# What does the solution to the transport equation describe?

I'm studying the transport equation

$$u_t + cu_x = 0$$

Where c is a constant.

What does the solution, $u(x,t)$, describe? Is it ... the height of the wave? The location of the wave? Does the transport equation only describe waves?

The source that I am studying from only says, "the wave, u(x,t) ... "

I know in the heat equation, the solution $u$ is temperature (dependent on position x and time t) and in the wave equation, the solution $u$ gives the height of the wave.

So, I'm just not sure of what the solution to the transport equation describes.

Thanks,

• I thought that, at position $x$, time $t$, the solution $u(x,t)$ gives the concentration. – Tom Ultramelonman May 30 '16 at 22:19

The quantity $u(x,t)$ can describe:
1. The height of a wave at point $x$ at time $t$
2. The temperature at point $x$ at time $t$
3. The concentration of some substance at point $x$ at time $t$
PDEs are not tied to physical quantities; they describe physical processes. The PDE $u_t+cu_x$ describes the process of propagation (of whatever) with constant velocity. The PDE $u_t = ku_{xx}$ describes the process of diffusion, of whatever is able to diffuse. It is not really correct to say that "in $u_t =k u_{xx}$, the function $u$ is temperature". It could be that, or it could be radioactivity level, etc. Conversely, the temperature could be modeled by $u_t=ku_{xx}$, or by $u_t = cu_x$, or by $u_t = cu_x + ku_{xx}$, or by many other equations. It depends on what processes one chooses to focus on.