Page 39 - Summer 2010

Fig. 2. The calculated amplitude of the string along its length, when plucked, and after about twelve reverberations showing the string “settling in” to a modal shape—a tone.
effect. The largest is that the strings are not completely flexible; they have some stiffness that causes the various frequency components in the wave to travel at different speeds.2 This dis- perses or spreads the wave shape as it propagates and is called “dispersion.”
Back to the jump-rope analogy. You may recall that it takes some force on your part to hold the end of the rope. For a plucked string, the force analogous to what your hand sup- plied is supplied by the membrane. I have calculated this force and show it in the lower part of Figure 2. This force, when transferred through the bridge to the membrane, caus- es the membrane to deflect and radiate sound. The force varies with time, and this variation depends on the shape of the wave hitting the bridge, and this, in turn, is determined by where the string is plucked. Figure 2 shows two illustrative cases: in the top case the string is plucked as shown in the top part of the figure. This is near where a player would normal- ly pluck. In the lower case the string is plucked mid-way
between the bridge and the nut. Plucked at the center, the ini- tial shape of the string is a little more like a (half) sine wave than the top curve; and so the imparted force and the shape settle in to a sine-wave shape more quickly, i.e., the string set- tles into vibrating at a specific frequency that we hear as a particular note. We call this a mode. The settling-in of the pluck to this mode is more apparent in the slow motion, accompanying, movie version of the evolution of the pluck in time. [See sidebar - Movie 1-StringResp] Finally, the settling- in process takes only a small fraction of a second and during that time, the vibration is not modal and, therefore, not a well-defined note. This is the “twang.”
The membrane
Like the string, the membrane is also under tension and has mass but in this case waves travel as circles expanding outwardly from their origin rather than along a line. This would be orderly enough except that their origin (the bridge) is not in the center of the circle. As a result, different parts of the expanding wave meet the boundary (the clamp) at differ- ent times. It is much easier to visualize this added complexi- ty by looking at the propagation of a very short wave. Such a wave would be impossible to generate experimentally, but in the world of mathematics, we can do pretty much anything. Thus, Fig. 3 shows a snapshot of such a calculated wave which has left the source and traveled to a point where part of the wavefront has reflected from the clamp. Note that the vertical scale is exaggerated and that the pulse goes from pos- itive amplitude (upward) to negative (downward) at the reflection. Every reflection at the clamped edge will cause such a reversal. Also, keep in mind that the wave loses ener- gy (and amplitude) as it travels and when it encounters either the clamp or the bridge. Again, the complexity can be seen more easily in the accompanying time evolution movie [Movie 2-ShortPulse].
The ability to generate such a non-physical pulse and to readily change material properties illustrates several of the major benefits of modeling. In this case it illustrates the com-
Fig. 3. A “snapshot” of a hypothetical short pulse on a banjo head. The pulse originated at the bridge (the vertical line) and has traveled as an expanding circle, except for that part of the wavefront which has encountered and reflected away from the back edge of the membrane.
38 Acoustics Today, July 2010