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
Fifty years of Schallamach waves: from rubber friction to nanoscale fracture
TL;DR: In this article , the authors place Schallamach's results in a broader context and review subsequent investigations of stick-slip dynamics, before discussing recent observations of solitary Schallach waves.
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Subharmonic resonant interaction of a gravity–capillary progressive axially symmetric wave with a radial cross-wave
Meng Shen,Yuming Liu +1 more
TL;DR: In this article, the authors theoretically investigate the subharmonic resonant interaction of a progressive ring wave with a radial cross-wave in the context of the potential-flow formulation for gravity-capillary waves.
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Shock trains on a planar beach
TL;DR: In this article, the propagation of trains of shock waves on a planar beach is studied in the framework of the nonlinear shallow water equations, based on the use of a quasi-analytical solution valid for a shock wave which is fed by a constant Riemann invariant.
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The investigation of exact solutions of Korteweg-de vries equation with dual power law nonlinearity using the expa and exp (− Φ (ξ)) methods
Ghazala Akram,Naila Sajid +1 more
TL;DR: In this paper, the methodologies of expa and exp(−Φ(ξ)) methods are used to investigate the solutions of Korteweg-de Vries (KdV) equation with dual power law nonlinearity.
Posted Content
Real-normalized Whitham hierarchies and the WDVV equations
TL;DR: In this paper, a new class of explicit solutions to the WDVV (or associativity) equations is presented, based on a relationship between the DWVV equations and Whitham (or modulation) equations.
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.