## A general solution for the celerity of long gravitational waves in a stratified liquid |

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approach infinity approaches zero arbitrary assumed Benton bottom boundary celerity of long channel class of density class of systems constant with depth continuous fluid system continuously stratified fluid critical flow density and velocity density difference density varies depths h differential equation discontinuities energy per unit exist fastest long gravitational fastest long wave fastest wave finite depth finitesimal flow relative fluid velocity free surface homogeneous fluids horizontal infinite density gradient infinite number infinitesimal amplitude internal gravity waves internal turbulence long gravitational waves modified Froude Number multi-layered system neutral long gravitational neutral long waves neutral waves non-dimensional number of layers numerical analysis numerical solution obtained parameters perfect fluid physical possible long gravitational remains finite result Richardson Number rigid surface shearing instability shown in Figure source of perturbations specified speed of long stability sub-case supercritical flow tribution unity upstream valid vertical coordinate vertical density wave is given wave of infinitesimal wave speed yields