The existence of longitudinal waves in vibrating piano strings has been previously established, as has their importance in producing the characteristic sound of the piano. Modeling of the coupling between the transverse and longitudinal motion of strings indicates that the amplitude of the longitudinal waves are quadratically related to the transverse displacement of the string, however, experimental verification of this relationship is lacking. In the work reported here this relationship is tested by driving the transverse motion of a piano string at only two frequencies, which simplifies the task of unambiguously identifying the constituent signals. The results indicate that the generally accepted relationship between the transverse motion and the longitudinal motion is valid. It is further shown that this dependence on transverse displacement is a good approximation when a string is excited by the impact of the hammer during normal play.

The importance of longitudinal waves in piano strings has been previously identified by several investigators. Recent experimental work has provided insight into the origin of these waves and their relationship to the transverse string motion. These measurements indicate that there are multiple regimes in which longitudinal waves are created through different processes.

No abstract

It is known that longitudinal waves in piano strings noticeably contribute to the characteristic sound of the instrument. These waves can be induced by directly exciting the motion with a longitudinal component of the piano hammer, or by the stretching of the string associated with the transverse displacement. Longitudinal waves that are induced by the transverse motion of the string can occur at frequencies other than the longitudinal resonance frequencies, and the amplitude of the waves produced in this way are believed to vary quadratically with the amplitude of the transverse motion. We present the results of an experimental investigation that demonstrates the quadratic relationship between the magnitude of the longitudinal waves and the magnitude of the transverse displacement for steady-state, low-amplitude excitation. However, this relationship is only approximately correct under normal playing conditions.

It has been proposed that axial vibrations of the bells of brass wind instruments can lead to audible effects in the sound [Kausel et al. (2010)]. Using both laser Doppler vibrometry and a novel implementation of electronic speckle pattern interferometry, we have demonstrated that these vibrations exist, and that the magnitude is of the order predicted. [Work supported in part by a grant from the National Science Foundation.]