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Assume a wave function $\psi = \psi(x,t)$ where $x$ is position from the starting point $(0,0)$ and $t$ is time. Two oscillating points A and B are located at $x_1$ and $x_2$ respectively with $x_2 \geq x_1$ and $x_1 > vt_1$ where $t_1$ is a moment in time. If I was to find their distance via using calculus could I do this? :

\begin{align}s(x_2,x_1) &= \int^{x_2}_{x_1} ds \\ &= \int^{x_2}_{x_1} \sqrt{(dx)^2 + (d \psi)^2} \\ &= \int^{x_2}_{x_1} \sqrt{(dx)^2 + \bigg(\frac{\partial \psi}{\partial x}dx \bigg)^2} \\ &=\int^{x_2}_{x_1} \sqrt{1 + \bigg(\frac{\partial \psi}{\partial x}\bigg)^2} dx\end{align}

Instead of finding $x = x_2 - x_1 $, $y = \psi(x_2,t_1) - \psi(x_1,t_1)$ and then $s(x_2,x_1) = \sqrt{x^2 + y^2}$

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Depends on what you are looking for. The integral will give you the (travel) distance along the (surface) of the oscillating material, and the second the span between two points regardless of the shape in-between. You can do both, but you have not provided enough information as to what the intent of this is?

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  • $\begingroup$ The exercise is to find the actual distance between two points (in the XY plane) in a wave but, for personal research purposes, I was thinking of a second way to do it. $\endgroup$ – bolzano Sep 17 '14 at 15:00
  • $\begingroup$ If you are an ant and want to go from A to B you care about the 1st number. If you have straight ruler you care about the 2nd number. It depends as I said. $\endgroup$ – ja72 Sep 17 '14 at 15:03
  • $\begingroup$ The line that joins A and B at this time moment. $\endgroup$ – bolzano Sep 17 '14 at 20:15
  • $\begingroup$ So then the 2nd only. $\endgroup$ – ja72 Sep 18 '14 at 1:02

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