In calculus we can read the "normal derivative", $\frac {df}{dx}$, as the rate of change of our function $f$ with respect to $x$. With partial derivatives of multivariate functions it is very much the same, where computing $\frac{\partial f}{\partial x}$ of say $f(x,y)$ would be the the rate of change of $f$ with respect to $x$, except here we assume that $y$ is fixed, or constant, and vice versa for computing $\frac{\partial f}{\partial y}$.
How does one move on from this concept to reading a partial differential equation? That is to say, if you were to write it out fully in english, what would be the correct way to do so? One example I am interested in is the wave equation,
$$\frac{\partial^2u}{\partial t^2}=c^2\frac{\partial^2u}{\partial x^2}$$
Where $c$ is the speed at which the wave travels.
Best Answer
If the wave equation represents the vibration of a string, $u_{tt}$ is the acceleration of a point in the string, and $u_{xx}$ represents the force on the point of the string coming from the curvature of the string. So the equation represents Newton's law.
For the heat equation $u_t=k^2u_{xx}$, $u_t$ represents the variation of temperature, and $u_{xx}$ heat diffusion.
In each case you need to know ther physical problem the equation represents to adequately interpret it.