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I've seen a couple derivations of wave equations, but I was never convinced that waves would actually exist.

Consider a spring with a compression pulse in a part of it (that is, assume it has an elongated region, followed by a compressed region, followed by an elongated region). It is rather surprising that the configuration of forces and speeds(*) in this situation is such that, after a small while, the pulse will be shaped the same but be a bit further on the spring.

I am looking for an intuitive explanation on why that would be so, or barring that, a derivation that is as clear as possible (preferably on a spring, or on the simplest medium of propagation possible, to avoid adding extra hypotheses).

(*) Speed seems to be important, since it probably decides which way a wave is traveling. I had never thought of that.

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Why don't derivations of wave equations convince you that they happen? If the laws of nature are equations whose solutions are wave-like, then waves happen. What about that argument fails to convince you? –  ACuriousMind Jun 30 at 21:18
what I mean is that I have no intuition about why they happen. The equations say so (and, for that matter, so do videos and simulations) but, usually, it is possible to antecipate the result of the equations with some intuition. In the case of waves, there is nothing of the sort (to me) –  josinalvo Jun 30 at 21:25
"I was never convinced that waves would actually exist" Surely you have looked at the surface of a pond when a rock has been tossed in; or played with a slinky; or strummed a guitar or ukulele? To expect intuition to proceed from math instead of from the real world is mostly doing it backwards. These things are, and you are free to demand that the math agree with that. The core notion is linear dependence of the return forces on the difference between the local value of something and it's value "next door". What "something" is depends on the nature of the wave. –  dmckee Jun 30 at 21:34
Adding to dmckee's response to "I was never convinced that waves would actually exist": josinalvo, don't treat physics as if it were philosophy. Philosophers still argue whether Zeno's paradoxes are valid. Physicists don't. Motion obviously happen, as so do waves. A physicist's job is to best describe that which is observed, preferably with mathematics. –  David Hammen Jun 30 at 22:48
I finding the task of imagining a world without wave phenomena remarkably difficult. Thus, I find it counter-intuitive that anyone would find waves counter-intuitive. Energy, initially stored here cannot get over there without being somewhere in between first. –  Alfred Centauri Jun 30 at 23:30

1 Answer 1

Picture a straight line of balls confined to a line with no gravity, friction or anything else. The balls are free to move around the line apart from the fact that every two neighboring balls are connected with an elastic spring which pushes them apart when they get closer and pulls them together when farther than a distance $a$.

When we take one ball on the left end of the line, and move it, it's neighbour on the right will copy it's movement due to the ellastic force between them. Force however only causes acceleration, so the neighbour of the ball we move will always reproduce the original movement only with a slight delay. But once the neighbour of our ball on the left end is disturbed from it's initial position, it also will cause his neighbour on the right side to copy his movement and so the movement is copied throughout the whole line of balls. When you think about it, this is practically a propagating pulse such as a wave and it arises solely from the fact that the spring forces two neighboring balls to copy their movement.

Now why does this converge to a wave equation? We call the axis of the original line $x$, the balls are put at equidistant $x_i,\, i=1,2,3...$ and the spring constant between them is always $k$. The equation of motion for the position $x_i$ is then $$\frac{d^2 \! x_i}{dt^2} = -k(x_i - x_{i-1}) -k(x_i-x_{i+1}) = k(x_{i+1}-2 x_i+x_{i-1})$$ Where we have accounted for the spring force of both neighbors and the last equality is just adding the two. Now we can investigate the displacement of the ball from it's initial position $\delta(i,t)=x_i(t)-x_i(0)$ and suppose we have so many tiny balls at so tiny distances we can actually just talk about the displacement of the ball at an original position $x$, so we have $\delta=\delta(x,t)$. Then our equation reads $$\frac{\partial^2 \delta}{\partial t^2} (x,t) = k(\delta(x+a,t)-2 \delta(x,t)+\delta(x-a,t))$$. $\delta(x+a,t)-2 \delta(x,t)-\delta(x-a,t)$ for $a$ supersmall converges to the second partial spatial derivative times $a^2$ (check this) $$\delta(x+a,t)-2 \delta(x,t)+\delta(x-a,t) \to a^2 \frac{\partial^2 \delta}{\partial x^2}(x,t)$$ So now we just have a wave equation for the super-tiny-ball-in-a-line displacement. $$\frac{\partial^2 \delta}{\partial t^2} (x,t) = ka^2 \frac{\partial^2 \delta}{\partial x^2}(x,t)$$ And all this just from a spring-like connection forcing the neighbours to copy each other. I feel it is pretty intuitive that this "pulse copying" or "pulse propagation" will happen also in this continuum limit and also in any processes governed by same equations.

It might seem surprising that the exact shape of the disturbance is propagated, i.e. that the wave-form does not change or decay in any way. In fact, it is just this spring-like "copy" force growing linearly with distance that allows the conservation of the wave-front, and most of the systems have wave equations like this only for small disturbances, i.e. neglecting non-linear forces.

Neglected dissipation also damps the disturbance, but may for example do so quicker for higher frequencies thus dumbing down sharp features of the wave-front quicker. I believe it is a right physical down-to-earth intuition that arbitrary wave-front conservation is just a very approximate concept.

Once non-linear forces become non-negligible and dissipation is introduced, there is no general wave-front conservation, but only special kinds of pulses may propagate without change which hit just the right combination of forces etc. - solitons.

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