Physics - Wave on String Question with Solution | TestHub

Resultant intensity of two periodic waves is given by $$I=I_1+I_2+2\sqrt{I_1{I}_2}\cos \delta$$ where $\delta$ is the phase difference between the waves. For maximum intensity, $$\delta =2n\pi ;n=0,1,2,\dots \text{ etc. }$$ Therefore, for zero order maxima, $\cos \delta =1$ $$I_{\max }=I_1+I_2+2\sqrt{I_1{I}_2}={\left(\sqrt{I_1}+\sqrt{I_2}\right)}^2$$ For minimum intensity, $$\delta =(2n-1)\pi ;n=1,2,\dots \text{ etc }$$ Therefore, for Ist order minima, $\cos \delta =-1$ $$\begin{array}{c}{I}_{\min }=I_1+I_2-2\sqrt{I_1{I}_2}\\ ={\left(\sqrt{I_1}-\sqrt{I_2}\right)}^2\\ \text{ Therefore, }{I}_{\max }+I_{\min }={\left(\sqrt{I_1}+\sqrt{I_2}\right)}^2\\ +{\left(\sqrt{I_1}-\sqrt{I_2}\right)}^2\\ =2\left(I_1+I_2\right)\end{array}$$