what is the intuition behind the one dimensional wave equations?

Question

In our PDE course, we learned one dimensional wave equation as $u_{tt}=cu_{xx}$(we ignore the initial conditions first.), where $u$ is the displacement function $u(x,t)$. My question is what is the physical meaning of $u_{tt}$ and $u_{xx}$? Can I understand $u_{tt}$ as the acceleration? What about $u_{xx}$? What is the physical meaning of this equation?

Answer 1

Imagine a tube with liquid pumped through it, for example, blood in a vessel. Denote $\,u\left(x\right)\,$ pressure at a point $\,x\,$ in the tube. Note that "pressure at a point" is, basically, equivalent to the amount of fluid being stored within an infinitesimally small portion of tube $\,\big[\, x-\Delta, \, x+ \Delta\,\big], \ \; \Delta \ll 1.\,$

If the liquid is pumped irregularly, like, for example, blood is pumped via pulses, than the pressure at a point (i.e. amount of a fluid within the neighborhood of a point) changes in time. This rate of change at a certain point as time goes is represented by temporal partial derivative $\,u_t = \partial u \,/ \partial t$.

In the example with blood in a vein it is clear that the first temporal partial derivative of pressure cannot be constant: otherwise pressure would keep increasing or decreasing non-stop, so the vein would either burst or collapse, which does not normally happen to us very often. Lucky us, the raise of blood pressure accelerates and decelerates regularly in our veins, allowing us to stay alive. This acceleration (positive or negative) is reflected by the second temporal derivative $\,u_{tt} = \partial^2 u \,/ \partial t^2$.

On the other hand, if we trace pressure along the length of at a fixed moment of time (kind of like is you froze the tube with liquid at an instant) then pressure would also change from one position to another. This is called spatial rate of change of pressure and is denoted $\,\partial u\,/\partial x\,$ Corresponding spatial acceleration is related to second spatial derivative $\,u_{xx} = \partial^2 u \,/ \partial x^2$.

Now, what the equation $\,u_{tt}=cu_{xx} \, $ tells you is that the spacial acceleration is proportional to temporal acceleration. This turned out to be the case for waves – thus the name "wave equation". In case of our example of blood pressure in a vessel we are talking about waves of pressure caused by the heart beats.