Energy conservation in wave interference

[O.J.]

Energy conservation in wave interference...

Hey folks,

Let's get right into the subject: we know that two waves of the same phase, frequency, type and wavelength traveling alongside each other interfere constructively and their intensity is 4 times that if it were one wave..

Now let's take it this way, suppose the two waves are traveling from a source parallel to each other, each wave carrying a specific amount of enrergy (E). Now you'd think that TWO of that wave would have an energy = 2E and hence assuming they fall on the same area the intensity is doubles, but it's 4 times now. How come??

 

[topsquark]

Hey folks,

Let's get right into the subject: we know that two waves of the same phase, frequency, type and wavelength traveling alongside each other interfere constructively and their intensity is 4 times that if it were one wave..

Now let's take it this way, suppose the two waves are traveling from a source parallel to each other, each wave carrying a specific amount of enrergy (E). Now you'd think that TWO of that wave would have an energy = 2E and hence assuming they fall on the same area the intensity is doubles, but it's 4 times now. How come??

Since when is the waveform amplitude for constructive interference 4 x the amplitude? Where are you getting that number from?

-Dan

 

[pallidin]

Right. Constructive interference is only additive of their individual amplitude. Otherwise, we would have "free energy"

 

[jtbell]

O.J. is probably getting the 4 from the fact that the energy carried by a wave is proportional to the square of the amplitude. So if two waves with equal amplitude interfere constructively to produce a resultant wave with twice the amplitude, the resultant does carry four times the energy of either original wave.

The "catch" here is that when real-world waves interfere (like sound waves or water waves or light waves), there are regions of both constructive and destructive interference, and the regions of destructive interference have zero energy! Overall, interference just redistributes the energy from regions of destructive interference to regions of constructive interference.

suppose the two waves are traveling from a source parallel to each other

How do you get two waves from the same source? If they're from different sources in different locations, how do you superpose them so they're going in exactly the same direction along the same path, everywhere? Try to come up with a real physical setup.

 

[Claude Bile]

Nonlinear optics frequently encounters scenarios where the same wave is regarded as two separate sources. Just a remark that this setup is physically feasible, if a little artificial .

In any case, you do get intensities 4x that of the original signal, but as jtbell has already pointed out, the 4x multiplication only applies to parts of the waveform. If you average out the increase over the whole waveform you get a 2x increase.

Claude.

 

[O.J.]

just for the record, there can be two light waves traveling froma source... as in lasers probably?

 

[ZapperZ]

If you wish to pursue this further, a very detailed explanation can be found from this paper:

W. N. Mathews "Superposition and energy conservation for small amplitude mechanical waves", Am. J. Phys. v.54, p.233 (1986).

Zz.

 

[choon_min]

hi, I am choon min
recently have some problem with interference,
may i ask where can get the article?
as i search from google, it need to pay
thanks