Question

Three waves of equal frequency having amplitudes $10 \mu m$, $4 \mu m$ and $7 \mu m$ arrive at a given point with a successive phase difference of $\pi/2$. What is the amplitude of the resulting wave in $\mu m$?
A. $7$
B. $6$
C. $5$
D. $4$

Answer 1

Hint: When two waves of same frequency superimpose on each other, they may either interfere constructively or destructively. Interference will be constructive or destructive depending on the phase difference between the waves. Resultant amplitude of the waves can be calculated by first calculating the resultant amplitude of the two waves and then resultant of the wave left and wave resulted from those two waves.

Complete step by step answer:
When two waves of the same frequency superimpose on each other, they may either interfere constructively or destructively. Interference will be constructive or destructive depending on the phase difference between the waves. If the phase difference between the waves is $2n\pi $ then the waves interfere constructively and the magnitude of resultant wave is the sum of their individual amplitudes. Generally, resultant amplitude of the wave can be given as
$A=\sqrt{a_{1}^{2}+a_{2}^{2}+2{{a}_{1}}{{a}_{2}}cos\phi }$
Where ${{a}_{1}},\,{{a}_{2}}$ are the amplitude of the waves and $\phi$ is the phase difference.
We are given three waves of amplitudes $10 \mu m$, $4 \mu m$ and $7 \mu m$ respectively. The phase difference between waves with amplitude $10 \mu m$ and $7 \mu m$ is $\pi$. Therefore, these interfere destructively and the resultant amplitude is given by $10-7\mu m=3\mu m$. Now this resultant wave and wave with amplitude $4 \mu m$ have phase difference $\pi /2$. Substituting the values in amplitude equation, we have
$A=\sqrt{{{3}^{2}}+{{4}^{2}}+2\times 3\times 4cos\dfrac{\pi }{2}}=\sqrt{25}$
That is, amplitude of the resultant wave due to interference of the three waves is $A=5\mu m$

So, the correct answer is “Option c”.

Note:
When the phase difference between the waves is $2n\pi $ then the waves interfere constructively and the magnitude of resultant wave is the sum of their individual amplitudes.
If waves with amplitude $10 \mu m$ and $4 \mu m$ were added first, we had to find the phase difference between the resultant wave and the wave with amplitude $7 \mu m$. As waves with amplitude $10 \mu m$ and $7 \mu m$ have phase difference $\pi$, the resultant wave also remains at phase difference $\pi/2$ from waves with amplitude $4 \mu m$.