Experimental measurements have been made of the motion of a red brass harpsichord wire driven electromagnetically in a fixed direction perpendicular to the equilibrium position of the wire. The motion is complex compared to that predicted by simple linear theory because of effects due to tensional changes and longitudinal motions. Optoelectronic detectors are used to measure amplitude and phase of the transverse motions as functions of the driving frequency, both in the driving direction y and the direction z perpendicular to y. Near the free-vibration fundamental frequency f0 the z and y amplitudes are comparable even for a very low driving force and amplitude. Amplitude jumps and hysteresis effects are observed for large amplitudes. The z–y phase difference is measured as 0°, 90°, and 180° in different frequency regions, yielding both planar and whirling or tubular motion. As the driving frequency increases, the phase difference between the driving force and the y motion varies steadily from 0° to 90° before jumping to 180°. There is no evidence of a critical frequency of onset of the z motion as is predicted in some theoretical treatments. Similar effects are observed near 3f0, for which amplitude measurements have been made down to 0.01 μm for a 0.71-m-long wire.

A stationary dew drop assumes a shape that minimizes the sum of its gravitational and surface energies. This shape, however, cannot be described by a mathematical function in closed form because the governing differential equations of the drop do not admit of an analytical solution but must be integrated numerically to yield the drop profile. Hence, this problem provides a good case for using the calculus of variations to obtain the governing differential equations and presents advanced students with an opportunity to gain insight and experience in numerical computations involving a real and interesting physical problem. In this paper we obtain the differential equations governing the shape of the drop in two ways: (i) by considering the balance of forces on the drop in static equilibrium; (ii) by seeking the profile which minimizes the total energy of the drop through the use of the calculus of variations. The resulting differential equations are then integrated numerically to obtain the theoretical profile. Two computational strategies, one for each of the two classes of wetting and nonwetting drops, are discussed. Further, it is shown how one may obtain the values of the surface tension and the contact angle for a liquid drop by matching the computed profile to the experimentally observed shape.

Some effects of the asymmetries causing a splitting of the fundamental natural vibrating frequency of a wire have previously been reported [Hanson et al., J. Acoust Soc. Am. 103, 2873 (1998)]. It has been demonstrated in this work on brass harpsichord wire that the splitting of the frequency is due to intrinsic properties of the wire itself and not of asymmetries in the end clamps. The two frequencies are associated with two definite orientations with respect to the wire. These two vibrational directions have been determined to be orthogonal within one degree experimental uncertainty. This orthogonality is in agreement with predictions of a simple model which assumes that, for small-amplitude free vibrations, the observed portion of the wire moves under the action of a linear anisotropic conservative restoring force. Measured splittings for several samples of harpsichord wire have ranged from 0.12 to 0.30 Hz for a frequency of about 70 Hz. The nonlinear effects of generation of motion perpendicular to the driving direction [Hanson et al., J. Acoust. Soc. Am. 96, 1549–1556 (1994)] and generation of higher harmonics are profoundly influenced by the orientation of the driving direction with respect to these vibrational orientations for low-driving forces. Related effects for plucked strings will be discussed.

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It is shown how a simple electrical circuit may be made to operate as an impulsively excited damped harmonic oscillator having independently variable natural frequency and damping. The oscillator’s response is developed in a particularly transparent way through impulse analysis, a technique that is wholly within the time domain and provides an alternative to Fourier analysis in the frequency domain. Resonance is exhibited and interpreted using impulse analysis, and through resonance, contact is made with Fourier analysis.