On the sound field radiated by a tuning fork Daniel A. Russell Science and Mathematics Department, Kettering University, Flint, Michigan 48504 ͑Received 14 June 1999; accepted 25 April 2000͒ When a sounding tuning fork is brought close to the ear, and rotated about its long axis, four distinct maxima and minima are heard. However, when the same tuning fork is rotated while being held at arm’s length from the ear only two maxima and minima are heard. Misconceptions concerning this phenomenon are addressed and the fundamental mode of the fork is described in terms of a linear quadrupole source. Measured directivity patterns in the near field and far field of several forks agree very well with theoretical predictions for a linear quadrupole. Other modes of vibration are shown to radiate as dipole and lateral quadrupole sources. © 2000 American Association of Physics Teachers. I. INTRODUCTION II. THE TUNING FORK AS A LATERAL QUADRUPOLE If one rotates a sounding tuning fork once about its long A typical explanation of the sound field produced by a axis while holding the fork close to the ear, one finds four tuning fork goes something as follows.2,3,5,8 positions where the sound is loud, alternating with four po- 1 When the tines move outward, a compression is sent out in sitions where the sound is very quiet. The loud regions are the directions of A in Fig. 1͑b͒ and simultaneously a rarefac- indicated as A and B in Fig. 1͑b͒ and the quiet regions fall tion in the directions of B. As the tines move inward they approximately along the dotted lines. By rotating the fork send out a rarefaction in the directions of A and a compres- close to the end of a tuned resonance tube this phenomenon sion in the directions of B. These sets of waves are always in may be demonstrated to a group of people.2,3 If one listens opposite phase, and along the directions 45° from the plane very carefully, one finds that maxima in the plane of the fork of the tines ͑dotted lines͒ the compressions of one set of ͓regions A in Fig. 1͑b͔͒ are noticeably louder than those waves and the rarefactions of the other will coincide, and perpendicular to the fork ͑regions B͒. there will be silence. A very different pattern of loud and quiet regions is heard, As an example of a sound source which follows the above description, consider a cylinder whose radius alternately ex- however, when a sounding tuning fork is held at arm’s length 9 from the ear and rotated once about its long axis. Now, only pands and contracts according to two maxima are heard, both in the plane of the fork ͑regions r͑␪,t͒ϭRϩcos͑2␪͒sin͑␻t͒, ͑1͒ A͒, while minima are heard perpendicular to the fork ͑re- ͒ where R is the mean radius of the cylinder. Figure 2 shows gions B . The differences between near- and far-field sound one cycle of the radial oscillation of the cylinder. The cylin- patterns may be effectively demonstrated to a larger audience der simultaneously elongates in the horizontal direction, by using an inexpensive microphone and preamp connected pushing air outward, and contracts in the vertical direction to an oscilloscope. drawing air inward. Half a cycle later the cylinder contracts While several older acoustics texts describe the sound in the horizontal direction drawing air inward and expands in field radiated by a tuning fork,1–6 discussion of this phenom- the vertical direction pushing air outward. Figure 3 shows enon is absent from recent acoustics texts. Unfortunately, two frames from an animation created with those older texts which attempt to explain the sound field MATHEMATICA,10,11 which shows the sound field resulting close to the fork do so in terms of constructive and destruc- from Eq. ͑1͒. The animation is available as an animated GIF 12 tive interference effects. This would seem an implausible movie on the WWW. The two still frames differ by one explanation since the tines of a tuning fork are almost always half-period of the cylinder motion. It is clear to see that separated by a distance much smaller than half a wavelength waves propagating in the horizontal and vertical directions of the sound emitted, which means that interference effects have opposite phase, and that the waves completely cancel along lines 45° from the horizontal. should be noticed only at significant distances from the fork 7 A similar sound field is produced when four identical om- tines. Sillitto has shown that close to the fork it is path- nidirectional sources are placed in a lateral quadrupole ar- length-dependent amplitude differences