Two harmonic travelling waves are described by the equations $y_1=a\sin \left(kx-\omega t\right)$ and $y_2=a\sin \left(-kx+\omega t+\varphi \right)$ The amplitude of the superposed wave is

$y_1=a\sin \left(kx-\omega t\right) ...\left(1\right)$

$y_2=a\sin \left(-kx+\omega t+\varphi \right) ...\left(2\right)$

On adding $\left(1\right)$ and $\left(2\right)$,

$y_1+y_2=2\mathrm{asin}\left(\frac{\left(kx-\omega t\right)+\left(-kx+\omega t+\varphi \right)}{2}\right)$

$\cos \left(\frac{kx-\omega t+kx-\omega t-\varphi }{2}\right)$

$=2\mathrm{asin}\left(\frac{\varphi }{2}\right)\cos \left(\frac{2(kx+\omega t)-\varphi }{2}\right)$

Amplitude, $A\left(\varphi \right)$ is a function of phase difference $\varphi$ between constituent of waves

$\therefore \left(\varphi \right)=2a\sin \left(\frac{\varphi }{2}\right)$