Sound/accoustics - guitar string question

[sophiecentaur]

Excel will do most things if I'm determined enough! Cheers.

 

[DragonPetter]

I had to jump out of this thread and pull my parachute string a while ago. There is too much of person A talks apples and person B talks oranges and person C tells person A he is wrong because his apple is not an orange or that he thought his apple looked like an orange but didn't see the difference, and it started to feel like I was eating an apple and it tasted like an orange. And in the middle of all of this, some people were talking mashed potatoes and claimed it tasted like a fruit salad.

 

[sophiecentaur]

I think at least two of us have sorted our ideas out now. I am pretty omnivorous when it comes to eating fruit.

 

[rbj]

How about this:
A guitar string can be quite accurately thought of as vibrating at a number of distinct frequencies, each at possibly different amplitudes. The lowest frequency is called the fundamental frequenciy, and the higher frequencies are integer multiples of the fundamental frequency. These various frequencies that the string vibrates at are called harmonics, the first harmonic being the fundamental, the second harmonic being twice that, etc. If you pluck a string, all of its harmonics are excited, the lower harmonics being the strongest.

that is not necessarily the case. if you pluck your guitar string in the very center of it, only the odd-numbered harmonics are excited. if you pluck the string while constraining the nodal point in the very center of the string, then only even-numbered harmonics will be excited. if you pluck the string while constraining the nodal point 1/3 of the string length from either side, only harmonics that are integer multiples of 3 will be excited.

do you understand that, while the wave equation of the string supports all of these harmonics possibly existing (with non-zero amplitude), there is more to the complete solution of a differential equation than just solving the diff. eq.: there are also the initial conditions that determine the complete result.

 

[Rap]

that is not necessarily the case. if you pluck your guitar string in the very center of it, only the odd-numbered harmonics are excited. if you pluck the string while constraining the nodal point in the very center of the string, then only even-numbered harmonics will be excited. if you pluck the string while constraining the nodal point 1/3 of the string length from either side, only harmonics that are integer multiples of 3 will be excited.

Well, I wouldn't call raising one point while constraining another "plucking". But even if you define plucking as simply raising one point and letting go, you are still correct - when the distance to the pluck point divided by the length of the string is a rational number, not all harmonics will be excited. Let's amend the statement to read "If you pluck a string, then various harmonics are excited ..."

Of course, the probability of plucking exactly a rational number is zero...

To SophieCentaur - I changed the problem to one where the first oscillator has a fixed velocity at time zero, rather than a fixed offset. The math seems simpler. The easiest way to express the solution is $$x_1=\sum_{n=1}^2 \sum_{m=1}^2 A_{mn}e^{i \omega_{mn} t}$$$$x_2=\sum_{n=1}^2 \sum_{m=1}^2 B_{mn}e^{i \omega_{mn} t}$$ where we are using four terms instead of two terms with phases. The natural frequency of the first oscillator is $1-\delta/2$, the natural frequency of the second is $1+\delta/2$, $\gamma$ is the damping constant, and $\epsilon$ is the coupling constant (same for both oscillators). The four frequencies are: $$\omega_{mn}=\frac{(-1)^m}{2}\sqrt{1-\gamma^2+(\delta/2)^2+\epsilon+(-1)^n\sqrt{\delta^2+\epsilon^2}}$$ You can see that when the damping and coupling go to zero, you get back the two resonant frequencies, and their negatives. The amplitudes are: $$A_{mn}=\frac{1}{2\omega_{mn}}\sqrt{i-\frac{(-1)^n i\delta}{\sqrt{\delta^2+\epsilon^2}}}$$ and $$B_{mn}=\frac{-i\omega_{mn}(-1)^n}{\sqrt{\delta^2+\epsilon^2}}$$
Note I might have made a mistake, so I will keep going over this until I am sure its right, but this is a start. To get the real signals, just take the real part of the $x_1$ and $x_2$. I use $\delta=0.1, \gamma=0.05, \epsilon=0.02$ to get a nice plot from 0 to 50 seconds. (I use the direct solution without simplification, and the errors I might have made are in the simplification).

 

[Rap]

I kind of figured those numbers were wrong. The four frequencies are: $$\omega_{mn}=-(-1)^n\sqrt{1-\gamma^2+(\delta/2)^2+\epsilon+(-1)^m\sqrt{\delta^2+\epsilon^2}}$$ and the amplitudes are: $$A_{mn}=\frac{-i}{4\,\omega_{mn}}\left(1-\frac{(-1)^m\delta}{\sqrt{\delta^2+\epsilon^2}}<br /> \right)$$ and $$B_{mn}=\frac{-i}{4\,\omega_{mn}}\left(\frac{(-1)^m\epsilon}{\sqrt{\delta^2+\epsilon^2}}\right)$$

 

[PhilDSP]

Hi all,

Just looking at this thread for the first (and probably last) time. But I thought it would be worth mentioning that body resonances are very important for acoustic and hollow body electric guitars. And of course for other string instruments. If I remember correctly, all are pretty much configured so that the primary body resonance occurs near the second lowest open string frequency or one octave away. In Violins, especially, a mark of quality is how close the body resonance matches that characteristic. If the resonance becomes overly pronounced it is called a wolf tone and can be difficult for a player to control.