Contradiction in Wave Amplitude, intensity and Conservation of Energy?

[SecretSnow]

Hi guys, let's say we have a wave where the power P is proportionate to the square of its amplitude, which is A^2. If now we have 2 identical waves in superposition in phase, then we have an amplitude of 2A am i right?

Next, we realize that because of the amplitude of the superposed waves is 2A, it seems only natural that the power of the superposed waves together become 4P, since (2A)^2= 4A^2. In this in case there seem to be a contradiction in the conservation of energy. If these waves are left alone by themselves, the total power emitted would be 2P I think, not 4P. Why is this the case?

Then we consider intensity as well, if I=P/s (s is the surface area of sphere); the intensity in this case would be 4I if the 2 waves are superposed. Would the amplitude of the superposed wave, however, affect the surface area of the sphere s? If it doesn't, why would the intensity of the wave be 4I instead of 2I when these 2 waves are left alone, if the surface area s doesn't change? (because intensity is W/m^2!)

I don't get this apparent contradiction, please help, thanks guys!

 

[Simon Bridge]

That would be like the energy in the waves on a string.
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/powstr.html

Did you include the potential energy? Anyway:

You have noticed that it takes 4x the work to make a wave with twice the amplitude.
Conservation of energy has not been violated in this observation - the two situations are not equivalent.

A 2A wave is the same as two 1A waves on top of each other - but it is not the same as making the two 1A waves separately.

This is for the same reason it takes 4x the work to compress a spring by 2x, but you can compress two springs by x with only twice the work.

Making two waves on the same string, with equal amplitude, phase, and direction ... basically means making them one after the other. Twice the work. Making them at the same time is four times the work.

 

[SecretSnow]

hmm..but why would the situations be different? Why would the work done, because of superposition, be more than original? What's so special about superposition?

 

[Simon Bridge]

I told you - same reason as with the spring.
Each point on the string acts as a mass on a spring.
Pull the mass twice as far you do four times the work.

You can imagine you already have the first 1A wave - then you add a second one to it to make a 2A wave. This means you have to, somehow, pull each bit of the string an extra bit on top of what's already there. It's harder to pull the peak (for eg) from 1A to 2A than it was to pull it from 0 to 1A because the string is already pulling back. Give it a go sometime.