Shock waves

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$\large \frac{v}{c}=$

A wave source travels to to the right at velocity $v$ while it emits waves that travel at a velocity $c$. The wavefronts are closer together in front of the source and they are farther apart behind the source.If the speed of the source is faster than the wave velocity $(\frac{v}{c} < 1)$, a conical shock wave is produced. The cone forms an angle

$$\theta=\sin^{-1}\left(\frac{c}{v}\right)$$

with the velocity vector of the source. The wavefronts add constructively on the cone so there is a loud noise.

It is relatively easy to drive a boat faster than the speed of water waves. The characteristic V-form wake behind a fast-moving boat is a shock wave.

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