain why the mean level rises inside the surf zone although it fell outside it.

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