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Based on the analogy with sound waves, can we say that the speed of a transverse wave in a solid medium is, $$v = \sqrt {\frac {G}{\rho}}$$ Where, $G$ is the shear modulus of elasticity of the medium and $\rho$ is its density? If yes, is there an elementary derivation of the expression as we have for sound waves or for transverse wave in strings? Any help would be greatly appreciated.

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To find the possible waves in a solid you really need the full theory tensor theory of elasticity. The transverse shear wave is the simplest case though. This is because, unlike the general elastic response, the force is parallel to the transverse displacements. You can find the wave equation by just equating the force gradient $$ \frac {\partial F_y} {\partial x}=\mu \frac {\partial^2 y}{\partial x^2} $$ to the acceleration $y$ component $$ \rho \frac {\partial^2 y}{\partial t^2}. $$ Thus $c^2_{\rm transverse}= \mu/\rho$. Here $\mu$ is the shear modulus. (This analysis does require that you have an infinite plane wave)

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