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
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Journal ArticleDOI
Multicomponent long-wave-short-wave resonance interaction system: Bright solitons, energy-sharing collisions, and resonant solitons.
TL;DR: This work constructs the exact bright multisoliton solutions by applying the Hirota's bilinearization method and investigates the head-on and overtaking collisions of bright solitons and explores two types of energy-sharing collisions as well as standard elastic collision.
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Nonlinear higher-order spectral solution for a two-dimensional moving load on ice
TL;DR: In this paper, the nonlinear response of an infinite ice sheet to a moving load in the time domain in two dimensions, using a higher-order spectral method, was calculated, and it was shown that the non-linearity is due to the moving boundary, as well as the non linear term in Bernoulli's equation and the elastic plate equation.
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A comparison of high-order polynomial and wave-based methods for Helmholtz problems
TL;DR: The high-order polynomial method (p-FEM with Lobatto polynomials) and the wave-based discontinuous Galerkin method are compared for two-dimensional Helmholtz problems, indicating that the differences in performance, accuracy and conditioning are more nuanced than generally assumed.
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Multilevel Monte Carlo Finite Volume Methods for Shallow Water Equations with Uncertain Topography in Multi-dimensions
TL;DR: The multilevel Monte Carlo (MLMC) algorithm is extended to numerically approximate the random shallow water equations efficiently and is suitably modified to deal with uncertain data on each node of the underlying topography grid by the use of a hierarchical topography representation.
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Multiple solutions for granular flow over a smooth two-dimensional bump
Sylvain Viroulet,James Baker,Andrew Edwards,Christopher Johnson,C. Gjaltema,P. Clavel,John Gray +6 more
TL;DR: In this paper, a depth-averaged terrain-following avalanche theory that is formulated in curvilinear coordinates is used to model the system and small-scale experiments are performed to investigate the flow of a granular avalanche over a two-dimensional smooth symmetrical bump.
References
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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.
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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.
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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.
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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.