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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)?

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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

fig1

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).

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  • $\begingroup$ Tnx. I like your first equation for f_drag, but I find statements later on somewhat unusual, because finger is not a mathematical point. This seems more like holding the string with a nail or something. Shouldn't it be better to multiply the original equation for f_drag with the Gaussian f(x) to get f_drag(x)=f(x)f_{yourdrag} and solve the pde numerically? $\endgroup$
    – dpistalo
    Commented Feb 21, 2022 at 14:45
  • $\begingroup$ @dpistalo - that is a next-level representation of the interaction using continuum mechanics. I am not sure what this level of fidelity is going to achieve, but I bet it is going to be very close to the idealized model described above. Oh and the area of contact is finite and thus a Gaussian distribution is not a good fit. I would venture a half-cosine would be better, $\endgroup$ Commented Feb 21, 2022 at 15:19
  • $\begingroup$ @dpistalo see updates. $\endgroup$ Commented Feb 21, 2022 at 15:35
  • $\begingroup$ Updates are fine, tnx. I think I will go with the continuous distribution of the damping. It is for sure different: if you put a nail on the middle of the string it will destroy the odd harmonics but will not affect the even. On the other hand, a fat finger will be muting all the guys, but it will still mute the odd ones much faster $\endgroup$
    – dpistalo
    Commented Feb 21, 2022 at 15:56
  • $\begingroup$ A sound answer :) $\endgroup$
    – Arc
    Commented Feb 22, 2022 at 10:46

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