Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing. It only takes a minute to sign up.
Sign up to join this community
I am having a difficult time understanding the basic principles when sampling the function describing the vibration of a string. Mainly I am confused on given a sampling rate how many samples to take and what size the increments when sampling should be.
I am using the solution to the wave equation from this article (in german). If my explanation is unclear let me know and I'll try to give more information.
There's two kinds of sampling here, time sampling and spatial sampling, since the string equation is a PDE. I cannot read German, so I cannot understand your article. I will be presenting the discussion from the book, Physical Audio Signal Processing.
The ideal string PDE is $Ky'' = \epsilon\ddot{y}$, where $K$ is the string tension, $\epsilon$ is the mass density and $y$ is the string displacement.
Now if you use the FDTD method to solve this differential equation, the first and second order partial derivatives are $$y'(t,x) = \frac{y(t,x+X) - y(t,x)}{X} \\ y''(t,x) = \frac {y(t,x+X) - 2y(t,x) + y(t, x-X)}{X^2} \\ \dot{y}(t,x) = \frac{y(t+T,x) - y(t,x)}{T} \\ \ddot{y}(t,x) = \frac{y(t+T,x) - 2y(t,x) + y(t-T,x)}{T^2} $$
Here, $T$ is the time-domain sampling rate, and it can be chosen according to Keegs' answer, i.e., twice the frequency of the highest frequency string harmonic. To model all audible string modes, choose $T = \frac{1}{44,100}$s. $X = cT$ is the spatial sampling rate. Here $c = \sqrt{K/\epsilon}$ is the speed of sound.
Now, using $t = nT$ and $x = mX$, and using $c^2 y'' = \ddot{y}$, we have $$c^2 \frac{y(nT, mX+X) - 2y(nT, mX) + y(nT, mX-X)}{X^2} = \frac{y(nT+T,mX) - 2y(nT,mX)+y(nT-T,mX)}{T^2} \\ c^2\frac{y(n,m+1) - 2y(n,m) + y(n,m-1)}{c^2 T^2} = \frac{y(n+1,m) - 2y(n,m) + y(n-1,m)}{T^2} \\ y(n, m+1) = y(n+1,m) + y(n-1,m) - y(n,m-1) $$ We can solve this iteratively using some initial and boundary conditions, such as $y(t,0) = 0, y(0,x) = $ triangle function etc.
Assuming I understand the question correctly, you need to sample the string's position at least twice as fast as the maximum frequency you anticipate the string can move or the highest frequency you care about. So if the highest overtones the string produces are 16kHz, you need to measure at at least 32kHz if you want to be able to reconstruct the original motion faithfully. This is the Nyquist-Shannon sampling theorem and you can probably find several hundred descriptions of it here and elsewhere on the internet.
If you know the lowest frequency the string can produce or that you care about, there are other tricks you can do (look up heterodyne) to reduce your sample rate requirements assuming you have the equipment to do so on the analog side.
