Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/437

“Say thus:—Divide the wave train into lengths each containing unit mass. The time taken by each of these lengths to pass a given point is proportional to the length. Also the time during which, etc. ... or something of the sort. But it appears to me that this is not the whole story, but that the motion docs communicate momentum. If the velocity forward is $u\ \longrightarrow ,$ momentum $\rho \ u\times u=\rho \ u^{2}$ crosses the plane $\longrightarrow .$ If the velocity is $u\ \longleftarrow ,$ momentum $\rho \ u^{2}$ crosses the plane $\longleftarrow ,$ and both of these give an addition of momentum $\longrightarrow$ to the region on the forward side of the plane.”^[1]

For the purpose of reference p. 3 of the author's original MS. is given in the accompanying footnote, the paragraph marked by Prof. Poynting being italicised. The initial and final paragraphs are completed as on pp. 2 and 4 of the MS.

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Considering now a supposititious wave in a medium obeying Boyle's law,—
The volume occupied by any small unit of mass is by Boyle's law inversely as the pressure. Therefore the linear distance in the direction of propagation occupied by any small unit of mass is inversely as its pressure.
But the time during which pressure acts across the imaginary plane is by (4) proportional to this linear distance. Therefore the time during which any pressure acts across the imaginary plane is inversely as that pressure, or $p\ t=$ constant for any small unit of mass. But $p\ t$ is the momentum communicated across the imaginary plane by pressure per unit area, and we have shown the total units of mass in any wave is the same as in undisturbed air. Consequently in a plane wave in a fluid obeying Boyle's law the momentum communicated by the pressure of the wave is precisely that communicated by the undisturbed fluid.
And since the sum of the translation of mass $(m\ l)$ by the wave is zero, the sum of the communication of momentum by motion $(m\ v)$ is also zero.
That is to say, the plane wave in an elastic fluid obeying Boyle's law carries no momentum.
If the adiabatic wave is examined by the foregoing method an excess of mean pressure is found to exist, and without doubt, if the source of sound emits a continued succession of waves, momentum accompanies such waves as an ever-spreading field of excess mean pressure, but it is not clear that if the source ceases to emit, this pressure region will be confined to and move with the advancing waves; it appears more probable to the author that the air contained within the wave sphere shares in the excess pressure.”

1. This argument appears to involve a fallacy similar to that mentioned in footnote, p. 399. The note in question is the answer to an argument actually used by Poynting in conversation with the author.

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