Question

Three waves of equal frequency having amplitudes $10\mu m,4\mu m,7\mu m$ arrive at a given point with successive phase difference of $\pi /2$, the amplitude of the resulting wave in $\mu m$ is given by

• A. 4
• B. 5
• C. 6
• D. 7

Answer 1

Answer: (B)

The amplitudes of the waves are
$a_1=10\mu m,a_2=4\mu m$ and $a_3=7\mu m$
and the phase difference between 1st and 2 nd wave is
$\frac{\pi }{2}$ and that between 2 nd and 3 rd wave is $\frac{\pi }{2}$.
Therefore, phase difference between 1st and 3rd is $\pi$. Combining 1st with 3rd, their resultant amplitude is given by
$A_1^2=a_1^2+a_3^2+2a_1{a}_3\cos \varphi$
or $A_1=\sqrt{10^2+7^2+2\times 10\times 7\cos \pi }$
$=\sqrt{100+49-140}$
$=\sqrt{9}=3\mu m$ in the direction of first.
Now combining this with 2 nd wave we have, the resultant amplitude
$A^2=A_1^2+a_2^2+2A_1{a}_2\cos \frac{\pi }{2}$
or $A=\sqrt{3^2+4^2+2\times 3\times 4\cos 90^{\circ }}$
$=\sqrt{9+16}$
$=5\mu m$