The Ideal Struck String

The ideal *struck string* [#!Morse!#] can be modeled by a *zero*
initial string displacement and a nonzero initial *velocity*
distribution. In concept, an inelastic ``hammer strike'' transfers an
``impulse'' of momentum to the string at time 0
along the striking
face of the hammer. (A more realistic model of a struck string will be
discussed in §9.3.1.)
An example of ``struck'' initial conditions is
shown in Fig.6.10 for a striking hammer having
a rectangular shape. Since
,^{7.6}the initial velocity distribution can be integrated with respect to
from
, divided by
, and negated in the upper rail to obtain
equivalent initial displacement waves [#!Morse!#]. Interestingly,
the initial displacement waves are not local (see also Appendix E).

The hammer strike itself may be considered to take zero time in the ideal case. A finite spatial width must be admitted for the hammer, however, even in the ideal case, because a zero width and a nonzero momentum transfer sends one (massless) point of the string immediately to infinity under infinite acceleration. In a discrete-time simulation, one sample represents an entire sampling interval, so a one-sample hammer width is well defined.

If the hammer velocity is
, then the force against the
hammer due to pushing against the string wave impedance
is
. The factor of
arises because driving a point in
the string's interior is equivalent to driving two string endpoints in
``series,'' *i.e.*, the reaction forces sum. If the hammer is itself a
dynamic system which has been ``thrown'' into the string,
as discussed in §9.3.1 below, the reaction
force slows the hammer over time, and the interaction is not impulsive, but
rather the momentum transfer takes place over a period of time.

The hammer-string collision is ideally *inelastic* since the
string provides a reaction force that is equivalent to that of a
dashpot. In the case of a pure mass striking a single point on the
ideal string, the mass velocity decays exponentially, and an
exponential wavefront emanates in both directions. In the musical
acoustics literature for the piano, the hammer is often taken to be a
nonlinear spring in series with a mass, as discuss further in
§9.3.2. A commuted waveguide piano model including a
linearized piano hammer is described in §9.4-§9.4.4.
``Wave digital hammer'' models, which employ a traveling-wave
formulation of a lumped model and therefore analogous to a wave
digital filter [#!FettweisMain!#], are described in
[#!VanDuynePierceAndSmithWDH!#,#!BilbaoEtAlASA02!#,#!BensaT!#]. The ``wave
digital'' modeling approach is introduced in §F.1.

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