At left in the photograph above, is a toy xylophone, which you can play by striking its bars with the two mallets behind it. Next is a tuning fork, whose frequency is 256 Hz, which you can hit with the mallet beside it. At right of the tuning fork is a cowbell, which has a ring by which you can hold it, and a clapper inside, which sounds the bell as you swing it. The two rods are singing rods. You coat your thumb and fingers with gum rosin, in the pouch behind the rods, hold a rod at one of the score lines, and then stroke it between your thumb and fingers. At right is a “Space Phone,” with which you can make interesting noises merely by holding the two horns and setting the spring in motion by giving one a jerk.

Idiophones are a class of percussion instrument, so named because the instrument, or the part that is struck, is itself the vibrating medium. Technically, the space phone shown above is probably not really an idiophone, but when you use it as described above, which is the easiest way, the spring provides the vibration. We have therefore kept it in with the other instruments, which clearly are idiophones.

When one sets such an instrument vibrating, its vibrations cause the air around it to vibrate in similar fashion, causing a compression wave to propagate through the air, which we hear as sound. In cases where the sound produced this way is too weak, the instrument can be made with some kind of resonator – a box or tube having such dimensions that a sound wave at that particular frequency undergoes constructive interference and is thus reinforced – appropriately placed with respect to the vibrating medium. This is done, for example, in marimbas and most xylophones, in which rows of hollow tubes of appropriate length are set underneath the rows of rectangular bars that sound when you strike them with mallets.

The xylophone is a bar instrument; that is, an instrument in which the vibrating medium is a set of rectangular bars. In an actual xylophone, the bars are made of rosewood or some kind of plastic, and each has a slight arch cut into its underside, to tune the overtones of the bar to provide the desired tone quality. The bars of this toy xylophone more closely resemble those of, say, a glockenspiel, whose steel bars are flat. When you strike one of these bars on the top surface, it undergoes transverse oscillation, that is, it vibrates in an up-and-down motion. (It could also undergo torsional, or twisting, motion, but these modes are usually not harmonically related to the fundamental.) The exact manner in which this oscillation occurs depends on where the bar is suspended, and whether or not the ends are free. (In the instruments mentioned above, and in this toy xylophone, the ends are free.) The frequency of oscillation depends (among other things) on the length and thickness of the bar, and on its elasticity, and for the transverse modes can be expressed as f_n = [(πvK)/(8L²)] m², where v is the speed of sound in the medium (= √(Y/ρ); Y, Young’s modulus, the ratio of the change in applied pressure to the resulting change in length, is a measure of the elasticity), K is the radius of gyration, which, for a flat bar is the thickness divided by 3.46, L is the length of the bar, and m equals 3.0122, 5, 7, . . . , (2n + 1). (So for the fundamental, n = 1 and m = 3.0122.) The range of this particular toy xylophone is from the C two octaves above middle C, to the C four octaves above middle C. Middle C (in an equal-tempered scale with A = 440 Hz) is 261.63 Hz. So this xylophone plays from the C at 1,046 Hz to the C at 4,186 Hz.

The tuning fork is quite useful, mostly for pitch reference. Though many musicians now use electronic tuners for their pitch references, there still many who use tuning forks. For its symmetrical modes of vibration, the lowest of which is the fundamental, its frequency is f_n = [(πvK)/(8L²)] m², where m equals 1.194, 2.988, 5, 7, . . . , (2n - 1). v, K and L are the same as for the previous equation, but refer to the characteristics of the individual tines. Because the tines of the fork, in the symmetrical modes, vibrate out of phase, the sound emitted by the fork is quite weak. When you place the handle of a vibrating tuning fork on a large piece of wood, or on a resonating box, like the one in the photograph, the vibrations of the fork are greatly amplified, and the sound is easily audible.

The cowbell in the photograph sounds at about 460 Hz, just shy of the B-flat above middle C (466.16 Hz), with a rather interesting set of harmonics mixed in.

The singing rods are rather interesting. As noted above, you coat your fingers and thumb with gum rosin, and then, holding a rod at one of the scored lines, stroke it between thumb and fingers. When you do this, your thumb and fingers alternately grip and slide on the rod as you move your hand along it, and you set up a longitudinal wave in the rod. This wave is similar to one in an organ pipe open at both ends. If you hold the rod in the center, you set a node there, and there is an antinode at each end of the rod. The rod sounds the fundamental. If you hold the rod one-quarter of its length from one end, the wave has an antinode at either end of the rod, one in the middle, and a node at one-quarter rod length from either end; the rod sounds the second harmonic. These longitudinal waves depend only on the speed of sound in the rod, which, as noted above, is v = √(Y/ρ).

The rods in the photograph are 3/8″ in diameter, or about 10 mm. One is 50 cm long, and the other is 75 cm long. The shorter rod has one score mark at the center, and the longer rod has three score marks equally spaced along the rod, so one at every quarter rod length. The short one sounds at about 4,975 Hz, which is close to the D-sharp just over four octaves above middle C (4978.1 Hz). The long rod, when held at the center, sounds at about 3,315 Hz, close to the G-sharp just over three octaves and a fifth above middle C (3,322.4 Hz). When held at either mark one-quarter of its length from the end, it sounds at about 6,630 Hz, an octave above that. The speed of sound, its wavelength and its frequency are related by v = fλ. Since these rods have resonances at all half wavelengths, the wavelengths associated with the above frequencies are, respectively, 1.0 m, 1.5 m and 0.75 m. Within significant figures, all three of these wavelengths, with their associated frequencies, give a speed of sound of 5.0 × 10³ m/s. Resnick and Halliday give 5,100 m/s as the speed of sound in aluminum (Resnick, Robert and Halliday, David. Physics, Part One, Third Edition (New York: John Wiley and Sons, 1977), Table 20-1 on p. 436), quite close to what we observe.

You can use the space phone in two ways. With sufficient tension on the spring, you should be able to communicate with a person at the other end by talking into, and listening with, your respective horns. Speech transmitted this way, unfortunately, is rather faint. What are clearly audible, however, are the interesting sounds that this device makes when you hold the horns with the spring suspended between them, and then you give one a shake. These sounds arise from longitudinal vibrations in the spring, which cause the diaphragm in each horn to vibrate, which vibration is amplified by the horn.

References:

1) For vibrations in bars and singing rods, see Rossing, Thomas D. The Physics Teacher 14, 546 (1976).
2) For the operation of a tuning fork, see Rossing, Thomas D., Russell, Daniel A. and Brown, David E. American Journal of Physics 60, 620 (1992).
3) Resnick, Robert and Halliday, David. Physics, Part 1, Third Edition (New York: John Wiley and Sons, 1977), pp. 435-6.