The Propagation of Planar Air Waves of Finite Amplitude
Even though the differential equations for determining the motion of gases have long been known, their integration has been carried out almost exclusively for the case when the pressure differences can be viewed as infinitesimally small fractions of the entire pressure, and until recently one had to be satisfied with taking account of only the first powers of these fractions. Until quite recently, Helmholtz was able to bring the second order terms into the calculation and in this way he clarified the objective generation of combination tones. But still, the precise differential equations can be completely integrated for the case where the initial motion takes place in the same direction and where the velocity and pressure are constant in every plane perpendicular to this direction. And even though the former treatment is entirely sufficient even for explaining the experimental phenomena found to date, still in all, it is possible-given the tremendous progress made most recently by Helmholtz in the experimental treatment of acoustic problems—that the results of this more precise calculation will provide additional insight into experimental research in the not too distant future. Regardless of its purely theoretical interest pertaining to the treatment of non-linear, partial differential equations, this might contribute toward a further elucidation of the problem.
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