In making certain investigations on the properties of the sound-wave, transmitted through a small horizontal tube of uniform bore, I found reason for thinking that the equation dy/dt = F(dy/dx) . . . . . . . . . . . . (1.) must always be satisfied; F being a function of a form to be determined. Differentiating this equation with regard to t, we find d2y/dt2 = {F'(dy/dx)}2·d2y/dx2 . . . . . . . . . (2.) which by means of the arbitrary function F can be made to coincide, not only with the ordinary dynamical equation of sound, but with any dynamical equation in which the ratio of d2y/dt2 and d2y/dx2 can be expressed in terms of dy/dx. Equation (1.) is a partial first integral of (2.), and by means of it we shall be able to obtain a final integral of (2.), which will be shown to be the general integral of (2.) for wave-motion, propagated in one direction only in such a tube as we have supposed, by its satisfying all the conditions of such wave-motion.
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Samuel Earnshaw (1860) studied this question.