String Networks

I've always been interested in the vibrations of networks of strings. Take this as an example:

http://www.acoustics.org/press/151st/Leger.html

Obviously the strings will exhibit some sort of inharmonic spectra when they are plucked. I am not sure how to model this using equations though, as the normal equations of string vibrations (e.g. frequency is inversely proportional to length) do not apply here.

Any ideas on how to predict the inharmonic spectra of a given network of strings attached at a single point, if we are only given 3 lengths?

Ryan

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:
>
> I've always been interested in the vibrations of networks of strings. Take this as an example:
>
> http://www.acoustics.org/press/151st/Leger.html
>
> Obviously the strings will exhibit some sort of inharmonic spectra when they are plucked. I am not sure how to model this using equations though, as the normal equations of string vibrations (e.g. frequency is inversely proportional to length) do not apply here.
>
> Any ideas on how to predict the inharmonic spectra of a given network of strings attached at a single point, if we are only given 3 lengths?

If you want to solve this yourself, first read about the *derivation* of the usual string vibration equations. For example, http://www.people.fas.harvard.edu/~djmorin/waves/transverse.pdf . "Frequency is inversely proportional length" is not a fundamental law, but "an infinitesimal portion of the string satisfies F = ma" is fundamental. The starting assumptions of the derivation are the same for more complex string networks; only the conclusions resulting from them (i.e. the spectrum of vibrational modes) are different.

To begin with let's assume these are purely transferse modes of a planar string network, i.e. the strings are all in the x-y plane at equilibrium but they only move along the z direction when they vibrate.

Depending on the angles at which the strings meet, the tension in each string might have to be different. Actually if you work it out you get the beautifully simple relation that the tension in each string is proportional to the sine of the angle between the two other strings. (This comes from the fact that the intersection point isn't going anywhere, so the net force on it must be zero.)

The linear mass density and tension in each string give you the wave propagation speed, c = sqrt(T/mu), which is really all you need to characterize the part of the standing wave in that string.

Each normal mode of vibration has a sinusoidal time dependence, with a single frequency omega, and it also has a sinusoidal shape *in each uniform length of string*. The dispersion relation is omega = c_n k_n so the wavenumber k_n in string number n is inversely proportional to the propagation speed in that string.

The only additional facts you need to form a complete set of equations are the boundary conditions, but some of those are a little tricky.

One boundary condition is whenever a string ends at some fixed point, the amplitude of the wave there is zero. It's easy to build this into your ansatz by writing everything as sines of the distances from these endpoints.

Another boundary condition is that the strings are actually tied together at the intersection point, which means the amplitudes there must be equal. So you'd have some equation like
A1 sin(omega/c1*L1) = A2 sin(omega/c2*L2) = A3 sin(omega/c3*L3)
where A1, A2, and A3 are the amplitudes of the standing waves in each string, c1, c2, c3 are the propagation speeds, and L1, L2, L3 are the lengths.

This might seem like all the boundary conditions, but it's not enough. The missing one you need is that the intersection point between the strings has no mass, so the net force on it in the z direction must be zero. If there were only two strings meeting at a point this would mean their slopes would be equal, but when there are three strings their slopes can be different as long as the three strings are still in a plane.

You have the known parameters c1,c2,c3,L1,L2,L3 and want to solve for the amplitudes A1,A2,A3 and most importantly the spectrum of frequencies omega. This is four unknowns and three boundary conditions, but the overall normalization of A1:A2:A3 is meaningless; only the ratios matter. Therefore this is all the boundary conditions you need and you can get a bunch of discrete frequencies omega with their corresponding mode shapes.

Now, at the beginning I said assume purely transverse modes, but this is a major oversimplification. A simple guitar string actually has *two* modes at the fundamental frequency, two modes at the octave overtone, and so on. Each pair of modes is degenerate with each other because the string has circular symmetry. If the string goes along the x direction then vibrating in the y direction and vibrating in the z direction look exactly the same. With a network of three strings this is obviously not the case. If the network is in the x-y plane, you're going to have some purely transverse modes where all the vibration is in the z direction, but other modes where the vibration is in the x or y directions, causing some of the strings to change length as part of the vibration. For this reason you'd need to know more physical parameters to solve the problem, in particular the elastic modulus (Young's modulus times cross-sectional area) that describes how the tension changes in response to changing the length.

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> This might seem like all the boundary conditions, but it's not enough. The missing one you need is that the intersection point between the strings has no mass, so the net force on it in the z direction must be zero. If there were only two strings meeting at a point this would mean their slopes would be equal, but when there are three strings their slopes can be different as long as the three strings are still in a plane.

I don't get this part - are you saying that the intersection point doesn't move in the z direction?

> Now, at the beginning I said assume purely transverse modes, but this is a major oversimplification. A simple guitar string actually has *two* modes at the fundamental frequency, two modes at the octave overtone, and so on. Each pair of modes is degenerate with each other because the string has circular symmetry. If the string goes along the x direction then vibrating in the y direction and vibrating in the z direction look exactly the same. With a network of three strings this is obviously not the case. If the network is in the x-y plane, you're going to have some purely transverse modes where all the vibration is in the z direction, but other modes where the vibration is in the x or y directions, causing some of the strings to change length as part of the vibration. For this reason you'd need to know more physical parameters to solve the problem, in particular the elastic modulus (Young's modulus times cross-sectional area) that describes how the tension changes in response to changing the length.

I thought the additional mode could be negligible. If it doesn't do anything except change the length/tension, then doesn't that mean it creates sort of a chorus/tremolo effect?

Ryan

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
> > This might seem like all the boundary conditions, but it's not enough. The missing one you need is that the intersection point between the strings has no mass, so the net force on it in the z direction must be zero. If there were only two strings meeting at a point this would mean their slopes would be equal, but when there are three strings their slopes can be different as long as the three strings are still in a plane.
>
> I don't get this part - are you saying that the intersection point doesn't move in the z direction?

No, I'm not saying anything about its motion at all.

F = m a

m = 0

therefore F = 0

This doesn't depend on the acceleration a at all. The point could be moving or remaining still, doesn't matter. The net force on it has to be zero regardless.

> > Now, at the beginning I said assume purely transverse modes, but this is a major oversimplification. A simple guitar string actually has *two* modes at the fundamental frequency, two modes at the octave overtone, and so on. Each pair of modes is degenerate with each other because the string has circular symmetry. If the string goes along the x direction then vibrating in the y direction and vibrating in the z direction look exactly the same. With a network of three strings this is obviously not the case. If the network is in the x-y plane, you're going to have some purely transverse modes where all the vibration is in the z direction, but other modes where the vibration is in the x or y directions, causing some of the strings to change length as part of the vibration. For this reason you'd need to know more physical parameters to solve the problem, in particular the elastic modulus (Young's modulus times cross-sectional area) that describes how the tension changes in response to changing the length.
>
> I thought the additional mode could be negligible. If it doesn't do anything except change the length/tension, then doesn't that mean it creates sort of a chorus/tremolo effect?

You must be confused. The things I'm talking about are real physical modes of vibration that produce sounds on their own. Each mode produces a sine wave at its corresponding eigenfrequency.

Keenan