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Question: A sound wave is travelling in a medium in which the velocity is v. It is incident in the second medi...

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.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∘60^\circ
B. 45∘45^\circ
C. 30∘30^\circ
D. 15∘15^\circ

Explanation

Solution

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.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,=sin⁡isin⁡r=velocity  in  first  medium  velocity  in  sec⁡ond  medium   = \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
⇒sin⁡isin⁡r=velocity  in  first  medium  velocity  in  sec⁡ond  medium  =v2v ⇒sin⁡isin⁡r=12......eq.(1)  \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∘r = 90^\circ
On putting the value of r=90∘r = 90^\circ , in eq. (1), we get
⇒sin⁡isin⁡90∘  =12  \Rightarrow \dfrac{{\sin i}}{{\sin 90^\circ }}\; = \dfrac{1}{2}\;
⇒sin⁡i=12..........(∵sin⁡90∘=1)\Rightarrow \sin i = \dfrac{1}{2}..........(\because \sin 90^\circ = 1)
⇒i=30∘........(∵sin⁡30∘=12)\Rightarrow i = 30^\circ ........(\because \sin 30^\circ = \dfrac{1}{2})
So, the minimum angle of incidence in the first medium is 30∘.30^\circ .
Thus, option (C) 30∘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.