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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?

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

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

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