Starting from the one-dimensional continuity and momentum equations for isentropic…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Waves, acoustics and compressible flow

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

Problem

Starting from the one-dimensional continuity and momentum equations for isentropic flow of an ideal gas,

\rho_t+u\rho_x+\rho u_x=0,\qquad u_t+uu_x+\frac{1}{\rho}p_x=0,

with

p/\rho^\gamma

the same constant everywhere (isentropic flow), so that

c^2=\mathrm{d}p/\mathrm{d}\rho=\gamma p/\rho

, derive the Riemann invariants

J_\pm=u\pm\frac{2c}{\gamma-1},

each constant along its characteristic

dx/dt=u\pm c

. Then set

\gamma=2

and show the invariants reduce to the shallow-water Riemann invariants

u\pm2\sqrt{gh}

used for long waves.

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