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
Set-theoretic analysis of the isolated ramp metering problem
TL;DR: The notions of maximal robust controlled invariant set, as well as t-step robust controllable set are defined and used for analyzing the ramp metering problem independently of the control policy applied, and algorithms are developed to compute these sets.
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Miracles, misconceptions and scotomas in the theory of solitary waves
TL;DR: In an age of billion dollar particle accelerators and Mars rovers, it is surprising that solitary waves were first discovered by a man on horseback with no tools but his own eyes as discussed by the authors.
Journal ArticleDOI
A lattice Boltzmann model for two‐dimensional sound wave in the small perturbation compressible flows
TL;DR: In this paper, a multi-entropy-level lattice Boltzmann model for two-dimensional sound wave equation in the small perturbation flows is presented, and numerical results show that this model can be used to simulate sound wave propagation.
Dissertation
Instabilities and modeling of falling film flows
TL;DR: In this paper, a synthesis of research activities devoted to the study and low-dimensional modeling of long-wave instabilities in general and to falling film flows in particular is presented, and different methodologies are discussed in Chapter~1 and applied to the modeling of inertial flows in a Hele-Shaw cell.
Journal ArticleDOI
Riemann solution for a class of morphodynamic shallow water dam-break problems
Fangfang Zhu,Nicholas Dodd +1 more
TL;DR: In this paper, the authors investigated a family of dam-break problems over an erodible bed, where the hydrodynamics are described by the shallow water equations, and the bed change by a sediment conservation equation, coupled to the hydrogynamics by sediment transport (bed load) law.
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
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.