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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A well-balanced scheme to capture non-explicit steady states in the Euler equations with gravity
TL;DR: In this paper, a numerical discretization of the compressible Euler equations with a gravitational potential is presented, which is a finite volume method, whose Riemann solver is approximated by a so-called relaxation RiemANN solution that takes all hydrostatic equilibria into account.
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Solitary waves in abelian gauge theories
Vieri Benci,Donato Fortunato +1 more
TL;DR: In this article, the existence of solitary waves for Abelian gauge theories was analyzed and it was shown that the lower order term W is positive and that the coupling between matter and electromagnetic field is small.
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Numerical study of a nonlocal model for water-waves with variable depth
TL;DR: In this paper, the authors studied the propagation of solitary waves in a Hamiltonian nonlocal shallow water model for bidirectional wave propagation in channels of variable depth and showed that solitary waves propagate robustly in channels with rapidly varying bottom topography, and their speed is predicted accurately by an effective equation obtained by the homogenization theory.
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Two-dimensional modal method for shallow-water sloshing in rectangular basins
TL;DR: In this article, a two-dimensional model for the analysis of sloshing phenomena in shallow-water conditions has been defined using Boussinesq-type depth-averaged equations.
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Global weak solutions for a modified two-component Camassa–Holm equation
Chunxia Guan,Zhaoyang Yin +1 more
TL;DR: In this paper, the Helly theorem and some a priori one-sided supernorm and space-time higher integrability estimates on the first-order derivatives of approximation solutions were obtained.
References
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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
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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.