Question

A sound wave is travelling in a medium in which the velocity is v. It is incident in the second medium in which the velocity of the wave is \[2v.\] What should be the minimum angle of incidence in the first medium, so that the wave fails to cross the surface of separation of the two media?
A. \[60^\circ \]
B. \[45^\circ \]
C. \[30^\circ \]
D. \[15^\circ \]

Answer 1

Hint:To solve this question, i.e, to find the minimum angle of incidence in the first medium, we will apply the Snell’s law here, as it gives the relationship between the angles of incidence and angles of refraction and also with the velocities of the two medium. So, on putting the given values in the formula, we will get our required answer.

Complete step by step answer:
We have been given that a sound wave is travelling in a medium in which the velocity is v. It is given that the wave is incident in the second medium in which the velocity of the wave is \[2v.\] We need to find the minimum angle of incidence in the first medium, so that the wave fails to cross the surface of separation of the two media.
We know that, Snell's law,$ = \dfrac{{\sin i}}{{\sin r}} = \dfrac{{velocity\;in\;first\;medium\;}}{{velocity\;in\;\sec ond\;medium\;}}$
On putting the values in the above formula, we get
 $
\Rightarrow \dfrac{{\sin i}}{{\sin r}} = \dfrac{{velocity\;in\;first\;medium\;}}{{velocity\;in\;\sec ond\;medium\;}} = \dfrac{v}{{2v}} \\
\Rightarrow \dfrac{{\sin i}}{{\sin r}} = \dfrac{1}{2}......eq.(1) \\
$
Since, it is given that the wave fails to cross the surface of separation of the two media. Therefore, \[r = 90^\circ \]
On putting the value of \[r = 90^\circ \], in eq. (1), we get
\[\Rightarrow \dfrac{{\sin i}}{{\sin 90^\circ }}\; = \dfrac{1}{2}\;\]
\[\Rightarrow \sin i = \dfrac{1}{2}..........(\because \sin 90^\circ = 1)\]
\[ \Rightarrow i = 30^\circ ........(\because \sin 30^\circ = \dfrac{1}{2})\]
So, the minimum angle of incidence in the first medium is $30^\circ .$
Thus, option (C) \[30^\circ \], is correct.

Note: In the solutions, we have mentioned about Snell’s law. Let us understand about it in detail. Snell's law gives a formula, describing a relationship between the angles of incidence and angles of refraction.It states that the ratio of the sines of the angles of incidence and angle of refraction is equal to the ratio of phase velocities in the two given media.