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 $e^{-yx}$ and $e^{-vr}$ respectively appear in the formulas, where y 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 (v11) and transverse waves in a string $(v_{\perp})$:$$v_{\parallel }=\sqrt{ \frac{E}{\rho}}, v_{\perp}=\sqrt{\frac{T}{\rho_1}} $$where E is Young's modulus, p is the density of the medium, T is the tension of the string, Pi is its linear density.
  • Volume density of energy of an elastic wave:$$ \omega = \rho a^2\omega^2 \sin^2( \omega t- kx), (\omega)= \frac{1}{2 \rho a^2\omega^2 .}$$
  • Energy flow density, or the Umov vector for a travelling wave:$$j=\omega v, (j)=\frac{1}{2 \rho a^2 \omega^2 v} .$$
  • Standing wave equation:$$\xi = a \cos kx. \cos \omega t $$
  • Acoustical Doppler effect:$$ v= v_0 \frac{v+v_{ob}}{v-v_s}.$$
  • Loudness level (in bels):$$L=log(\frac{I}{I_0})$$
  • Relation between the intensity I of a sound wave and the pressure oscillation amplitude $(4p)_m$:$$ I=\frac{(p)^2_m}{2 \rho v} .$$


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