All Problems Elastic Waves. Acoustics.
Problem 4.185
A plane longitudinal harmonic wave propagates in a medium with density \(\rho\). The velocity of the wave propagation is \(v\). Assuming that the density variations of the medium, induced by the propagating wave, \(\Delta \rho \ll \rho,\) demonstrate that (a) the pressure increment in the medium \(\Delta p=-\rho v^{2}(\partial \xi / \partial x)\), where \(\partial \xi / \partial x\) is the relative deformation (b) the wave intensity is defined by Eq. (4.3i).
4.185. (a) Let us consider the motion of a plane element of the medium of thickness \(d x\) and unit area of cross-section. In accordance with Newton's second law \(\rho d x \ddot{\xi}=-d p,\) where \(d p\) is the pressure increment over the length \(d x\). Recalling the wave equation \(\ddot{\xi}=\) \(=v^{2}\left(\partial^{2} \xi / \partial x^{2}\right),\) we can write the foregoing equation as \[ \rho v^{2} \frac{\partial^{2} \xi}{\partial x^{2}} d x=-d p \] Integrating this equation, we get \[ \Delta p=-\rho v^{2} \frac{\partial \xi}{\partial x}+\text { const. } \] In the absence of a deformation (a wave) the surplus pressure is \(\Delta p=0 .\) Hence, const \(=0\)