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As the theory of superposition of waves express the amplitudes of the interfering waves do algebraically sum up. But when we sum up the the total energies of a particle doing a harmonic motion due to each wave, we get a different result from the value of energy that we get when when we calculate it with the resultant amplitude of the interfered wave. How can we combine these two expressions we get?

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  • $\begingroup$ Does the energy conservation hold in interference? $\endgroup$ – drvrm May 18 '16 at 18:15
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I don't think that energy considerations are particularly useful if you want to know the amplitude of the resultant wave. For example for a standing wave, depending on the location of the particle it can get different potential energies. The correct way is as you suggest, to sum up the waves. For example, if two counter-propagating waves interfere $$A(x,t) = \cos(kx-\omega t)$$ $$B(x,t) = \cos(-kx-\omega t)$$ The sum can be calculated with one of the trigonometric theorems: $$\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta.$$ So, $$A(x,t)+B(x,t) = \cos(kx)\cos(\omega t).$$

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You must first add the amplitudes and then square them to find the intensity.

It is easier at first to assume that the amplitudes of the superposing waves are the same $A$.
Then as the intensity is proportional to the amplitude squared the total intensity from the two sources individually is proportional to $2A^2$.
The interference pattern has a cosine squared variation ranging from $(2A)^2 = 4A^2$ at a maximum to zero at a minimum.
The average of a cosine squared function over a period is one-half the maximum so in this case the average intensity is proportional to $\frac 12 \times 4A^2 = 2A^2$.
So all that is happening is that the energy is being channel along preferred directions rather that travelling uniformly in all directions.

An equivalent analysis can be done if the amplitudes of the waves from the two sources is not the same.

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I think the question is in regards to energy conservation as a function of position. The power (time average of the amplitude squared) of each individual wave:

$\langle A(x, t)^2 \rangle + \langle B(x, t)^2 \rangle $

is not the same as the time average power of their superposition:

$\langle (A + B)^2 \rangle$

as a function of position. For instance, with counter propagating waves, it is always ZERO at nodes. One must then average over position to recover all the energy (and none more).

Thus, an interference pattern moves energy around spatially, while still conserving it globally.

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