Question

Two identical travelling waves, moving in the same direction, are out of phase by $\pi /2rad.$ What is the amplitude of the resultant wave in terms of the common amplitude $y_m$ of the two waves.

Answer 1

Let the waves be-

$y_1=A\sin (kx-\omega t)$

And $y_2=A\sin (kx-\omega t+{90}^o)=A\cos (kx-\omega t)$

$y=y_1+y_2$

$⟹y=A\left[\sin \right(kx-\omega t)+\cos (kx-\omega t\left)\right]$

$⟹y=\sqrt{2}A\left[\frac{1}{\sqrt{2}}\sin \right(kx-\omega t)+\frac{1}{\sqrt{2}}\cos (kx-\omega t\left)\right]$

$⟹y=\sqrt{2}A\left[\sin \right(kx-\omega t+{45}^o\left)\right]$

Hence $Amplitude=\sqrt{2}A=\sqrt{2}{y}_m$