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Acoustics: Sound Fields, Transducers and Vibration, 2nd Edition by Tim Mellow, Leo Beranek

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Part IV: Solutions of the wave equation in one dimension

2.3. General solutions of the one-dimensional wave equation

The one-dimensional wave equation was derived with either sound pressure or particle velocity as the dependent variable. Particle displacement, or the variational density, may also be used as the dependent variable. This can be seen from Eqs. (2.4a) and (2.13a) and the conservation of mass, which requires that the product of the density and the volume of a small box of gas remain constant. That is,
ρ V = ρ 0 V 0 = constant
image (2.28)
and so
ρ d V = V d ρ .
(2.29)
Let

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