which determine the rangement as shown in Fig. 4͑a͒. Such an arrangement of sound field and that path-length-dependent phase differences sources may be effectively demonstrated by passing a low only become dominant at large distances from the fork. frequency signal through four small boxed loudspeakers with The goals of this paper are to address some of the miscon- opposite polarities.13 The expression for the complex sound ceptions concerning the sound field radiated by a vibrating pressure amplitude produced by a lateral quadrupole may be tuning fork, to discuss the quadrupole nature of the tuning derived as14,15 fork sound field, and to present experimental measurements A 3 i3k of the near-field and far-field radiated patterns. While pri- ͑ ␪͒ϭ ͩ Ϫ 2Ϫ ͪ ␪ ␪ ͑ ͒ p r, 2 k sin cos , 2 mary attention will focus on the sound field produced by the r r r principal mode of the tuning fork, as shown in Fig. 1͑a͒, where A is an amplitude factor which depends on the size, radiation patterns from other less frequently observed vibra- strength, and frequency of the quadrupole source. The pres- tional modes will also be discussed. ence of the ‘‘i’’ in the third term in parentheses indicates that 1139 Am. J. Phys. 68 ͑12͒, December 2000 http://ojps.aip.org/ajp/ © 2000 American Association of Physics Teachers 1139 Fig. 3. Frames from an animated GIF movie ͑Ref. 12͒ animating the sound field produced by an oscillating cylinder according to Eq. ͑1͒. Fig. 1. ͑a͒ Principal mode of a tuning fork. ͑b͒ End view of a tuning fork showing regions of loud and quiet. III. THE TUNING FORK AS A LINEAR QUADRUPOLE this term is 90° out of phase from the other terms. The vari- When a tuning fork vibrates in its fundamental mode, the ϭ ␲ ␭ able k is the wave number (k 2 / ) and r is the distance to tines oscillate symmetrically in the plane of the fork, as the observer. The expression in Eq. ͑2͒ is valid for all dis- shown in Fig. 1͑a͒. Each individual tine might be modeled tances r. At large distances r ͑termed the far field͒ the pres- by a dipole source,17,18 much like an unbaffled loudspeaker19 sure amplitude may be approximated by and a transversely oscillating sphere14 are treated as a dipole Ak2 sources. Dipole sources are discussed further in Sec. V. p͑r,␪͒ϭ sin ␪ cos ␪. ͑3͒ A combination of two dipole sources with opposite phase, r such that the dipole axes lie along the same line, is called a ͑ ͒ ͑ ͒ linear, or longitudinal quadrupole source. The source distri- The angular dependence is the same for Eqs. 2 and 3 . ͑ ͒ Moving from near field to far field has no effect on the an- bution for a linear quadrupole is shown in Fig. 5 a . The exact expression for the pressure field radiated by a linear gular dependence of the directivity pattern. 14,15 Figure 4͑b͒ shows a polar plot of the directivity pattern for quadrupole may be derived as a lateral quadrupole source. For this and all following plots, A ik 1 k2 k2 the pressure amplitude is plotted on a logarithmic scale, with p͑r,␪͒ϭ ͫ͑1Ϫ3 cos2 ␪͒ͩ Ϫ ϩ ͪ Ϫ ͬ. ͑4͒ r r r2 3 3 units of decibels. In addition, all plots have been normalized to the maximum value, as is the accepted practice for direc- The result by Sillitto7 gives an equivalent expression in terms tivity plots.16 The directivity plot shows that the lateral quad- of Legendre polynomials and spherical Bessel functions. rupole pressure field exhibits four directions where sound is Figure 5͑b͒ is a polar plot showing the directivity pattern for radiated very well alternating with four directions in which a longitudinal quadrupole with an observer distance r no sound is radiated. This directivity pattern matches nearly ϭ0.05 m from a 426-Hz source, so that krϭ0.39. The prod- exactly the sound field predicted by the cylindrical source as uct kr is typically used to define the boundary between the shown in Fig. 3. near field and far field of a source. As a general rule of A lateral quadrupole source model has been used to de- thumb, if krϽ1 the observer is in the near field, and if kr 3,8 scribe the sound field of the tuning fork. While this might Ͼ1 the observer is in the far field. appear to explain what one hears in the near field of a tuning The near-field directivity pattern in Fig. 5͑b͒ matches what fork, there are several problems. First, this model does not is heard close to a tuning fork. There are four maxima, two explain why one hears only two maxima and minima when in the plane of the fork and two perpendicular. However, the the fork is held at arm’s length from the ear.