The Externally Excited String

Sections 6.5 and 6.6 illustrated plucking or striking the string by means of initial conditions: an initial displacement for plucking and an initial velocity distribution for striking. Such a description parallels that found in many textbooks on acoustics, such as [320]. However, if the string is already in motion, as it often is in normal usage, it is more natural to excite the string externally by the equivalent of a ``pick'' or ``hammer'' as is done in the real world instrument.

Figure 6.14 depicts a rigidly terminated string with an external excitation input. The wave variable $w$ can be set to acceleration, velocity, or displacement, as appropriate. (Choosing force waves would require eliminating the sign inversions at the terminations.) The external input is denoted ${\Delta w}$ to indicate that it is an additive incremental input, superimposing with the existing string state.

Figure 6.14: Discrete simulation of the rigidly terminated string with an external excitation.

For idealized plucked strings, we may take $w=a$ (acceleration), and ${\Delta w}$ can be a single nonzero sample, or impulse, at the plucking instant. As always, bandlimited interpolation can be used to provide a non-integer time or position. In the latter case, there would be two or more summers along both the upper and lower rails, separated by unit delays. More generally, the string may be plucked by a force distribution $f_p(t_n,x_m)$ . The applied force at a point can be translated to the corresponding velocity increment via the wave impedance $R$ :

$$\Delta v = \frac{f_p}{2R}$$ (7.15)

where $R=\sqrt{K\epsilon }$ as before. The factor of two comes from the fact that two string endpoints are being driven physically in parallel. (Physically, they are in parallel, but as impedances, they are formally in series. See §7.2 for the theory.)

Note that the force applied by a rigid, stationary pick or hammer varies with the state of a vibrating string. Also, when a pick or hammer makes contact with the string, it partially terminates the string, resulting in reflected waves in each direction. A simple model for the termination would be a mass affixed to the string at the excitation point. (This model is pursued in §9.3.1.) A more general model would be an arbitrary impedance and force source affixed to the string at the excitation point during the excitation event. This is a special case of the ``loaded waveguide junction,'' discussed in §C.12. In the waveguide model for bowed strings (§9.6), the bow-string interface is modeled as a nonlinear scattering junction.