Linear and nonlinear waves

By Graham W. Griffiths, William E. Schiesser. Scholarpedia, 2009

3,412 citations recorded by the source corpus.

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.

Subfields: Biomedical Engineering, Computer Networks and Communications, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/9eb4c07dc323f1cd52f950b6d412bd763147c065

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Biomedical Engineering

In Computer Networks and Communications

In Statistical and Nonlinear Physics

Citations in the corpus

Analysis on the blow-up of solutions to a class of integrable peakon equations (2016, background): “Examples can be found in the Whitham equation [9, 30], Camassa-Holm (CH) equation [5, 13], the Degasperis-Procesi (DP) equation[16, 13], and the Novikov equation [24], etc.”
Conservation laws on complex networks (2009, methodology): “models with two equations, were considered, see for example [1, 13, 24, 26, 35, 37, 40, 41], while a third order model was presented in [27].”
Analytical solutions for tsunami runup on a plane beach: single waves, N-waves and transient waves (2010, background): “Exact solutions to the linear KdV equation on a constant depth According to Whitham (1974), his § 13.6, an exact solution to the linear KdV equation reads ηδ(χ, τ ) = ( 2 τ )1/3 Ai (Z(χ, τ )), (7.1) where Ai is the Airy function, and where χ ≡ x̂ h0 , τ ≡ t √ g h0 , Z(χ, τ ) ≡ (χ − τ ) ( 2 τ )1/3 .”
On the Blow-up Phenomena for the Degasperis–Procesi Equation (2010, background): “It is also noted that the KdV equation, unlike the CH equation or DP equation, does not have wave breaking phenomena, that is, the wave profile remains bounded, but its slope becomes unbounded in finite time [34].”
Multi-species simple exclusion processes (2009, background): “(6)) is a nonlinear convection term that is associated with models of traffic flow [30] and the Burgers equation [23,24].”
Coherent structures in wall-bounded turbulence (2017, background): “The propagation velocity of wave packets is the group velocity (Whitham 1974), cgχ(k) = ∂k1(k1cχ) = cχ + k1∂k1cχ.”
Control of Homodirectional and General Heterodirectional Linear Coupled Hyperbolic PDEs (2015, background): “Although this approach to solving hyperbolic equations is classical (see [23] and [29]), the particular geometry of the problem and the coupling between kernels, both in-domain and at the boundaries, could lead the equations to explode in finite space on the triangle.”
Non-Uniform Dependence for the Periodic CH Equation (2010, background): “Furthermore, they discovered that the equation models breaking waves: there are smooth solutions developing singularities in finite time in the form of the slope becoming unbounded while the solution remains bounded (see Whitham [29]).”
Fundamentals of Geophysical Fluid Dynamics (2011, background): “For further analysis of this and many other nonlinear wave problems see Whitham (1999).”
New class of solutions for water infiltration problems in unsaturated soils (2010, methodology): “…and numerical solutions to the Richards equation exist based on Brooks and Corey’s representation of D and K. Replacing n = 0 and k = 2 in Equations (10) and (11) yields the classic Burgers equation extensively studied by many researchers (Whitman 1974, Broadbridge and Rogers 1990, Basha 2002).”
TT¯-deformed 2D quantum field theories (2016, background): “4) describes an incompressible fluid in the absence of viscosity and pressure [42].”
A second Wronskian formulation of the Boussinesq equation (2009, background): “It also appears in a wide variety of physical systems [15–22], for example, nonlinear waves in the evolution of perturbations with the dispersion relation close to that for the sound waves [15], electromagnetic waves interacting with transversal optical phonons in nonlinear dielectrics [16], magneto”
Asymptotic equivalence of homogenisation procedures and fine-tuning of continuum theories (2008, methodology): “5 4 Exact Rayleigh-Love Mindlin-Herrmann [4,8] (Rayleigh limit) [4,8] (modified inertia)”
KP solitons in shallow water (2010, background): “It is well known that a general initial data decaying rapidly in the spatial variable evolves to a number of individual solitons and weakly dispersive wave trains separate from the solitons (see for examples, [1, 31, 34, 44]).”
Finite difference methods for ordinary and partial differential equations - steady-state and time-dependent problems (2007, background): “See, for example, [55], [102] for more on dispersive waves.”
Discrete breathers — Advances in theory and applications (2008, background): “Originally this approach was designed for the study of waves in continuous media — some older papers also use the term Benjamin-Feir instability instead [402].”
Stability of peakons for the Degasperis‐Procesi equation (2007, background): “Under wave breaking we understand that development of singularities in finite time by which the wave remains bounded but its slope becomes unbounded [31].”
Infragravity waves: From driving mechanisms to impacts (2018, positive): “Following the concept of weak-solution (Whitham, 1974), the broken-wave fronts can be approximated by discontinuities or shocks.”
Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy (2007, background): “Next, following the above considerations, and according to the actual structure of many systems arising in physical models [36, 28, 37, 13, 31], we focus our investigation on a slightly restricted class of entropy dissipative non-degenerate systems, namely the systems such that (2.”