The analysis of a vibrating string can be found in a variety of texts under a variety of subjects (music1, acoustics2, vibration3, general physics4). The problem provides an excellent introduction to second order dynamical systems.
Observe Fig. 1. This shows a string that is fixed at both ends (a boundary-value problem). As there is no applied force on the string, the net force f(x,t) must be zero. Notice that f(x,t) is a function of both the transverse location x and time t. We desire to derive a model for the displacement y(x,t) of the string. If the mass density &vepsilon; is distributed uniformly along the x-axis, and the tension T is constant along x, then if the maximum displacement y is small « length l, the resulting oscillation will be identical in space and time. Put another way, the resulting 'wave' (the shape of the string) can be viewed as a constant shape that 'moves' along x-axis.
This property allows us to express the displacement of the string as
y = -F(x-ct) + F(-x-ct)
(Morse p. 76), which is equal to 0 at x=0 (for our boundary conditions). With that general property in mind, we can go on to derive the actual function.
At any point x along the string, a free-body diagram can be drawn of a differential element of the string (see Fig. 2). If we look at the net force on the string we find that the force required to maintain dynamic equilibrium is T(Δsinϕ). Because the displacement is assumed small, sinϕ can be approximated by tanϕ, which is equal to dy/dx. The net force on this section of string is dx * Tδ(Δtanϕ)/δx. If we equate this force to the actual force of the string (mass &vepsilon;dx*acceleration), we obtain the resulting wave equation:
In our case, the string is fixed at both ends (x=0 and at x=l). Plugging that into our first equation for y(x,t) results in a periodic function. Because the solution can be written in the form y(x,t) = Y(x)F(t), the time-function can be expressed as a complex exponential: ejωt. Plugging that into the wave equation, then reducing the exponentials results in the following equation:
By applying the boundary condition at x=0, we can reduce that equation to a sin(ax)cos(bt-Φ) product. Applying the second boundary condition adds the constraint that the frequency must be an integer multiple of (c/2l), where c is the speed of the wave along the string (c2 = T/&vepsilon;), and l is the length of the string.
We can now write a general form for a single mode of vibration of the string. An example figure is shown below (Fig. 3)
This shows the first 5 harmonics for the guitar string vibration. Notice the points at which all waves equal zero. These are nodes.
Returning to the solution of the wave equation. Each mode of vibration (called a harmonic) is orthogonal, and each is a solution to the equation. So the final solution can be written as the sum of each mode as follows:
By observation, this is nothing more than a Fourier Series. The coefficients are determined by plugging in the initial conditions. We need to know the displacement of the string y0(x) and the velocity v0(x) at time t=0. This results in the following equations for coefficients Bm and Cm:
where the integrals are evaluated over the length of the string.
For our analysis, the guitar string is plucked at some point along the string x0. This can be treated with an initial displacement as shown below (Fig. 4), with an initial velocity of zero. Evaluating the integral for Bn, and observing that C = 0 (since initial velocity is zero), after working through, the displacement of the string y(x,t) results in the below equation:
For an indication of how well these equations model a real guitar signal, Fig. 5 shows
two waves: and actual signal, and the approximation generated by the above formula (taken
to a finite number of terms, of course). Notice that the equations do an excellent job
of indicating the shape of the wave.
A rigourous analysis of a non-conservative vibration will not be undertaken here. The mathematics are extremely complicated, and were not really vital to the goals of this project. However, there are a couple points of note.
The general effects of friction on the guitar string are not large. Observe Fig. 6. The overall amplitude of the vibration decays by little more than half over the length of the signal. Hence, in the global sense, we can assume that the time-constant of the wave is greater than the duration of the signal itself. The player is much more likely to stop the wave than to let it decay to a significantly smaller level.
On a more detailed level, though, friction has a profound impact on the shape of the wave over time. As with any damping, friction is proportional to the velocity of the string, so the wave equation above has an added velocity term. Additionally, the friction 'factor' is a function of the frequency of vibration. Higher frequencies are damped much faster than lower frequencies. Fig. 7 shows two portions of the wave from Fig. 6. One image is taken from the beginning of the wave, and the other is taken near the end. The difference is due to the fact that early on, there is a lot of energy in the higher harmonics, but by the time of the second, the lower harmonics contain all the energy.