Where does extra energy come from in superposed waves?

[Chris 197]

Where does "extra" energy come from in superposed waves?

1. I am needing to understand how, when two identical waves are superposed in phase, and the amplitude doubles, how is it that the energy of the resulting wave is greater than the sum of the two superposed waves.



2. Given that E = A^2, and my example waves are measured in meters and use this formula: E=100 J/m^2, then,

a wave with an amplitude of 2m therefore has an energy of E = 100 J/m^2 * 2^2 = 400 J

Two 2m waves are superposed in phase and the amplitude is now 4m. The new wave now has an energy of E = 100 J/m^2 * 4^2 = 1600 J

Where did the extra 800J come from? It seems to have come out of thin air, which violates the conservation of energy.

 

[rl.bhat]

You cannot write E1 + E2 = A1^2 + A2^2
It should be E1 + E2 = (A1 + A2)^2

 

[Chris 197]

Thank you rl.bhat.

It seems to me that I did add the amplitudes in the way you described:

"The new wave now has an energy of E = 100 J/m^2 * 4^2 = 1600 J." To get the "4^2" I added two 2m amplitudes of the original waves that were then superposed, and then squared 4 to get 16.

Thanks for the help to understand where I am missing something.