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
Modulation analysis of nonlinear beam refraction at an interface in liquid crystals
TL;DR: In this paper, a modulation theory based on a Lagrangian formulation of the governing optical solitary wave equations is developed, and the resulting low-dimensional equations are found to give solutions in excellent agreement with full numerical solutions of governing equations, as well as with previous experimental studies.
Journal ArticleDOI
Local Dirichlet-type absorbing boundary conditions for transient elastic wave propagation problems
TL;DR: A new collocation technique for constructing time-dependent absorbing boundary conditions (ABCs) applicable to elastic wave motion is devised and contributes to developing a consistent meshless framework for the solution of unbounded elastodynamic problems in time domain.
Journal ArticleDOI
Exact Discontinuous Solutions of Exner’s Bed Evolution Model: Simple Theory for Sediment Bores
TL;DR: In this paper, generalized solutions of Exner's classic bed evolution model were considered, and a simple theory for the formation and propagation of discontinuities in the bed or so-called sediment bores was developed.
Journal ArticleDOI
Study on the amplitude inversion of internal waves at Wenchang area of the South China Sea
TL;DR: In this article, a corrected nonlinear Schrodinger (NLS) equation was proposed to inverse the amplitude of the internal wave in Wenchang area east of Hainan Island (19°35'N, 112°E) of China.
Journal ArticleDOI
Error estimates for Galerkin approximations of the “classical” Boussinesq system
TL;DR: This work discretizes an initial-boundary-value problem for these systems in space using Galerkin-finite element methods and proves error estimates for the resulting semidiscrete problems and also for their fully discrete analogs effected by explicit Runge-Kutta time-stepping procedures.
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.