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
$A_{1}^{2}=a_{1}^{2}+a_{3}^{2}+2 a_{1} a_{3} \cos \phi$
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
$A^{2}=A_{1}^{2}+a_{2}^{2}+2 A_{1} a_{2} \cos \frac{\pi}{2}$
or $A=\sqrt{3^{2}+4^{2}+2 \times 3 \times 4 \cos 90^{\circ}}$
$=\sqrt{9+16}$
$=5 \,\mu m$