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
More filters
Journal ArticleDOI
Hitchhiker's guide to the fractional Sobolev spaces
TL;DR: In this article, the authors deal with the fractional Sobolev spaces W s;p and analyze the relations among some of their possible denitions and their role in the trace theory.
Journal ArticleDOI
Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).
TL;DR: To the best of our knowledge, there is only one application of mathematical modelling to face recognition as mentioned in this paper, and it is a face recognition problem that scarcely clamoured for attention before the computer age but, having surfaced, has attracted the attention of some fine minds.
Journal ArticleDOI
The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations
Adrian Constantin,David Lannes +1 more
TL;DR: In this article, the authors prove that the nonlinear dispersive partial differential equations (NPDPDE) and Korteweg-de Vries (KDE) arise in the modeling of the propagation of shallow water waves over a flat bed.
Journal ArticleDOI
Discrete breathers — Advances in theory and applications
Sergej Flach,Andrey V. Gorbach +1 more
TL;DR: In this paper, the authors introduce the concept of localized excitations and review their basic properties including dynamical and structural stability, and focus on advances in the theory of discrete breathers in three directions.
Introduction to Physical Oceanography
TL;DR: In this paper, the authors present an overview of what is known about the ocean, including the equations of motion, the influence of earth's rotation, and viscosity of the ocean.
References
More filters
Journal ArticleDOI
Similarity and Dimensional Methods in Mechanics
TL;DR: Similiarity and Dimensional Methods in Mechanics, 10th edition as mentioned in this paper is an English language translation of this classic volume examining the general theory of dimensions of physical quantities, the theory of mechanical and physical similarity, and theory of modeling.
Book
Principles of Computational Fluid Dynamics
TL;DR: The basic equation of fluid dynamics: analytic aspects of the stationary convection-diffusion equation and Unified methods for computing incompressible and compressible flow.
Journal ArticleDOI
High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems
TL;DR: The history and basic formulation of WENO schemes are reviewed, the main ideas in using WenO schemes to solve various hyperbolic PDEs and other convection dominated problems are outlined, and a collection of applications in areas including computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, computational biology, and some non-PDE applications are presented.
Book
Handbook of Nonlinear Partial Differential Equations
TL;DR: In this paper, the authors present a general framework for nonlinear Equations of Mathematical Physics using a general form of the form wxy=F(x,y,w, w, wx, wy) wxy.