Hand damping of a vibrating string or membrane Problem is the following: If I have a guitar string or a drum membrane which is vibrating (and thus creating sound), when I place my hand or finger on it looses energy quickly, and  eventually silences.
Now, when I want to model this, I can add a drag force $f_{drag}$ to the wave equation
$$\frac{\partial^2 u}{\partial t^2}-c^2\nabla^2u=f_{drag}$$ which would be non-zero only on the place where my hand is put.
Question is: for this kind of damping, what would be the correct $f_{drag}$ (linear to the speed, quadratic to the speed, or something completely different)?
 A: So the mechanism is that your finger is acting like a dashpot (a spring-damper system) that reacts on the vibrating member.
I don't know the specific stiffness and damping coefficient for a finger, and I suppose there might be different values for light touch where only the tissue interacts, vs moderate touch where the bones interact also.
But you can start with
$$ f_{\rm drag} = -k\, u|_{\rm A} -d\, \dot{u}|_{\rm A} $$
where point A is the point of contact.
The second part of modeling this is the fact that in the equation you posted $f_{\rm drag}$ is distributed over the entire vibrating object. But this is not what is going on. The interaction causes a sort of node (one that can displace maybe) but changes the shape of the waveform on each side.
For example, if the drag force is upwards at some instant, the waveform should look like this

This is because the tension in a string is always tangent to the shape, and the forces must balance at the node.
So what you have is two equations of motion and a force balance equation no model your system
$$\begin{aligned}
  \frac{\partial^2 u_1}{\partial t^2}-c^2 \frac{\partial^2 u_1}{\partial x^2} & = 0 \\
  \frac{\partial^2 u_2}{\partial t^2}-c^2 \frac{\partial^2 u_2}{\partial x^2} & = 0 \\
  T_2 \frac{\partial u_2}{\partial x}|_{\rm A} - T_1 \frac{\partial u_1}{\partial x}|_{\rm A} + f_{\rm drag} & = 0 \\
\end{aligned}$$
The subject matter is rather deep, and finding a solution of the above requires you to understand the concept of mechanical impedance and what happens when the impedance of the string matches the impedance of the finger. You can read articles on computer simulations of strings that go further into this.
The main point of my answer is to indicate that the shape of the vibrating object is non-smooth when it interacts with discrete objects (that is the slopes/gradient is non-continuous).
