Energy of the summation of two sinusoidal waves

[O.J.]

Suppose we have two laser diodes that are made to transmit light at the same wavelength and intensity. Also, suppose that we place them in an open space and separate them by a distance such that when their generated beams intersect at one point in space and one point only. Further suppose that they add constructively at that point.

Laser 1 = A sin wt
Laser 2 = A sin wt

Laser 1 + Laser 2 = 2 A sin wt

However the sum of the energies of the two waves does not equal the energy of the sum of the total. Can someone clarify that? It's been bugging me for a while.

 

[IxRxPhysicist]

Long story short, energy from an optical (electromagnetic) source is proportional to the irradiance at the point of measurement, in your case the point of constructive interference. The irradiance at the point of measurement is given by I_total = I_1 + I_2 + 2*sqrt(I_1*I_2)*cos(phase_1 - phase_2) where I_1 and I_2 are the irradiance of source 1 and 2 respectively. So the energy can actually be 4 times the original power of the source if I_1 = I_2 and w_1 = w_2, this is due to the phenomena of superposititon.

Brief source: https://en.wikipedia.org/wiki/Superposition_principle

But a mechanics book like Taylor will have a pretty thorough expose on the topic.

Cheers,
IR

 

[O.J.]

The energy (or power really) at the point of interference is twice the total energy/power of the two waves. How does that work in light of conservation of energy?

 

[nasu]

The interference take place in a finite area of space not at a geometric point.
You will have maxima and minima of interference. In some areas the energy is less than the sum of the two, in others is larger.
The total energy (over the whole area) is just the sum of the two energies.

 

[IxRxPhysicist]

Lasers put out a certain definite (more or less) power over a certain amount of time or space (depending on the situation or what you are interested in). Think of the laser as a hose with a fixed flow, ignoring the incompressibility of water, you can think of interference as a repositioning of power sort of like taking the hose and running the water through a funnel which concentrates the flow of water. So a fixed amount gets concentrated to a more specific area, this is what interference does.

Cheers,
IR

 

[O.J.]

My question is very fundamental: adding two sinusoids in phase results in a sinusoid with twice the total energy of both sinusoids combined according to how we calculate power as A^2/2 with A being the amplitude. None of this answers the extremely fundamental question of how is energy conserved??!

 

[nasu]

The "fundamental" answer is that if you double the amplitude of a wave, the power increases by a factor of 4.
This has nothing to do with conservation of energy. It's a "fundamental" property of waves. You need to use 4 times more power to double the amplitude.

If you are thinking of "interference" of two waves that are in phase everywhere this simply means that you have a single source and you just double the amplitude.

If you have two separate waves, produced by two sources, you cannot overlap them in phase everywhere in space. You will have maxima as well as minima.

 

[jtbell]

To put it another way, interference doesn't destroy or create energy. It simply redistributes the energy. Some of the energy is "moved" to other parts of the wave, from where it would have been if there had been no interference.