Why does the irradiance of two intersecting waves not contradict the conservation of evergy?

Question

I am currently working through Optics by Eugene Hecht. When two waves intersect together, Hecht shows that the total irradiance $I$ is $$I=I_1+I_2+\sqrt{I_1I_2}\cos{(\delta/2)}$$ where $I_1$ and $I_2$ is the irradiance of each separate wave and $\delta$ is the phase difference between the two waves.

When we consider the irradiance of the two waves to be the same and we have no phase difference , we see that the total irradiance is $$I=4I_0$$ Why does this not contradict the conservation of energy. When the two waves are separate we have a total of $I_0+I_0=2I_0$ irradiance in the room. So how do we double that?

Answer 1

You are picking up just the phase point where the fields interfere constructively, meaning that the amplitude add up, but in the total volume where the field is present there are points where the electric field of the two waves interfere destructively and other constructively: $I = 2I_0(1 + \cos\chi) = 4I_0(\cos^2(\frac{\chi}{2}))$. Then the intensity oscillates between 0 and $4I_0$. Intensity is $cu_V$, where i call $u_V$ the energy density per volume. The total energy is obtained considering the total volume, not a single plane where the waves interfere in a particular manner. No conservation of energy is violated.

Answer 2

This is a long comment:

Here is a simple plane electromagnetic wave:

Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This 3D animation shows a plane linearly polarized wave propagating from left to right. The electric and magnetic fields in such a wave are in-phase with each other, reaching minima and maxima together

What happens to energy at the nodes, where both E and B fields are zero?

Let us look at the definition of energy in Maxwell's theory. It is found in the Poynting vector

Electromagnetic waves carry energy as they travel through empty space. There is an energy density associated with both the electric field E and the magnetic field B. The rate of energy transport per unit area is described by the vector

......

Finally

The rate of energy transport S is perpendicular to both E and B and in the direction of propagation of the wave. A condition of the wave solution for a plane wave is $Bm = Em/c$ so that the average intensity for a plane wave can be written

Note, the "average" intensity . That takes care of the problem of energy conservation at the nodes.

As irradiance is defined in CGS unit "erg per square centimetre per second" it cannot be compared with the power carried by the wave, one is instantaneous, the other is average.

Conservation of energy becomes simple when we understand that an electromagnetic wave is a marvelous mathematical confluence ( superposition)of the wave functions of a great number of photons. I am fond of this double slit experiment one photon at a time that shows how the frequency effects appear when the number of photons gets into the thousands.

No doubts about conservation of energy as each photon carries an $energy= hν$ where $ν$ is the frequency of the classical wave that is built up by the photons.