Why is the energy not conserved when adding waves?

[Daker]

Hi

There is something wrong with this logic. Anyone see the flaw.

If we have a wave in y traveling along the z axis y= y_0cos(kz-wt) then the energy it carries is proportional to y_0^2. If we superimpose a second wave with the same amplitude (and so the same energy) and in phase with the first wave then we add the amplitudes to get a wave described by y= 2y0cos(kz-wt). The energy in this wave is proportional to the amplitude squared i.e. 4y_0^2. But energy is not conserved between the two initial waves and the resultant wave i.e y_0^2+y_0^2 \ne 4y_0^2.

Similarly if the waves are pi out of phase the resultant is zero and in this case y_0^2+y_0^2 = 0!

Sorry for being dumb!

Daker

 

[jeppetrost]

I believe the reason is, that even though you can superposition waves, you can't simply add the enrgies - for the exact reason you mentioned.
We have the same problem with EM-fields, since the energy here is also proportional to E or B squared.
I think you just have to re-calculate the energy/intensity.

 

[AlephZero]

You need to include the work done by whatever is creating the waves.

If a single plane wave is transmitting power P, then whatever creates the wave also requires power P.

But if you then put a second "identical" wave generator in the path of the first wave, in general it requires a different amount of power to operate it, because it is also interacting with the first wave. The power required depends on the phase angle between the force and the motion of the wave.

 

[Daker]

Thank you AlephZero and jeppetrost for these comments - very helpful.

 

[PJC01]

Hi Daker,

There is no need to apologize for not understanding something, as science can often be complex and confusing. In this case, the flaw in the logic is that energy is not simply additive when it comes to waves. While adding two waves of the same amplitude may result in a wave with double the amplitude, the energy carried by that wave is not simply doubled. This is because energy in a wave is proportional to the square of the amplitude, not just the amplitude itself. So, while the resultant wave may have a larger amplitude, its energy is not necessarily equal to the sum of the energies of the two initial waves. This is why energy is not conserved when adding waves.

Additionally, when two waves are out of phase, they can cancel each other out and result in a wave with zero amplitude and therefore zero energy. This is why y_0^2 + y_0^2 = 0 in this case.

I hope this helps to clarify the concept of energy conservation in waves. Please let me know if you have any further questions.

Best,