# Elastic waves, Acoustic

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• Equations of plane and spherical waves:$$\xi = a\cos(\omega t - kx), \xi = \frac{a_0}{r} \cos (\omega t - kr).$$ In the case of a homogeneous absorbing medium the factors $e^{-\gamma x}$ and $e^{-\gamma r}$ respectively appear in the formulas, where $\gamma$ is the wave damping coefficient.
• Wave equation:$$\frac{\partial^2 \xi}{\partial x^2} + \frac{\partial^2 \xi }{\partial y^2} + \frac{\partial^2 \xi}{\partial z^2} = \frac{1}{v^2}\frac{\partial^2\xi}{\partial t^2}$$
• Equations of plane and spherical waves:$$\xi = a\cos(\omega t - kx), \xi = \frac{a_0}{r} \cos (\omega t - kr).$$ In the case of a homogeneous absorbing medium the factors $e^{-\gamma x}$ and $e^{-\gamma r}$ respectively appear in the formulas, where $\gamma$ is the wave damping coefficient.
• Wave equation:$$\frac{\partial^2 \xi}{\partial x^2} + \frac{\partial^2 \xi }{\partial y^2} + \frac{\partial^2 \xi}{\partial z^2} = \frac{1}{v^2}\frac{\partial^2\xi}{\partial t^2}$$
• Phase velocity of longitudinal waves in an elastic medium ($v_\parallel$) and transverse waves in a string $(v_{\bot})$: $$v_{\parallel }=\sqrt{ \frac{E}{\rho}}, v_{\bot}=\sqrt{\frac{T}{\rho_1}},$$where $E$ is Young's modulus, $\rho$ is the density of the medium, $T$ is the tension of the string, $\rho_1$ is its linear density.
• Volume density of energy of an elastic wave:$$\omega = \rho a^2\omega^2 \sin^2( \omega t- kx), \langle\omega\rangle= \frac{1}{2} \rho a^2\omega^2.$$
• Energy flow density, or the Umov vector for a travelling wave:$$\vec{j}=\omega \vec{v}, \langle \vec{j}\rangle=\frac{1}{2} \rho a^2 \omega^2 \vec{v}.$$
• Standing wave equation:$$\xi = a \cos kx. \cos \omega t.$$
• Acoustical Doppler effect:$$\nu= \nu_0 \frac{v+v_{ob}}{v-v_s}.$$
• Loudness level (in bels):$$L=log(\frac{I}{I_0})$$
• Relation between the intensity $I$ of a sound wave and the pressure oscillation amplitude $(\Delta p)_m$:$$I=\frac{(\Delta p)^2_m}{2 \rho v}.$$

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