Recently, it was interesting to see a blog post on the Energy Paradox of Coherent Waves in Science Network.The lecture just talked about plasma waves, and I was also very interested in it, so I specialized in research.Many foreign websites have been discussed 2-5.Then, with a clear understanding of the matter, I felt that it should be recorded.
If there are two columns of monochromatic simple harmonics (which can be represented by the same trigonometric function -- E=A cos (wt - kz), the propagation direction, frequency, wavelength and amplitude (A) are equal, and the energy density is ~A2/2.The addition of two waves corresponds to the wave-to-wave structural interference, here it is complete or everywhere long interference, so this contradiction is called the coherent wave energy paradox.
At first glance, it is indeed puzzling, and even violates the iron law of energy conservation.The above-mentioned blog provides an explanation and feels that appropriate additional explanation is needed, which is clearer and easier to refer to.
Let's be clear: plane monochromatic simple harmonics exist only in mathematical transformation and analysis, not in the real world.This is because, by definition, such waves exist in infinite space-time, from infinity to infinity, the total energy is infinite, and nature cannot exist.
There is no doubt that real disturbances exist in finite space-time.According to Fourier analysis theory, finite amplitude disturbance can be decomposed into several monochromatic simple harmonics under certain conditions, each simple harmonic amplitude, frequency and wavelength can be different.If the disturbance satisfies the linear equation(system), the mathematical processing is simplified because of the arbitrariness of amplitude, the monochromatic simple harmonic obtained by decomposition must satisfy the same linear equation(system).This shows that monochromatic simple harmonic is introduced by mathematical linearization, and caution is needed when further introducing it.The paradoxical confusion is caused by the fact that the amplitude is a linear parameter and the energy as an amplitude square is a nonlinear parameter.
A simple analysis shows that a wave with amplitude A (called A wave) consumes ~A2/2 energy, and a wave with amplitude 2A consumes more energy than ~A2, which is higher than the energy required to produce two A waves.This can be done through the "digging" analogy: digging a hole 1 meter deep, need to drain the soil or overcome gravity to do 1 unit of work; digging 2 meters deep, need to do 3 units of work; digging n meters, need to do (n+1) n/2 units of work.It is clear that digging a 2 meter hole requires more energy than digging two 1 meter holes, or that the 2 meter hole "carries" higher gravitational potential energy.To generate waves, you also need to store enough potential energy to achieve the desired amplitude, so you can compare the amplitude with the depth of the hole to understand why the 2A wave carries more energy than the two columns of A wave.In the generation of 2A waves, it is necessary to go through stages from 0 to A to 2A, and the latter stage will be affected by the first stage, which will indeed cost more energy.
The above text shows that under the requirement of energy conservation, a column of 2A waves cannot be decomposed into two columns of A waves, and two columns of A waves cannot be directly combined into one column of 2A waves.The reason it appears to be decomposable is only mathematically so-called yes (A + A = 2A), or just amplitude, which is physically impossible due to the destructive energy conservation.
Different sources cannot produce two columns of simple harmonics with exactly the same phase, and there will always be differences in the direction of propagation or other parameters, so that the complete coherence phenomenon does not occur.
In addition to the above-mentioned complete phase length interference paradox, the complete destructive interference paradox can also be proposed: the two columns of monochromatic simple harmonics (A – A) in space-time are added everywhere.?Where is the conservation of energy?
Or it can be understood that two columns of simple harmonics, only the opposite phase, are represented as A cos (wt - kz), - A cos (wt - kz), which exist at every point in time and space, from infinity to infinity.
In fact, since planar monochromatic simple harmonic does not exist physically at all, or any meaningful disturbance can only exist in finite space-time, neither of the above interference scenarios can exist.The energy density of the two columns of coherent waves increases four times as much as before coherent, and on average, the energy in the whole space is twice as much as before coherent, corresponding to the sum of the energy in the two columns.
The superposition of two columns of "differentiated" monochromatic waves, as shown in the formula below:
A exp (ik1*z) + A exp (ik2*z) = 2A cos((k1 - k2)/2 * z) exp (i (k1 + k2)/2 * z)
The amplitude of the synthesized wave is 2A cos((k1 - k2)/2 * z), so the energy paradox only occurs when k1 and k2 are strictly equal; and as long as k1 and k2 are not equal, the phase length and elimination interference follow the cosine function.There is nothing wrong with it.
Summary: The planar monochromatic simple harmonic is derived from mathematical simplification and linearization, which can be decomposed into multi-column waves in mathematical form, but because energy is a nonlinear parameter corresponding to amplitude square.
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