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vided until much more recently. Boocock and Maunder (1969) developed a theoretical analysis, supported by experimental results, indicating that the presence of the octave at the stem is due to longitudinal inertia forces. They were able to explain Rayleigh’s (1912) observation that bending the tines (thus offsetting the longitudinal imbalance) reduces the strength of the octave component. Sönnerlind (2018) developed a detailed computer model of a tuning fork and found that the octave motion in the stem is likely due to a nonlinear relationship between the vertical movement of the center of mass of the fork and the displacement of the tines. His model shows that a double frequency (octave) occurs because the center of mass of the fork reaches its minimum position twice per cycle, when the fork tines bend both inward and outward. Sönnerlind’s model also indicates that the octave from the stem is more prominent for forks with longer tines and forks with tines having a square cross section (rather than a circular cross section).
The presence of the octave at the stem could affect the results of the Rinne and Weber hearing tests and the Rydel-Seiffer vibration sensitivity test because the stem is placed in contact with the skull, hands, or feet. This is why forks for assessing conduction hearing loss and nerve response to vibration are often fitted with weights at the tip of the tines because this reduces the presence of the octave at the stem.
The presence of an octave at the stem also has implica- tions for piano tuners; touching a 440-Hz fork to the piano soundboard will produce a 440-Hz tone along with a stron- ger 880-Hz octave, and the 880-Hz octave from the tuning fork stem will beat with the A 880-Hz piano string that is tuned slightly sharp due to the intrinsic inharmonicity in piano strings. This very problem was posed as a ques- tion to me during my graduate school days, and answering the question was the beginning of my fascination with the acoustics of tuning forks (Rossing et al., 1992).
Directivity Patterns, Quadrupole Sources, and Intensity Maps
When a tuning fork vibrates in its fundamental mode, the tines oscillate in opposite directions, with each tine acting as a dipole source such that the two oppositely phased dipoles combine to form a linear quadrupole source (Rossing et al., 1992). The linear quadrupole is an interesting sound source because the sound field at
near and far distances from the source exhibits distinct differences in directivity patterns, vector intensity maps, and the phase between pressure and particle velocity.
Quadrupole and Dipole Directivity Patterns
The nature of the quadrupole radiation may be demon- strated by rotating a tuning fork about its long axis while holding it close to the ear or near to the opening of a quarter-wavelength resonator tuned to the fork funda- mental (Helmholtz, 1885, p. 161). During one complete rotation, there will be four positions where the resulting sound is loud, alternating with four regions where the sound is very quiet; the sound will be loud when the tines are in-line with the ear and also when the tines are perpendicular. However, if the fork is held at arm’s length from the ear and rotated, only two loud regions will be heard, when the tines are in-line with the ear, and the previously loud regions when the tines are perpen- dicular to the ear will now be quiet. This variation in the loudness means that care must be taken regarding the orientation of the tuning fork tines with respect to the external auditory canal during the air conduction por- tion of the Rinne test (Butskiy et al., 2016).
Figure 4, a-c, compares the measured directivity patterns at increasing distances from a 426-Hz tuning fork with theoretical predictions for a linear quadrupole source. Measured sound pressure levels around a 426-Hz tuning fork vibrating in its fundamental mode agree very nicely with theory at all distances (Russell, 2000; Froehle and Persson, 2014). These data explain why one hears four loud regions when a fork is rotated close to the ear but only two loud regions when the fork is rotated at arm’s length. It also explains why, if you listen very carefully, the sound is noticeably louder (about 5 dB) when the tines are aligned with the ear compared with when they are perpendicular.
If a fork is rigidly clamped at the stem, it may be forced into several other natural modes of vibration that radiate sound as a dipole source or as a lateral quadrupole source. Figure 4, d-f, shows measured directivity patterns for a 426-Hz tuning fork that was clamped at the stem and driven at an in-plane dipole mode at 257 Hz, an out-of- plane dipole mode at 344 Hz, and a lateral quadrupole mode at 483 Hz. The measured data agree well with the theoretical predictions for dipole and lateral quadrupole sources (Russell, 2000).
52 Acoustics Today • Summer 2020

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