On a water table the correspondence

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

Signal Speeds in Fluids, Ch Steepening and the normal shock wave

Problem

On a water table the correspondence h↔ρh \leftrightarrow \rho, gh↔c\sqrt{gh} \leftrightarrow c and Fr↔M\mathrm{Fr} \leftrightarrow M holds with γ=2\gamma = 2. A hydraulic jump with upstream Froude number Fr1\mathrm{Fr}_1 has depth ratio h2h1=12(1+8Fr12−1),\frac{h_2}{h_1} = \frac{1}{2}\left(\sqrt{1+8\mathrm{Fr}_1^2}-1\right), from mass and momentum conservation alone. A gas shock at the same upstream Mach number M1=Fr1M_1 = \mathrm{Fr}_1, with γ=2\gamma = 2, has density ratio ρ2ρ1=(γ+1)M12(γ−1)M12+2=3M12M12+2\frac{\rho_2}{\rho_1} = \frac{(\gamma+1)M_1^2}{(\gamma-1)M_1^2+2} = \frac{3M_1^2}{M_1^2+2} from mass, momentum and energy conservation together. ((a)) Evaluate h2/h1h_2/h_1 and ρ2/ρ1\rho_2/\rho_1 at Fr1=M1=1.2\mathrm{Fr}_1 = M_1 = 1.2 (a weak disturbance) and comment on the size of the difference between them. ((b)) Evaluate both ratios again at Fr1=M1=3.5\mathrm{Fr}_1 = M_1 = 3.5 (a strong disturbance). ((c)) State, in terms of what each type of jump conserves, why the two ratios agree closely in one case and differ sharply in the other, and identify which of the two ratios stays bounded as Fr1→∞\mathrm{Fr}_1 \to \infty.

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