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
Cascade model of wave turbulence.
TL;DR: The cascade model of wave turbulence is proposed, in which some modes are driven and damped and others are shared by pairs of quartets and transferring energy between them, mimicking the natural energy transfer mechanism in weakly turbulent waves.
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The Large‐Time Solution of Burgers' Equation with Time‐ Dependent Coefficients. II. Algebraic Coefficients
TL;DR: In this paper, the authors considered an initial value problem for Burgers' equation with variable coefficients, where x and t represent dimensionless distance and time, respectively, while, are given continuous functions of t ( > 0).
Journal ArticleDOI
On the relationship between nonlinear equations integrable by the method of characteristics and equations associated with commuting vector fields
TL;DR: In this paper, a scalar nonlinear PDE from another class of $S$-integrable PDEs was considered, which is commutativity condition of the pair of vector fields.
Posted Content
Special Polynomials and Exact Solutions of the Dispersive Water Wave and Modified Boussinesq Equations
TL;DR: In this paper, the dispersive water wave and modified Boussinesq equations are expressed in terms of special polynomials associated with rational solutions of the fourth Painleve´ equation, which arises as generalized scaling reductions of these equations.
Thermal Solitary Waves
TL;DR: In this paper, the authors derived the governing system of equations for an optical beam travelling in a nonlinear thermal medium, and provided an outline of the work undertaken in this thesis.
References
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Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method
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