Coupling of transverse and longitudinal waves in piano strings

Etchenique, Nikki; Collin, Samantha; Moore, Thomas R.

doi:10.1121/1.4916708

Cited by 7 publications

(5 citation statements)

References 16 publications

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“…An important effect of tension modulation is the creation of transverse motion in the bridge that has different consequences. First, it can excite the mode of the string initially absent from the vibration (typically, this happens over a time of order 1 s); second, it creates another transverse polarization that gains energy through coupling; lastly, the nonlinear coupling with the two possible transverse polarizations of the string vibration induces the creation of forced-response longitudinal waves (Etchenique et al, 2015).…”

Section: The Physics Of the Pianomentioning

confidence: 99%

Physics-informed differentiable method for piano modeling

Simionato,

Fasciani,

Holm

2024

Front. Signal Process.

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Numerical emulations of the piano have been a subject of study since the early days of sound synthesis. High-accuracy sound synthesis of acoustic instruments employs physical modeling techniques which aim to describe the system’s internal mechanism using mathematical formulations. Such physical approaches are system-specific and present significant challenges for tuning the system’s parameters. In addition, acoustic instruments such as the piano present nonlinear mechanisms that present significant computational challenges for solving associated partial differential equations required to generate synthetic sound. In a nonlinear context, the stability and efficiency of the numerical schemes when performing numerical simulations are not trivial, and models generally adopt simplifying assumptions and linearizations. Artificial neural networks can learn a complex system’s behaviors from data, and their application can be beneficial for modeling acoustic instruments. Artificial neural networks typically offer less flexibility regarding the variation of internal parameters for interactive applications, such as real-time sound synthesis. However, their integration with traditional signal processing frameworks can overcome this limitation. This article presents a method for piano sound synthesis informed by the physics of the instrument, combining deep learning with traditional digital signal processing techniques. The proposed model learns to synthesize the quasi-harmonic content of individual piano notes using physics-based formulas whose parameters are automatically estimated from real audio recordings. The model thus emulates the inharmonicity of the piano and the amplitude envelopes of the partials. It is capable of generalizing with good accuracy across different keys and velocities. Challenges persist in the high-frequency part of the spectrum, where the generation of partials is less accurate, especially at high-velocity values. The architecture of the proposed model permits low-latency implementation and has low computational complexity, paving the way for a novel approach to sound synthesis in interactive digital pianos that emulates specific acoustic instruments.

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Section: The Physics Of the Pianomentioning

confidence: 99%

Physics-informed differentiable method for piano modeling

Simionato,

Fasciani,

Holm

2024

Front. Signal Process.

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“…(3) Because of its radial motion, the arterial wall deforms along its longitudinal direction. The radial component from the longitudinal tension force of the wall segment is [22,23]…”

Section: A Radial Motion Of the Arterial Wallmentioning

confidence: 99%

Radial and longitudinal motion of the arterial wall: Their relation to pulsatile pressure and flow in the artery

2018

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The aim of this paper is to analyze the radial and longitudinal motion of the arterial wall in the context of pulsatile pressure and flow, and to understand their physiological implications for the cardiovascular system. A reexamination of the well-established one-dimensional governing equations for axial blood flow in the artery and the constitutive equation for the radial dilation of the arterial wall shows that two waves-a pulsatile pressure wave in the artery and a radial displacement wave in the arterial wall-propagate simultaneously along the arterial tree with the same propagation velocity, explaining why this velocity combines the physical properties and geometries of both the blood and the arterial wall. With consideration of their coupling, the governing equations for the radial motion and longitudinal motion of the arterial wall are derived separately. The driving force for the radial motion of the arterial wall arises from its longitudinal motion. The longitudinal motion of the arterial wall is the standard longitudinal elastic wave with two driving forces: one associated with the pulsatile flow rate and the other associated with the radial motion. These derived governing equations shed insights on some recent experimental findings in the literature, including the correlation of measured arterial wall stiffness (solely based on the radial motion) to its longitudinal motion, the decreasing trend of arterial wall distensibility from the aorta to the periphery, the underlying mechanism of the longitudinal motion pattern of the common carotid arterial wall, and the motion pattern variation in longitudinal motion with aging and diseases.

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“…The vibro-acoustic broadband behavior of this structure has been studied by several authors including Fletcher [2], Weinrech [3], and Kindel [4], then by Suzuki [5], [6], Conklin [7]- [9], Giordano [10]- [12], Dérogis [13], Berthaut [14], Bensa [15], Stulov [16], and more recently by Ege and Boutillon [17], [18], Chaigne [19], Chabassier [20], Rigaud [21], and Etcheniquel [22]. It is now the subject of software development for piano sound synthesis [23].…”

Section: Introductionmentioning

confidence: 99%

A modal approach to piano soundboard vibroacoustic behavior

Trévisan

Ege

Laulagnet

2017

The Journal of the Acoustical Society of America

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This paper presents an analytical method for modeling the vibro-acoustic behavior of ribbed non-rectangular orthotropic clamped plates. To do this, the non-rectangular plate is embedded in an extended rectangular simply supported plate on which a spring distribution is added, blocking the extended part of the surface, and allowing the description of any inner surface shapes. The acoustical radiation of the embedded plate is ensured using the radiation impedances of the extended rectangular simply supported plate. This method is applied to an upright piano soundboard: a non-rectangular orthotropic plate ribbed in both directions by several straight stiffeners. A modal decomposition is adopted on the basis of the rectangular extended simply supported plate modes, making it possible to calculate the modes of a piano soundboard in the frequency range [0;3000] Hz, showing the different associated mode families. Likewise, the acoustical radiation is calculated using the radiation impedances of a simply supported baffled plate, demonstrating the influence of the string coupling point positions on the acoustic radiated power. The paper ends with the introduction of indicators taking into account spatial and spectral variations of the excitation depending on the notes, which add to the accuracy of the study from the musical standpoint. A parametrical study, which includes several variations of soundboard design, highlights the complexity of rendering high-pitched notes homogeneous.

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Coupling of transverse and longitudinal waves in piano strings

Cited by 7 publications

References 16 publications

Physics-informed differentiable method for piano modeling

Physics-informed differentiable method for piano modeling

Radial and longitudinal motion of the arterial wall: Their relation to pulsatile pressure and flow in the artery

A modal approach to piano soundboard vibroacoustic behavior

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