Well-posedness, energy and charge conservation for nonlinear wave equations in discrete space-time

We consider the problem of discretization for the U(1)-invariant nonlinear wave equations in any dimension. We show that the classical finite-difference scheme used by Strauss and Vazquez \cite{MR0503140} conserves the positive-definite discrete analog of the energy if the grid ratio is $dt/dx\le 1/\sqrt{n}$, where $dt$ and $dx$ are the mesh sizes of the time and space variables and $n$ is the spatial dimension. We also show that if the grid ratio is $dt/dx=1/\sqrt{n}$, then there is the discrete analog of the charge which is conserved. We prove the existence and uniqueness of solutions to the discrete Cauchy problem. We use the energy conservation to obtain the a priori bounds for finite energy solutions, thus showing that the Strauss -- Vazquez finite-difference scheme for the nonlinear Klein-Gordon equation with positive nonlinear term in the Hamiltonian is conditionally stable.Comment: 10 page

Detection of keyboard vibrations and effects on perceived piano quality

Two experiments were conducted on an upright and a grand piano, both either producing string vibrations or conversely being silent after the initial keypress, while pianists were listening to the feedback from a synthesizer through insulating headphones. In a quality experiment, participants unaware of the silent mode were asked to play freely and then rate the instrument according to a set of attributes and general preference. Participants preferred the vibrating over the silent setup, and preference ratings were associated to auditory attributes of richness and naturalness in the low and middle ranges. Another experiment on the same setup measured the detection of vibrations at the keyboard, while pianists played notes and chords of varying dynamics and duration. Sensitivity to string vibrations was highest in the lowest register and gradually decreased up to note D5. After the percussive transient, the tactile stimuli exhibited spectral peaks of acceleration whose perceptibility was demonstrated by tests conducted in active touch conditions. The two experiments confirm that piano performers perceive vibratory cues of strings mediated by spectral and spatial summations occurring in the Pacinian system in their fingertips, and suggest that such cues play a role in the evaluation of quality of the musical instrument

The Role of Haptic Cues in Musical Instrument Quality Perception

We draw from recent research in violin quality evaluation and piano performance to examine whether the vibrotactile sensation felt when playing a musical instrument can have a perceptual effect on its judged quality from the perspective of the musician. Because of their respective sound production mechanisms, the violin and the piano offer unique example cases and diverse scenarios to study tactile aspects of musical interaction. Both violinists and pianists experience rich haptic feedback, but the former experience vibrations at more bodily parts than the latter. We observe that the vibrotactile component of the haptic feedback during playing, both for the violin and the piano, provides an important part of the integrated sensory information that the musician experiences when interacting with the instrument. In particular, the most recent studies illustrate that vibrations felt at the fingertips (left hand only for the violinist) can lead to an increase in perceived sound loudness and richness, suggesting the potential for more research in this direction