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The speed of wave in an elastic medium is given by

Velocity of Longitudinal Wave

1) Velocity of Sound in any Elastic Medium: It is given by $v=\sqrt{\frac{E}{\rho}}=\sqrt{\frac{Elasticity\,of\,the\,medium}{Density\,of\,the\,medium}}.$
a) In solids, $v=\sqrt{\frac{\gamma}{\rho}}$; Where $\gamma=$ Young's modulus of elasticity.
b) In a liquid and gaseous medium, $v=\sqrt{\frac{B}{\rho}}$; where $B = $ the Bulk modulus of elasticity of liquid of gaseous medium.

This velocity of wave is, I'm assuming, not turbulent but a regular wave.

But my question is, is it possible to propagate some kind disturbance in this elastic medium at a speed faster than this predicted velocity. Like stretching the medium and snapping it so some disturbance if not a wave will move faster than the predicted velocity of wave.

Is this possible in water? Like when Tsunami earthquakes happen does the disturbance initially travel faster than the speed of sound in water?

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You are looking for shock waves, which is generally defined as, "...a type of propagating disturbance that moves faster than the local speed of sound in the medium." Examples of shock waves include:

And so on.

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