Linear and nonlinear waves
TLDR
The study of waves can be traced back to antiquity where philosophers, such as Pythagoras, studied the relation of pitch and length of string in musical instruments and the subject of classical acoustics was laid down and presented as a coherent whole by John William Strutt in his treatise Theory of Sound.Abstract:
The study of waves can be traced back to antiquity where philosophers, such as Pythagoras (c.560-480 BC), studied the relation of pitch and length of string in musical instruments. However, it was not until the work of Giovani Benedetti (1530-90), Isaac Beeckman (1588-1637) and Galileo (1564-1642) that the relationship between pitch and frequency was discovered. This started the science of acoustics, a term coined by Joseph Sauveur (1653-1716) who showed that strings can vibrate simultaneously at a fundamental frequency and at integral multiples that he called harmonics. Isaac Newton (1642-1727) was the first to calculate the speed of sound in his Principia. However, he assumed isothermal conditions so his value was too low compared with measured values. This discrepancy was resolved by Laplace (1749-1827) when he included adiabatic heating and cooling effects. The first analytical solution for a vibrating string was given by Brook Taylor (1685-1731). After this, advances were made by Daniel Bernoulli (1700-82), Leonard Euler (1707-83) and Jean d’Alembert (1717-83) who found the first solution to the linear wave equation, see section (3.2). Whilst others had shown that a wave can be represented as a sum of simple harmonic oscillations, it was Joseph Fourier (1768-1830) who conjectured that arbitrary functions can be represented by the superposition of an infinite sum of sines and cosines now known as the Fourier series. However, whilst his conjecture was controversial and not widely accepted at the time, Dirichlet subsequently provided a proof, in 1828, that all functions satisfying Dirichlet’s conditions (i.e. non-pathological piecewise continuous) could be represented by a convergent Fourier series. Finally, the subject of classical acoustics was laid down and presented as a coherent whole by John William Strutt (Lord Rayleigh, 1832-1901) in his treatise Theory of Sound. The science of modern acoustics has now moved into such diverse areas as sonar, auditoria, electronic amplifiers, etc.read more
Citations
More filters
Journal ArticleDOI
Energy invariant for shallow water waves and the Korteweg -- de Vries equation. Is energy always an invariant?
TL;DR: The question of how the Korteweg-de Vries (KdV) equation has an infinite set of conserved quantities and also how these KdV quantities relate to those of the Euler shallow-water equations is answered.
Journal ArticleDOI
How the propagation of heat-flux modulations triggers E × B flow pattern formation.
TL;DR: A novel mechanism to describe E×B flow pattern formation based upon the dynamics of propagation of heat-flux modulations, which includes a flux response time, during which the instantaneous heat flux relaxes to the mean heat flux, determined by symmetry constraints.
Journal ArticleDOI
The Spatial Buildup of Compression and Suppression in the Mammalian Cochlea
TL;DR: This framework of spatial buildup of local effects unifies the widely different effects of overall intensity, low- side suppressors, and high-side suppressors on BM responses.
Book
The Sine-Gordon Equation in the Semiclassical Limit: Dynamics of Fluxon Condensates
TL;DR: In this article, the inverse problem for fluxon condensates elementary transformations of $\mathbf{J}(w)$ construction of $gw$ using of $w(w), and the inner parametrization of the outer parametrix.
Journal ArticleDOI
Sustained gravity currents in a channel
TL;DR: In this article, a two-layer shallow-water model was proposed to account for the flow of both the dense and the overlying less dense fluids in a horizontal channel, and the authors showed that a variety of flow-field patterns are feasible, including those with constant height along the length of the current and those where the height varies continuously and discontinuously, depending on the magnitude of the dimensionless flux issuing from the source and the source.
References
More filters
Book
The finite element method
TL;DR: In this article, the methodes are numeriques and the fonction de forme reference record created on 2005-11-18, modified on 2016-08-08.
Book
Mathematical Methods of Classical Mechanics
TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
Book
Linear and Nonlinear Waves
TL;DR: In this paper, a general overview of the nonlinear theory of water wave dynamics is presented, including the Wave Equation, the Wave Hierarchies, and the Variational Method of Wave Dispersion.
Journal ArticleDOI
Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method
TL;DR: In this article, a second-order extension of the Lagrangean method is proposed to integrate the equations of ideal compressible flow, which is based on the integral conservation laws and is dissipative, so that it can be used across shocks.
Book
Finite Volume Methods for Hyperbolic Problems
TL;DR: The CLAWPACK software as discussed by the authors is a popular tool for solving high-resolution hyperbolic problems with conservation laws and conservation laws of nonlinear scalar scalar conservation laws.