THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

5.8 Boundary Conditions for Transverse Vibrations 107Figure 5.9. The first four transverse modes of a vibrating bar.which in turn gives a discrete set of allowable frequencies of the transverse vibration.The frequencies are given byf = ξ πκc8L 2where ξ = 3.0112 2 , 5 2 , 7 2 , 9 2 ,....The overtones are not harmonics of the fundamental.Figure 5.9 illustrates the transverse modes of a clamped-free bar and a free–freebar. In the free–free bar the modes correspond to the fundamental frequency f 1and all additional odd-numbered frequencies. The odd-numbered frequencies f 3 ,f 5 , and so on, are symmetric about the center of the bar. The slope ∂y/∂x is alwayszero at the center, which is a true antinode. But the even-numbered frequencies f 2 ,f 4 , f 6 ,...yield asymmetric modes of vibrations with respect to the center. In allmodes the nodal points are found to be distributed symmetrically about the center.A bar may therefore be supported on a knife edge (or held by knife-edged clamps)at any nodal point without affecting the mode of vibration having a node at thatpoint. A knife edge or a knife-edged set of clamps disallows displacement but nota change in the slope that occurs at the node.The xylophone consists of metal bars that are supported at the nodal locationsof the fundamental. But the nodes of the associated overtones are unlikely to belocated at the same points, and the overtones quickly die out, leaving the pure toneof the fundamental. The concept of a free–free bar applies to a tuning fork whichis essentially a U-shaped bar attached to a stem (Figure 5.10). The geometry of thefork and the mass-loading effect of the stem cause the nodes of the fundamental tobe spaced closely near the stem. When the tuning fork is struck, overtones damp