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I see that the retarded/causal Green's function for the 1+1D wave equation is $$ G(x, t \,|\, x_{0}=0, t_{0}=0) = \frac{1}{2c} H(t - |x/c|), $$ (where $H$ is the Heaviside step function) which satisfies $$ \partial_{t}^{2}G - c^{2}\partial_{x}^{2}G = \delta(x)\delta(t). $$ I understand how to derive this result mathematically (which is why this post is not on MSE), but it seems to violate my intuition.

The way I visualize this solution is that it is essentially a "box function" whose width is expanding at $c$ in both directions. Ok, so, is this colloquial description accurate so far?

If the solution really is a rectangular waveform that is expanding, how can we have the waveform simply expand like that with no other parts collapsing. If I imagine an idealized string, how can the idealized string just raise itself up that way in a self-sustaining matter? This is where my intuition fails.

Note that I realize this is working with an idealized model. I am looking for an answer that primarily deals with the idealized model itself. However, if you have additional comments for how this idealized situation does or does not translate to real life scenarios, I would love to hear them as well. (Can we create a situation where a string somehow raises itself after an impulse? If not, what part of the idealized case fails to translate to the real-life case?)

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    $\begingroup$ It might be worth pointing out that for a stretched string, there's no energetic cost associated with a segment of string that is simply lifted up in the middle; the string is unstretched in that segment (no additional potential energy) and it's not moving up or down (no kinetic energy). $\endgroup$ Jan 19 at 13:05
  • $\begingroup$ It's hard to imagine on a string, since you're used to differentiable or at least continuous functions. If you think at a pressure disturbance in a 1D acoustic problem, it should be less disturbing. And if you want to solve the problem with a numerical method, this should be able to treat discontinuities (eg: FVM methods maybe could be a better choice than FEM or FDM) $\endgroup$
    – basics
    Jan 19 at 13:13
  • $\begingroup$ @Michael Seifert but at the two ends of the segment the string is stretched? $\endgroup$
    – user45664
    Jan 19 at 18:23
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    $\begingroup$ @user45664: True, but I wasn't talking about the ends of the segment, only its interior. In the picture I was talking about you have a substantial energetic contribution from the ends of the "lifted" segment and zero energy elsewhere. $\endgroup$ Jan 19 at 19:02

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Can we create a situation where a string somehow raises itself after an impulse?

'roll casting' a fly fishing line: if we start with a fly line laying straight on the water and properly flick the rod tip in a circular motion to the left it will create a pulse that travels down the fly line shifting the whole line to the left as it travels.{transverse wave} (search: roll cast fly fishing) With a different flick can cause the fly line to move further out along the straight line. {longitudinal wave}

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The problem with the wave function: Its second order in time. So the initial conditions are $$( \phi(t=0,x) , (\partial_t \phi)(t=0,x)) $$

The classical waves in 2d are simply any linear combinations of functions of traveling left or right, eg as a delta-like pulse wave, good for Hilbert space representations

$$e^{-\frac{(t\pm x)^2}{2 \sigma^2}}$$

or a steplike

$$\text{erf}\left(\frac{(t\pm x)}{\sigma}\right)$$

in the tsunami case with different boundary values at $\pm \infty$. Boxlike waves can be constructed a differences

$$\text{erf}\left(\frac{(t\pm (x-a))}{\sigma}\right) - \text{erf}\left(\frac{(t\pm (x+a))}{\sigma}\right) $$

What, if the initial conditions are $\phi(0,x)=0, \ \partial_t\phi(0) = f(x)$?

There classical real valued wave theory and complex quantum theory go different paths, classically one integrates along a path in the inverse Fourier transform, separating advanced and retarded solutions.

A different path of integration in the complex case separates positive and negative frequency spaces are promotes them to particle and and antiparticle representation spaces.

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