Ideal String Struck by a Mass

In §4.5, we discussed the ideal struck string modeled as a simple initial velocity distribution along the string, corresponding to an instantaneous transfer of linear momentum from the striking hammer into the transverse motion of a string segment at time zero. (See Fig.4.9 for a diagram of the initial traveling velocity waves.) In that model, we neglected any effect of the striking hammer after time zero, as if it had bounced away at time 0 due to a so-called elastic collision. In this section, we will consider the more realistic case of an inelastic collision, i.e., where the mass hits the string and sticks to it until something, such as a wave, pushes it away from the string.

For simplicity, let the string length be infinity, and denote its wave impedance by . Denote the colliding mass by and its speed by . It will turn out in this analysis that we may approximate the size of the mass by zero (a so-called point mass). Finally, we neglect the effects of gravity and drag by the surrounding air. When the mass collides with the string, our model must switch from two separate models (mass-in-flight and ideal string), to that of two ideal strings joined by a mass at , as depicted in Fig.4.20. The connections of the mass impedance with the two semi-infinite string endpoint impedances are formally in series because they all move together; that is, the mass velocity equals the velocity of each of the two string endpoints connected to the mass. (See §L.2 for a fuller discussion of impedances and their parallel/series connection.)

The equivalent circuit for the mass-string assembly after time zero is shown in Fig.4.21. Note that the string wave impedance appears twice, once for each string segment on the left and right. Also note that there is a single common velocity for the two string endpoints and mass. Impedances in series can be arranged in any order.

From the equivalent circuit, it is easy to solve for the velocity . Formally, this is accomplished by applying Kirchoff's Loop Rule, which states that the sum of voltages (``forces'') around any series loop is zero: