# Wave equation with spatial damping

I would like to understand "spatially damped wave equation", $$\frac{1}{v^2} \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} + \beta \frac{\partial u}{\partial x}.$$ I am interested in applying this equation to a transversely oscillating string of finite length $L$ and fixed ends. Here, $v$ denotes the wave speed in the string and $u(x,t)$ is the elongation function.

Now, we usually take $\beta = 0$ and get the classical oscillating string. If $\beta \neq 0$, however, we would expect the elongation function to be damped along the $x$ axis of the string. What kind of physical mechanism would cause $\beta \neq 0$ in real life?

• What do you mean 'how would it appear?' Wouldn't it be the same? Or are you asking what $\beta$ would be? – Kyle Kanos Dec 11 '15 at 1:21
• Ok, sure - what would physically cause the beta term? – user17116 Dec 11 '15 at 6:14

## 1 Answer

That should be something like damping proportional to the angle of the string displacement (angle between equilibrium state and actual state). I am not aware of any model like this - however, that's not a proof. If you are looking for a first generalizations for string equations, than the stiff string is more practical in my opinion:

$$a \frac{\partial^2 u}{\partial x^2} - b\frac{\partial^4 u}{\partial x^4} = c \frac{\partial^2 u}{\partial t^2}$$

with $a,b,c$ being material constants (see the link).

• I agree that your suggestion is more studied, but I think the OP wants a dissipative example. – WetSavannaAnimal Jun 14 '17 at 23:35