Investigating Energy Conservation with Sinusoidal Waves

[landaetaedwar]

Hi there. I am having trouble finding an explanation with waves.

Suppose that you have to sinusoidal waves
y1=Asin(kx-wt) and y2=Asin(kx-wt+phi)
If we add them up, the resulting wave will be y=2Acos(phi/2)sin(kx-wt+phi/2). Now, if phi equals pi then the resulting wave will have no amplitude.
We know that in order to cause both waves energy is needed, and that energy is proportional to the square of the amplitude.
I cannot find to explain what happens to the energy. Where does it go? Wouldn't this violate the principle of the energy conservation?

I appreciate any help
Thanks!

 

[Studiot]

Just how would you 'add them up' ?

 

[landaetaedwar]

Destructive interference occurs (superposition)

 

[kcdodd]

If they are both going in the same direction, then one wave was present when the second wave was created, which means whatever caused it had to do work against the first wave. So, the energy went into whatever was making the second wave.

 

[mordechai9]

Notice that for phi=pi, you just have y1=Asin(kx-wt) and y2=-Asin(kx-wt). So y1+y2=0 and you actually don't have a wave any longer.

The punchline is that you cannot choose whether or not to add up the waves. If two independent waves are both solutions for the same system, then the general solution of that system will be the sum of both waves. To put it differently, the "energy" of a single wave component is not the true energy of the system, which should be a combination of both waves.