The bed of a wide channel lies at height and the free surface at

PHYSICS · Subjective · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Waves, acoustics and compressible flow

Signal Speeds in Fluids, Ch Long waves of finite height: bores, jumps and solitary waves

Problem

The bed of a wide channel lies at height z=b(x)z = b(x) and the free surface at z=b+hz = b + h, so that h(x,t)h(x,t) is still the depth. Take the pressure as hydrostatic and the velocity u(x,t)u(x,t) as uniform with depth, and neglect friction on the bed. (a) Apply mass conservation to a vertical slice of width dx\mathrm{d}x and show that ∂th+∂x(hu)=0\partial_t h + \partial_x(hu) = 0 is unchanged by the slope. (b) Show that, besides the pressure forces on its two vertical faces, the slice is pushed by the bed with a horizontal force −ρgh b′(x) dx-\rho gh\,b'(x)\,\mathrm{d}x per unit width, and hence that ∂tu+u ∂xu+g ∂xh=−g b′(x)\partial_t u + u\,\partial_x u + g\,\partial_x h = -g\,b'(x). (c) For steady flow, show that h+u2/2g+bh + u^2/2g + b has the same value at every xx, so that the specific energy EE falls by exactly bb as a stream crosses a bump.

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