Question

Two sound waves, each of amplitude $A$ and frequency $\omega$, superpose at a point with a phase difference of $\frac{\pi }{2}$. The amplitude and the frequency of the resultant wave are, respectively

• A. $A^2,\omega$
• B. $2A,\omega$
• C. $A,\omega$
• D. $\sqrt{2}A,\omega$

Answer 1

The correct option is D $\sqrt{2}A,\omega$

Let the two waves be $A\sin wt,A\cos wt$
$\therefore y=y_1+y_2=A\sin wt+A\cos wt$
$=\sqrt{2}A(\frac{1}{\sqrt{2}}\sin wt+\cos wt\frac{1}{\sqrt{2}})$
$=\sqrt{2}A\sin (wt+\frac{\pi }{4})$
amplitude is $\sqrt{2}A$, frequency is same, i.e. $w$