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

The amplitudes of the waves are

${\mathrm{a}}_1=10\mathrm{\mu m},{\mathrm{a}}_2=4\mathrm{\mu m}\text{ and }{\mathrm{a}}_3=7\mathrm{\mu m}$

and the phase difference between 1st and 2nd wave is

$\frac{\pi }{2}$ and that between 2nd and 3rd wave is $\frac{\pi }{2}$.

Therefore, phase difference between 1st and 3rd is $\pi$. Combining 1st with 3rd, their resultant amplitude is given by

${\mathrm{A}}_1^2={\mathrm{a}}_1^2+{\mathrm{a}}_3^2+2{\mathrm{a}}_1{\mathrm{a}}_3\mathrm{cos\varphi }$

$\text{ or }{\mathrm{A}}_1=\sqrt{{10}^2+7^2+2\times 10\times 7\mathrm{cos\pi }}$

$=\sqrt{100+49-140}$

$=\sqrt{9}=3\mu \mathrm{m}$ in the direction of first.

Now combining this with 2nd wave we have, the resultant amplitude

${\mathrm{A}}^2={\mathrm{A}}_1^2+{\mathrm{a}}_2^2+2{\mathrm{A}}_1{\mathrm{a}}_2\cos \frac{\mathrm{\pi }}{2}$

$\text{ or }\mathrm{A}=\sqrt{3^2+4^2+2\times 3\times 4\cos {90}^{\circ }}=\sqrt{9+16}$

$=5 \mu \mathrm{m}$