9.5 DRUMS, CYMBALS, AND BELLS 165 FIGURE 9.15 An array of ideal springs and point masses, which serves as a model for a stretched membrane. Motions in and out of the page are the only modes of interest; There are 21 such modes here. As the number of masses and springs is made very large, this model becomes a good approximation to the continuous membrane. f2 = 1.59f, fy = 2.14f 's = 2.65/ 6 = 3.16/ 10 = 3.65/ A A = 2.30f 18 = 2.921, B = 3.50f. (c) fa = 3.60f FIGURE 9.16 (a) The first 10 natural modes of an ideal stretched membrane, arranged in families according to how many circular nodal lines are involved. (b) Side view of the shape of mode 4 along its diameter. (c) Corresponding pitches when model is By. Open note heads do not correspond closely to the equal-tempered scale. Modes 8, 9, and 10 all are within a semitone of the As shown, Higher modes contribute many additional pitches, continuing upward from these and crowded close together.166 CHAPTER 9 PERCUSSION INSTRUMENTS AND NATURAL MODES the "-" regions below; during the other half-cycle it is the other way around. In between there is a moment when the membrane is flat but adjoining regions are moving in opposite directions. There are now nodal lines where the membrane never moves up of down. These natural mode patterns can be confirmed experi- mentally, as shown in Figure 9.17. FIGURE 9.17 Photographs of a rubber membrane vibrating in several of its natural modes. Each pair represents the same mode at two times a half-cycle apart. (Courtesy of National Film Board of Canada and BFA Educational Media. )9.5 DRUMS, CYMBALS, AND BELLS 167 "The fundamental frequency for the membrane alone (with no air-loading effects) is fi=(0.765/d) vi/o, where d is the diameter, 7 the tension, and o the mass per unit area. Typical values for a Mylar tympani head (thickness 0.2 mm) are d == 0.6 m, o = 0.26 kg/m? and 7 = 2 x 103 N/m. For large values of n, a rough approximation of f, is provided by the expression 1.3 wi fi. Notice that higher modes of a drumhead crowd closer and closer together in frequency, unlike solid-bar modes, which spread farther and farther apart. To understand why drumhead vibrations do not generally sound very musical, suppose we translate the frequency ratios (with the help of Figure E and the Harmonic Series Slider) into musical notation. The results (Figure 9.16c) indicate a clamorous mixture for which it is difficult to assign any defi- nite pitch. Addition of a body to the drum creates a resonant cavity below the membrane; this alters both the mode frequencies and the efficiency with which they radiate so that the tone color changes. A judiciously shaped body even can make certain mode frequencies nearly harmonic so that a sound of definite pitch is produced. For further discussion of how this is done with tympani, see Box 9.1. BOX 9.1 Tympani Most drums do not produce a tone of definite initial thump, and thus is practically missing pitch, and we blame that on the lack of clear from the later persistent sound. That is good, musical relationships among the many natural because its frequency does not fit well into the mode frequencies shown in Figure 9.16. How desired pattern. is it, then, that in the special case of tympani Second, the usual striking point, half to we can obtain definite pitch? It turns out three-fourths of the way out from the center, is that this is the combined result of several near the circular nodal lines of many modes effects. such as modes 4, 6, 8, 9 in Figure 9.16). Thus First, the fundamental mode pushes all the the vibration recipe contains only small traces adjacent air in the same direction, unlike all of these modes; they contribute a little pun- the others that produce mainly a sloshing of gency to the flavor, but the burden of producing nearby air back and forth between the "+" a tone of definite pitch is left almost entirely to and " -" regions of the membrane. When the those modes that do not have circular nodal drumhead is mounted over a closed bowl, this lines (numbers 2, 3, 5, 7, 10, and 50 on). results in mode 1 radiating its sound to the Third, the frequency calculations must be outside world far more efficiently than any of corrected because those given in Figure 9.16 the others. Because of this strong radiation, mode 1 loses its energy rapidly in the strong (continued)