Physics - Sound Wave Question with Solution | TestHub

Let the equation of two waves are
$y_1=a\sin \left(\omega t-kx\right) \dots \left(i\right)$
And$y_2=a\sin \left(\omega t-kx+\mathrm{\varphi }\right) \dots \left(ii\right)$
When they superpose, the resultant wave is
$y=y_1+y_2$
$=a\left[\sin \left(\omega t-kx\right)+\sin \left(\omega t-kx+\mathrm{\varphi }\right)\right]$
$=a\left[2\sin \left(\omega t-kx+\frac{\mathrm{\varphi }}{2}\right)\cos \left(-\frac{\mathrm{\varphi }}{2}\right)\right]$
$=2a\sin \left(\omega t-kx+\frac{\mathrm{\varphi }}{2}\right)\cos \frac{\mathrm{\varphi }}{2}$
$=\left(2a\cos \frac{\mathrm{\varphi }}{2}\right)\sin \left(\omega t-kx+\frac{\mathrm{\varphi }}{2}\right) \dots \left(iii\right)$
Comparing Eq. (iii) with (i) or (ii), we get
$a=2a\cos \frac{\mathrm{\varphi }}{2}⟹\cos \frac{\mathrm{\varphi }}{2}=\frac{1}{2}$
$⟹\mathrm{ }\cos \frac{\mathrm{\varphi }}{2}=\cos \frac{\mathrm{\pi }}{3}\mathrm{ }\mathrm{ }$
$\therefore \mathrm{ }\mathrm{ }\cos \frac{\mathrm{\varphi }}{2}=\cos \frac{\mathrm{\pi }}{3}\mathrm{ }\mathrm{ }$
$\therefore \mathrm{ }\mathrm{ }\frac{\mathrm{\varphi }}{2}=\frac{\mathrm{\pi }}{3}$
$\mathrm{o}\mathrm{r}\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{\varphi }=\frac{2\mathrm{\pi }}{3}$