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Consider the wave equation,

$ y = A sin(kx-wt)$

I understand that taking the partial derivative of this function with respect to time gives the rate at which a particle at some distance 'x' oscillates.

However, I do not understand what it means to take the partial derivative of this function with respect to x. And, I'd also like to ask, what it means to take two derivatives of this equation with respect to the position and what it would mean to take one derivative with position then with time.

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Let’s first talk about partial derivatives. Say you have a function of two parameters $f(a,b)$. Now the parameters maybe independent or dependent by a constraint equation. What a partial derivative means is that you differentiate the function $f$ with respect to (one of) its variable keeping the other fixed.

So in the case of the wavefunction, taking a partial derivative with respect to time means you fix a position and look at how it varies in time. This can be seen as following a point in the gif.

enter image description here

Similarly a partial derivative with respect to space means you freeze an instant of time and look at the spatial variation of the function. Similarly this can be extended for multiple partial derivatives. This can be seen as freezing time and following the dots over space.

Coming to mixed partial derivatives, extending our way of thinking, we freeze time and look at the spatial variation rate then proceed to look at how that spatial variation varies at a particular point as time goes on. This can be seen as following how the line between the blue dots vary in time. That is if the blue dots are close enough (limiting behaviour).

The interactive graph can be found here.

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  • $\begingroup$ So a mixed derivative is like the time variation of the spatial variation? $\endgroup$
    – tryst with freedom
    May 14, 2020 at 11:31
  • $\begingroup$ Yup. In this case (as well as most cases in classical physics) the order of the partials don’t matter. So you could equally think of it as spatial variation of time variation. $\endgroup$
    – Superfast Jellyfish
    May 14, 2020 at 11:36
  • $\begingroup$ This may sound stupid but could you have like derivative by nudging both inputs instead of partial with time then partial with distance? Like, $f(x+\Delta x, t + \Delta t)$ $\endgroup$
    – tryst with freedom
    May 14, 2020 at 11:59

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