Question

• two monochromatic waves amplitude a superimpose with random phase difference,what is the resultant amplitude?

Answer 1

$$\begin{gathered} Letusconsidertwowaveis \\ y=A\sin (\omega t-\phi )andy\text{'}=A\sin (\omega t-\psi ) \\ where\phi and\psi istotallyarbitary \\ soresul\tan tY=y+y\text{'}=A\sin (\omega t-\phi )+A\sin (\omega t-\psi ) \\ =2A\sin \left(\frac{\omega t-\phi +\omega t-\psi }{2}\right)\cos \left(\frac{\omega t-\phi -\omega t+\psi }{2}\right) \\ =2A\sin \left[\omega t-\left(\frac{\phi +\psi }{2}\right)\right]cos\left[\frac{\psi -\phi }{2}\right] \\ thereforecomparingwiththes\tan dardformofwaveequationlikeX=P\sin (\omega t-\chi ) \\ wherePistheamplitudedoesn\text{'}tcontaintimeinformation,wecanwrite \\ Sotheresul\tan tamlitudewillbe=2Acos\left[\frac{\psi -\phi }{2}\right] \end{gathered}$$