${y_1} = a\sin \,\left( {\omega \,t - \frac{\pi }{4}} \right)$; ${y_2} = a\sin \omega \,t$ and
${y_3} = a\sin \,\left( {\omega \,t + \frac{\pi }{4}} \right)$.
On superimposing, resultant SHM will be $y = a\;\left[ {\sin \,\left( {\omega \,t - \frac{\pi }{4}} \right) + \sin \omega \,t + \sin \,\left( {\omega \,t + \frac{\pi }{4}} \right)} \right]$
$ = 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$
Resultant amplitude =$(1 + \sqrt 2 )a$
Energy is $S.H.M.$ $\propto$ (Amplitude)$^2$
$\frac{{{E_{{\rm{Resultant}}}}}}{{{E_{{\rm{Single}}}}}} = {\left( {\frac{A}{a}} \right)^2} = {(\sqrt 2 + 1)^2} = (3 + 2\sqrt 2 )$
==> ${E_{{\rm{Resultant}}}} = (3 + 2\sqrt 2 ){E_{{\rm{Single}}}}$
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