Three simple harmonic motions in the same direction having the same amplitude a and same period are superposed. If each differs in phase from the next by 45^o, then

Let simple harmonic motions be represented by

$y_1=a\sin \left(\omega t-\frac{\pi }{4}\right);y_2=a\sin \omega t \text{and }{y}_3=a\sin \left(\omega t+\frac{\pi }{4}\right)$

$\text{ Resultant will be }y=a\left[\sin \left(\omega t-\frac{\pi }{4}\right)+\sin \omega t+\sin \left(\omega t+\frac{\pi }{4}\right)\right]$

$\begin{array}{l}=a\left[2\sin \omega t\cos \frac{\pi }{4}+\sin \omega t\right]\\ =a[\sqrt{2}\sin \omega t+\sin \omega t]=a(1+\sqrt{2})\sin \omega t\end{array}$

Resultant amplitude $=(1+\sqrt{2})a$

Energy is S.H.M. $\propto (\text{ Amplitude }{)}^2$

$\begin{array}{l}\therefore \frac{E_{\text{Resultant }}}{E_{\text{Single }}}={\left(\frac{A}{a}\right)}^2=(\sqrt{2}+1{)}^2=(3+2\sqrt{2})\\ \Rightarrow E_{\text{Resultant }}=(3+2\sqrt{2})E_{\text{Single }}\end{array}$