On The Mathematical Theory of Sound

Summary

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

$$ \frac{{dy}}{{dt}} = F\left( {\frac{{dy}}{{dx}}} \right) $$

((1))

must always be satisfied; F being a function of a form to be determined. Differentiating this equation with regard to t, we find

$$ \frac{{{d^2}y}}{{d{t^2}}} = {\left\{ {F'\left( {\frac{{dy}}{{dx}}} \right)} \right\}^2} \cdot \frac{{{d^2}y}}{{d{x^2}}}, $$

((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 $\frac{{{d^2}y}}{{d{t^2}}}$ and $\frac{{{d^2}y}}{{d{x^2}}}$ can be expressed in terms of $\frac{{dy}}{{dx}}$.

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.

It will be convenient to begin with the simplest case of sound,—that in which the development of heat and cold is neglected.