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Let two waves moving in the direction from $D$ to $B$ and $C$ to $A$. These waves are out of phase at $O$. Then will the particles along $OA$ and $OB$ oscillate? If they do oscillate,then can you please explain why they will do so intuitively?

Both of the waves are of same frequency and amplitude.

The reason behind this question is that suppose the particles involved in the wave OC is pulling the particle O upwards whereas the wave OD is pulling it downwards,then the particle O would come to standstill.then how would the both waves propagate then? Will it get stopped there since these are cancelling each other?

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  • $\begingroup$ Which kind of wave is it? Does it only travel on the black lines? $\endgroup$
    – noah
    May 29, 2021 at 22:00
  • $\begingroup$ Any kind of transverse wave or electromagnetic wave propagating along the diagonals only $\endgroup$
    – MSKB
    May 30, 2021 at 4:08
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    $\begingroup$ Most/all mediums we use in science (air, water, EM field, etc) are assumed lossless. Too many math teachers teach waves adding and then cancelling, in physics superposition is temporary, a brief moment or second in time, the waves actually continue forever until absorbed. All waves are eventually absorbed: water waves crash oh the beach, sound is absorbed by materials, EM waves (can be called photons)are produced by electrons in individual atoms/molecules and are only absorbed by atoms/molecules. MUST conserve energy! $\endgroup$ Jun 3, 2021 at 21:12

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Discrete medium

Let's analyze the setup by placing masses at fixed intervals along the diagonals $\overline{AC}$ and $\overline{BD}$, such that one mass lands exactly on $O$. To make it easier to visualize, imagine the masses being mounted on guides such that they can only move up and down, so we consider transversal waves. Now connect all the masses to their nearest neighbors with springs. Every mass will be connected to two others, except at the points $A$, $B$, $C$, and $D$, which have only one neighbor, and $O$, which as four.

In this setup, it is quite obvious that if two waves are coming from $D$ and $C$, respecitvely, perfectly out of phase, one of the neighbors of $O$ (in the direction of $D$) will try to pull it up, while the other (from $C$) wants to pull it down with the same force. They cancel out and the mass at $O$ does not move. Because $O$ does not move, and all masses between $A$ and $O$, and between $B$ and $O$ are only connected to all other masses via $O$, none of these masses move.

This is one of the rare situations where the intuitive solution is correct, and one must think a bit more carefully about the mathematical reason. One might expect that, because of the principle of superposition for waves, that they might just pass through each other without interference.

The reason is that if there is only a wave travelling from $D$ towards $B$, it will split up at $O$ and travel to $A$, $B$, and $C$ with equal amplitudes. The same goes for a wave starting at $C$. If there are waves starting and $C$ and $D$, but out of phase, the waves coming from $D$ and the waves coming from $C$ completely cancel between $A$, $O$, and $B$ (superposition out of phase, or destructive interference).

Continuous media

The above conclusions do not straightforwardly translate to electromagnetic waves. If we imagine the lines as waveguides, as long as they have some finite width, the crossover region at $O$ is also of finite size, and the amplitude in that region will no longer exactly cancel, allowing waves to propagate through to $A$ and $B$. The very same argument also holds for water waves or any other setup where the structure of the medium is much smaller than the wavelength.

Only if we make the waveguides infinitely narrow, will we recover the above situation. This is consistent with the setup of masses connected by springs, because for infinitely small slits, the diffracted intensity becomes direction independent, and we again have cases of destructive interference on the sections $\overline{OA}$ and $\overline{OB}$.

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  • $\begingroup$ When Huygen formulated his wave theory of light as far as I know he basically was dependend on mechanical waves since the inclusion of aether atleast supports that. If we consider this fact then how will the concept of interference look like? Shouldn't there be some points between the screen and slit where a wave will meet with other ones in out of phase? So how would that intereference look like then? $\endgroup$
    – MSKB
    Jun 1, 2021 at 3:16
  • $\begingroup$ I am considering that the concept of electromagnetic wave is yet to be discovered for the above situation @noah $\endgroup$
    – MSKB
    Jun 1, 2021 at 7:56
  • $\begingroup$ Also it is to be mentioned that interference seems to happen in case of water waves too. So why does a water wave propagate even though it had met another wave in out of phase? $\endgroup$
    – MSKB
    Jun 1, 2021 at 8:02
  • $\begingroup$ I have elaborated on the case of a continuous medium, that should cover all your follow-up questions. $\endgroup$
    – noah
    Jun 1, 2021 at 10:37
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    $\begingroup$ I don't quite know what you mean. The question was whether the waves will propagate through to $A$ and $B$, and why. If you have more questions, feel free to post new ones, but don't tack them onto old ones. $\endgroup$
    – noah
    Jun 1, 2021 at 18:24
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If you are talking about sound waves of the same frequency from speakers at D and C, When out of phase the molecules at point O would be moving in a circle (vector sum of two longitudinal displacements changing with time). At other points within the square area the vector sum would be more complex as the amplitudes and phase differences change.

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  • $\begingroup$ If we think of a mechanical wave then at point O (out of phase) particles won't oscillate and this will cause the next particles to not to oscillate too. I could intuitively think of this. Then why would the motion of the particles in OA and OB be a bit complex? $\endgroup$
    – MSKB
    May 29, 2021 at 18:49

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