Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/428

App. II. A. treated, as a matter of ordinary dynamics, it is evident that it is the displacement of matter (if any) that gives rise to the momentum of a wave, and it is the momentum of the wave that gives rise to the pressure at reflection, and by equating the two, as has been done in the foregoing demonstration, we have the simplest known method of obtaining the expression for the velocity of sound.

The nature of the flaw in Larmor's theorem is discussed in Addendum B of the present Appendix.

The simplicity of the present method of the determination of the velocity of sound is largely due to the form in which Boyle's law is presented. It is usual to write the isothermal law (Boyle's law), for a perfect gas $PV =$ constant; now this presumes mass constant. It would be quite as correct to write $P / m =$ constant, taking the volume to remain unchanged. It is obviously best to include both mass and volume as variables and write $P / \rho =$ constant, as has been done.

The present method has much in its favour. The argument not only covers waves of small amplitude, but waves of any amplitude and any form; we may regard a wave in a fluid obeying Boyle's law as built up of a number of superposed elements, each of which conforms to the pressure-momentum equation giving the same value of $$U$$ for each element alone or in superposition. Consequently waves in a fluid obeying Boyle's law have no tendency to travel faster in one part than in another part; their form is permanent and velocity uniform.

In Poynting and Thomson's "Sound," a method is given for the theoretical determination of the velocity of sound, on the assumption that the pressure changes are proportional to the volume changes, and the usual well-known expression $$U = \sqrt{\frac{E}{\rho}}$$ is obtained. A foot-note is given in connection with this demonstration, as follows:—

If the pressure changes are too considerable to justify the assumption that they are proportional to the volume changes,