Energy conservation in identical destructive interference.

[siddharth5129]

Here's a thought experiment : Two identical spherical waves exactly 180 degrees out of phase destructively interfere throughout all of space. Is the energy contained in them just lost ?

 

[Chronos]

You must compute this in spherical coordinates to arrive at a sensible answer.

 

[siddharth5129]

Isn't it even theoretically possible to set up two identical spherical waves to destructively interfere throughout all of space?

 

[Dale]

No..

 

[sophiecentaur]

Isn't it even theoretically possible to set up two identical spherical waves to destructively interfere throughout all of space?

It's a bit like asking to connect two batteries up so there is no PD across their terminals - i.e. head to tail. In practice, loads of smoke and a PD across the terminals that is near to but not zero.

A practical / thought experiment: When you try to place two or more antennae near each other and feed them with the same signal (which is what you are implying) and try to adjust the phase to produce a null (or low level field strength in 'all' directions, you find that they affect each other and the impedance you need to drive them with gets lower and lower. This causes progressively more and more power loss in your driving source. You can get a narrow beam, channelling most of the power into a small range of directions, which is basically limited by the spacing ('aperture'). You can do a bit better than this with a 'super gain' array, in which the individual elements (of a multiple element array) have very high currents in them and are driven in near-opposite phases but it's a law of diminishing returns and, as with the two batteries, you end up chucking more and more power away in your power sources.
The Energy Conservation Law still rules, so relax.

 

[Malverin]

Here's a thought experiment : Two identical spherical waves exactly 180 degrees out of phase destructively interfere throughout all of space. Is the energy contained in them just lost ?

Destructive interference through all space is possible only if the two sources are at the same place and with same frequency, but with 180 degrees phase shift.

It is not possible, that sources are at exactly the same place in space. So there will never be full destructive interference.

http://www.cyberphysics.co.uk/topics/light/A_level/interference.htm

 

[sophiecentaur]

Destructive interference through all space is possible only if the two sources are at the same place and with same frequency, but with 180 degrees phase shift.

It is not possible, that sources are at exactly the same place in space. So there will never be full destructive interference.

Image.

http://www.cyberphysics.co.uk/topics/light/A_level/interference.htm

I nearly always like to look at this sort of problem in terms of RF antennae; the situation is more concrete and practical when approached that way. Also, you don't have to worry about coherence, if you use the same driver oscillator.
It is possible to set up a spherical wave from other than a point source. All antennae (whether directional or not, the wave is spherical) have a point phase origin when viewed from infinity, so the 'geometrical' - based objection is not necessarily valid.

You could envisage two antennae, interleaved or concentric which could produce identical radiation patterns. The clincher is that two such sources would need to have the same phase origin and be driven in anti phase so they would mutually interact to dissipate each other's power (they would both 'see' a zero impedance load). It's another of those 'irresistible force and immoveable object' situations, like my two DC voltage sources, above.

 

[siddharth5129]

A practical / thought experiment: When you try to place two or more antennae near each other and feed them with the same signal (which is what you are implying) and try to adjust the phase to produce a null (or low level field strength in 'all' directions, you find that they affect each other and the impedance you need to drive them with gets lower and lower. This causes progressively more and more power loss in your driving source. You can get a narrow beam, channelling most of the power into a small range of directions, which is basically limited by the spacing ('aperture'). You can do a bit better than this with a 'super gain' array, in which the individual elements (of a multiple element array) have very high currents in them and are driven in near-opposite phases but it's a law of diminishing returns and, as with the two batteries, you end up chucking more and more power away in your power sources.
The Energy Conservation Law still rules, so relax.

What you're saying is that in trying to set up two identical sources of spherical waves exactly out of phase with each other, they affect each other in such a way that the power dissipation happens at the source of the spherical waves itself? I don't see how your analogy with batteries works out. If you set up two batteries against each other, wouldn't the circuit just not draw any current, in which case there energy is right where it should be, stored in the chemical potential energy of the battery.
I suppose my question is a purely theoretical one. I get that it's practically impossible to set up two spherical waves to destructively interfere throughout all of space. But if somehow you could do it, would you have violated energy conservation? Or would it just affect the sources and cause power dissipation there, like you said? If so, how does that work out for, say, two oscillating point charges ?

 

[Dale]

I get that it's practically impossible to set up two spherical waves to destructively interfere throughout all of space. But if somehow you could do it, would you have violated energy conservation?

It is not just practically impossible, it is impossible even in theory. Poynting's theorem follows directly from Maxwell's equations and the Lorentz force law:
http://en.wikipedia.org/wiki/Poynting's_theorem

It guarantees that you cannot have destructive interference everywhere in space. In free space, if there is destructive interference in one spot then there must be constructive interference elsewhere in order to keep energy conservation. You can only gain or lose EM energy through doing work on matter.

The thread is closed. We don't allow speculation about violating Maxwell's equations as a pretext for speculating about violating the conservation of energy.