How to calculate wave equation from a stretched string?

Question

I am reading "Introduction to Electrodynamics" [Griffiths] and in section 9.1.1, there is an explanation for why a stretched string supports wave motion. It begins as follows:

It identifies 2 points, $z$ and $z + \Delta z$ in in the direction that the wave is propagating, and states the net transverse force on the segment between $z$ and $z + \Delta z$ is:

$\Delta F = T (sin\Theta' - sin\Theta)$

so far so good. Then, provided that the distortion of the string is not too great, tan can replace sine because the angles will be small so:

$\Delta F \cong T (tan\Theta' - tan\Theta) = T(\frac{\partial f}{\partial z} - \frac{\partial f}{\partial z})$

where each of the above partial derivatives are taken at points $(z + \Delta z)$ and $z$ respectively.

The next step I am unclear on the reasoning for. It says:

$T(\frac{\partial f}{\partial z} - \frac{\partial f}{\partial z}) \cong T \frac{\partial ^2 f}{\partial z^2} \Delta z$

I suppose I follow this in the sense that both the functions are going to be close to 0, but I feel that there's a more nuanced reasoning for this approximation that I'm not following.

Or perhaps it is that simple and I'm just more accustomed to a more easily geometrically visible reason for an approximation?

Answer 1

I'm going to notate the partials with subscripts and restore the points of evaluation. The step you're having trouble with is then written $$T(f_z(z+\Delta z)-f_z(z))\approx Tf_{zz}(z)\Delta z.$$

This is just from the definition of the derivative! When you compare a function (here $f_z$) at two nearby points (here separated by $\Delta z$), you approximate the derivative of the function: $$f_{zz}(z)\approx\frac{f_z(z+\Delta z)-f_z(z)}{\Delta z},$$ with the approximation becoming better as $\Delta z\to0.$ Multiply the above line by $\Delta z,$ then plug it in to your expression.

There's no need for $f_z(z)$ or $f_z(z+\Delta z)$ to be near zero for this step. This step is justified by

1. smallness of your choice of $\Delta z,$
2. smoothness of (the displacement of) the string, i.e. that $f$ is $C^2$

Answer 2

This is simply using

$$\frac{\frac{\partial f}{\partial z}\big{|}_{z+\Delta z}-\frac{\partial f}{\partial z}\big{|}_{z}}{\Delta z}\approx\frac{\partial^2 f}{\partial z^2}$$ which follows from the definition of the derivative. For simplicity call $g = f'$ and the formal definition of the derivative is$$ \lim_{\Delta z\rightarrow 0}\frac{g(z+\Delta z)-g(z)}{\Delta z}=\frac{\partial g}{\partial z}=\frac{\partial^2 f}{\partial z^2}$$ and so Griffiths removed the limit and called the formula approximate (where the approximation is better and better the smaller $\Delta z$ is). I assume later he will take $\Delta z \rightarrow 0$.