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
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
to indicate
that it is an additive incremental input, superimposing with the
existing string state.
For idealized plucked strings, we may take
(acceleration), and
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
.
The applied force at a point can be translated to the corresponding
velocity increment via the wave impedance
:
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(7.15) |
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.