Solved: For any type of wave that reaches a boundary

Chapter 11, Problem 87GP

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For any type of wave that reaches a boundary beyond which its speed is increased, there is a maximum incident angle if there is to be a transmitted refracted wave. This maximum incident angle  \(\theta_{i M}\) corresponds to an angle of refraction equal to \(90^{0}\). If \(\theta_{i}>\theta_{i M}\) all the wave is reflected at the boundary and none is refracted, because refraction would correspond to \(\theta_{r}>1\) (where  is the angle of refraction), which is impossible. This phenomenon is referred to as total internal reflection. (a) Find a formula for \(\(\theta_{i M}\)\) using the law of refraction, Eq. 11–20. (b) How far from the bank should a trout fisherman stand (Fig. 11–61) so trout won’t be frightened by his voice (1.8 m above the ground)? The speed of sound is about 343 m/s in air and 1440 m/s in water.

Equation transcription:

Text transcription:

90^{0}

theta{i}>theta{i M}

theta{i M}

theta{r}>1