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
Creation of Cadastral Maps of Flooding Based on Numerical Modeling
A. Yu. Klikunova,Alexander Khoperskov,E. O. Agafonnikova,A.S. Kuz'mich,Tatyana Dyakonova,S. S. Khrapov,I.M. Gusev +6 more
TL;DR: In this article, the authors considered the possibilities of numerical hydrodynamic modeling for the construction of cadastral maps of dangerous flood zones on the example of the Volgograd region and proposed a numerical model based on shallow water equations and Combined Smooth Particle Hydrodynamics -Total Variation Diminishing (CSPH-TVD) method for integrating hyperbolic differential equations.
Journal ArticleDOI
Charge-varying sine-Gordon deformed defects
TL;DR: In this paper, a triggering sine-Gordon model simultaneously supports kink-and lump-like defects, whose topological mass values are closed by trigonometric or hyperbolic successive deformations.
Journal ArticleDOI
Semiclassical Determination of Exponentially Small Intermode Transitions for 1 + 1 Spacetime Scattering Systems
Alain Joye,Magali Marx +1 more
TL;DR: In this article, the authors consider the semiclassical limit of systems of autonomous PDEs in 1 + 1 spacetime dimensions in a scattering regime, where the matrix-valued coefficients are analytic in the space variable and the corresponding dispersion relation admits real-valued modes only with one-dimensional polarization subspaces.
Journal ArticleDOI
Chaos in positive ion-negative ion magnetized plasmas
TL;DR: In this paper, the dynamical behavior of weakly nonlinear, low-frequency sound waves in a plasma composed of only positive and negative ions incorporating the effects of a weak external uniform magnetic field was investigated.
Journal ArticleDOI
A Crank–Nicolson linear difference scheme for a BBM equation with a time fractional nonlocal viscous term
Xue Shen,Ailing Zhu +1 more
TL;DR: In this paper, a Crank-Nicolson linear finite difference scheme for a Benjamin-Bona-Mahony equation with a time fractional nonlocal viscous term is presented.
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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