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I have looked everywhere for an answer but no one can provide an obvious account that explains all scenarios. Specifically, I want to know how the two interfering (constructive or destructive) waves can pass through each other, and end up on the other side unchanged in amplitude and "shape".

Is it possible to think of this as a one-dimensional wave instead of a real electromagnetic of mechanical wave, without losing any of the important parts of the process? I feel like this simplifies it for me and gives me a way to understand how physical waves can pass through another wave of the same nature and remain unchanged on the other side.

So let's consider two waves on a string. Say "Wave1" of an "amplitude" of +5, at its peak, is on the left, moving to the right. A second wave, "Wave2", of an "amplitude" of -2 is on the right, moving to the left. In other words, Wave1 is at a peak while Wave2 is at a trough. When the two waves, traveling towards each other, meet at the same location, a superposition is formed and their individual displacements add to form a net displacement. In this case, the combined wave will have an "amplitude" of +3 (... +5 + -2). After the waves go through each other, it is not clear to me how (after having passed through each other) the sum, "+3", can reform Wave1 on the right as +5 and Wave2 on the left as -2. Why don't they reform as something like +6 and -1 or +3 and -4, or as a function of an average of the two waves?

I hope I didn't provide an example that was too simplified. Also, I used to think I understood superposition but perhaps my confusion exists only because I simply don't understand it. Thanks!

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  • $\begingroup$ "Also, I used to think I understood superposition but perhaps my confusion exists only because I simply don't understand it." ::chuckles:: A 'maybe I don't understand this after all" moment is often associated with acquiring an appreciation for a deeper level of understanding. It's possible to have such moments repeatedly over the same topic, and our teaching pattern has some of them built in. $\endgroup$ – dmckee Jan 23 '17 at 22:59
  • $\begingroup$ Yes, it's another demonstration about how important it is that "learning" is accompanied with practical uses and not just definitions. The more practical applications, the more learned. $\endgroup$ – aiwyn Jan 26 '17 at 2:59
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Wave mechanics can be understood in 1 dimension and then extended to more than one dimension, as you are trying to do. However, you can't just look at the amplitude of the wave. You also have to look at the velocity as well.

The typical way of visualizing waves at first is that of a string which oscillates up and down. At any instant in time, every point on the string has a position, up or down. If we have a single sine wave on that string, it might look like $p(x,t)=A\cos(kx+\omega t)$ where $A$ is the amplitude of the wave, $k$ is a constant used to capture how "wide" the wave is, and $\omega$ is the frequency the wave is oscillating at. This equation is where you get your "amplitude 5" or "amplitude -2," the latter of which would be viewed as an "amplitude 2" which is 180 degrees out of phase at the time.

We can also take the derivative with respect to time, $\frac{\delta p}{\delta t}=A\omega\sin(kx+\omega t)=A\omega\cos(kx+\omega t - \frac{\pi}{2})$. We can use this to see that the velocity and position are always 90 degrees out of phase. We can think of this as what drives waves forward. If I have a point where the string is at a maximum amplitude, and look at the string to the left and right of it, I see one side which is currently moving towards maximum displacement, and the other side is currently moving away from it. If you think of the tension in the string, the tensions from this point are pulling one side towards its maximum displacement, and arresting the other side as it moves towards maximum negative displacement.

That's just how waves work. All of this and much more is captured by the general equation of waves: $\frac{\delta^2u}{\delta t^2}=c^2\frac{\delta^2u}{\delta x^2}$. Anything which is called a "wave" has an equation like that (or its multidimensional version $\frac{\delta^2u}{\delta t^2}=c^2\nabla^2u)$. As it happens to be, waves exhibit a mathematical property known as "superposition." This means if I have two waves, $f(x, t)$ and $g(x, t)$, I can know that $f\circ g(x, t) = f(x, t) + g(x, t)$, that is the result of combining two waves is just adding them. You just have to run the numbers to show that this property is true for all waves.

So for our waves in particular, we can say that your sum of two waves of +5 and -2 may lead to a +3 at the moment, but we also know that the velocities near that +3 point will be such that the waves will "pass through" each other. We won't see any +6 and -1, or any other combination. We know this because it is a wave, which means it obeys this superposition principle. We could run the numbers, sure, but it's easier to say "each of these waves can be handled on its own, and we can sum them in the end."

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The energy remains conserved at all times. Assuming a non-attenuated wave and a non dissipative medium, waves will have a constant amplitude and velocity along the entire string (1D), surface (2D), or volume (3D). Taking this as the basis, when constructive or destructive interference happens, the potential energy of compressing and stretching the medium, adds up. Think of it as two separate (signals) which are independent of each other. They propagate in medium by a kind of stretch and pull force which at time of constructive interference adds up and at times of destructive interference subtracts. This causes the amplitude to change, but the potential energy (and kinetic energy) still adds up. Once the waves have passed through the interference region, they are again showing their intrinsic frequency and amplitude.

One simpler way to think of this is by the following anology of a centrally fixed spring with two forces acting on it from opposite sides. Though the net force (amplitude) is lesser, the potential energy stored is the same, and when either of the force is stopped, and spring will get back to its initial length on that side while being compressed only on the other side, thereby allowing individual response to both forces (waves). It can be said that the medium itself divides into two parts for the waves to traverse through it, but we get to see the final composite at the time of interference.

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