Meaning of the terms in Wave Equation for a Transverse Wave I read in the book that partial derivative of the wave function, $y(x,t)$ with respect to $x$ (here $x$ is the position of the particle on the $x$-axis) corresponds to studying the shape of the string at one instant.
The first partial derivative $\partial y/\partial x$ is the slope of the string and the second partial derivative $\partial ^2x/\partial y^2$ is the curvature of the string (in case of a transverse wave).
But what does the slope and curvature of the string actually mean? What do they represent? A diagrammatic explanation would be better.
 A: To understand the wave equation one has to both understand where it comes from and what it's consequences are; much like the heat equation.
To understand where it comes from, you must consider some string (for the 1D case), with some mass density on the string, and then assume that the string has some arbitrary curvature/shape on an interval (small interval) with tension. From there we can apply Newton's Second law, and represent the Tension on the string in terms of the the shape of the string on that small interval, and then use the small angle approximation to find the Tension on the string in terms of the the first derivative w.r.t position of the string at the endpoints (this is because the endpoints are "pulling" on the string). Of course this "infinitesimal variation in the first derivative" is the second derivative and so we find that the Tension in the string is proportional to the curvature of the string.
Moreover by Newton's second law we also know that the Tension is proportional to the "acceleration" of the string, in other words the second partial derivative w.r.t time. And obviously there are proportionality constants which are carried through all the math in the form of the mass density of the string and the applied tension.
Here is a link that describes the derivation of the Wave Equation: https://www.animations.physics.unsw.edu.au/jw/wave_equation_speed.htm#:~:text=Deriving%20the%20wave%20equation,-Let's%20consider%20a&text=It%20is%20stretched%20by%20a,the%20forces%20acting%20on%20it.&text=Fy%20%3D%20T%20sin%20%CE%B82%20%E2%88%92%20T%20sin%20%CE%B81.
In essence the wave equation does not start from "waves" it is merely Newton's Second law applied to a curved string, with a restoring force(Tension) present. Where the wave comes into play is through solving this PDE one obtains wave solutions.
This is really cool because it takes the idea of a "wave" and generalizes to any system with a natural restoring force contained in its field of propagation. And in fact it is from this that we can say this level of generality which Maxwell states "Light is an wave", he did not discover the laws of E&M, but noticed that the fields defined in the laws of E&M have restorative forces and so as a result variations in the field are essentially the same as variations on an infinite string
