The Ideal Struck String

The ideal struck string [297] involves a zero initial string displacement but 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. An example of ``struck'' initial conditions is shown in Fig. 4.9 for a striking hammer having a rectangular shape. Since $v^\pm = \pm f^\pm /R= \mp c y'^\pm$,^5.3the initial velocity distribution can be integrated with respect to $x$ from $x=0$, divided by $c$, and negated in the upper rail to obtain equivalent initial displacement waves [297].

Figure 4.9: Initial conditions for the ideal struck string in a velocity wave simulation.

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 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 $v_h$, the wave impedance force against the hammer is $-2Rv_h$. The factor of $2$ arises because driving a point in the string's interior is equivalent to driving two string endpoints in ``series,'' i.e., their reaction forces sum. If the hammer is itself a dynamic system which has been ``thrown'' into the string, 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 momentum transferred is given by the integral of the contact force with respect to 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 [467,59,167,55,463,154]. A commuted waveguide piano model including a linearized piano hammer is described in [439,490]. ``Wave digital hammer,'' models, which employ a traveling-wave formulation of a lumped model and therefore analogous to a wave digital filter [127], are described in [493,51,39]. The ``wave digital'' modeling approach is introduced in §N.1.