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 0 , then
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(a, c)
From superposition principle
= a sin ꞷt + a sin (ꞷt + 45º) + a sin (ꞷt + 90º)
= a {sin ꞷt + sin (ꞷt + 90º)} + a sin (ꞷt + 45º)
= 2 a sin (ꞷt + 45º) cos 45º + a sin (ꞷt + 45º)
= ( 
+ 1) a sin (ꞷt + 45º)
= A sin (ꞷt + 45º)
Therefore, resultant motion is simple harmonic of amplitude
A = ( 
+ 1) a
and which differ in phase by 45º relative to the first.
Energy in SHM (amplitude) 2 [E = 
m A 2 ꞷ 2 ]
∴ 
∴ E resultant = (3 + 
) E single
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