Newton also made contributions in the Principia with regard to resistance to motion, also that the minimum cross-section of a stream issuing from a hole in a reservoir is reached just outside the wall the vena contracta.

### mathematics and statistics online

This subject now goes under the name of fluid dynamics and has many branches such as multi-phase flow, turbulent flow, inviscid flow, aerodynamics, meteorology, etc. The study of electromagnetism was again started in antiquity, but very few advances were made until a proper scientific basis was finally initiated by William Gilbert in his De Magnete. However, it was only late in the 18th century that real progress was achieved when Franz Ulrich Theodor Aepinus , Henry Cavendish , Charles-Augustin de Coulomb and Alessandro Volta introduced the concepts of charge, capacity and potential.

It was in this work that all electromagnetic phenomena and all optical phenomena were first accounted for, including waves, see section Electromagnetic wave. It also included the first theoretical prediction for the speed of light. At the end of the 19th century, when some erroneously considered physics to be very nearly complete, new physical phenomena began to be observed that could not be explained. These demanded a whole new set of theories that ultimately led to the discovery of general relativity and quantum mechanics ; which, even now in the 21st century are still yielding exciting new discoveries.

However, as this article is primarily concerned with classical wave phenomena, we will not pursue these topics further. Waves occur in most scientific and engineering disciplines, for example: fluid mechanics, optics, electromagnetism, solid mechanics, structural mechanics, quantum mechanics, etc. The waves for all these applications are described by solutions to either linear or nonlinear PDEs. We do not focus here on methods of solution for each type of wave equation , but rather we concentrate on a small selection of relevant topics.

However, first, it is legitimate to ask: what actually is a wave?

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This is not a straight forward question to answer. Now, whilst most people have a general notion of what a wave is, based on their everyday experience, it is not easy to formulate a definition that will satisfy everyone engaged in or interested in this wide ranging subject. In fact, many technical works related to waves eschew a formal definition altogether and introduce the concept by a series of examples; for example, Physics of waves [Elm] and Hydrodynamics [Lam]. Nevertheless, it is useful to at least make an attempt and a selection of various definitions from normally authoritative sources is given below:.

The variety of definitions given above, and their clearly differing degrees of clarity, confirm that 'wave' is indeed not an easy concept to define! Because this is an introductory article and the subject of linear and non-linear waves is so wide ranging, we can only include sufficient material here to provide an overview of the phenomena and related issues. Relativistic issues will not be addressed. To this end we will discuss, as proxies for the wide range of known wave phenomena, the linear wave equation and the nonlinear Korteweg-de Vries equation in some detail by way of examples.

To supplement this discussion we provide brief details of other types of wave equation and their application; and, finally, we introduce a number of PDE wave solution methods and discuss some general properties of waves. Where appropriate, references are included to works that provide further detailed discussion.

A non-exhaustive list is given below of physical wave types with examples of occurrence and references where more details may be found. Linear waves are described by linear equations, i.

This means that the superposition principle applies, and linear combinations of simple solutions can be used to form more complex solutions. Thus, all the linear system analysis tools are available to the analyst, with Fourier analysis : expressing general solutions in terms of sums or integrals of well known basic solutions, being one of the most useful.

The classic linear wave is discussed in section The linear wave equation with some further examples given in section Linear wave equation examples.

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The following represents the classical wave equation in one dimension and describes undamped linear waves in an isotropic medium. Consequently, equations 4 , 5 and 6 constitute a complete description of the PDE problem. Because the Laplacian is co-ordinate free, it can be applied within any co-ordinate system and for any number of dimensions. These equations occur in one form or another, in numerous applications in all areas of the physical sciences; see for example section Linear wave equation examples.

The solution to equations 4 , 5 and 6 was first reported by the French mathematician Jean-le-Rond d'Alembert in in a treatise on Vibrating Strings [Caj] [Far]. D'Alembert's remarkable solution, which used a method specific to the wave equation based on the chain rule for differentiation , is given below. It can also be obtained by the Fourier Transform method or by the separation of variables SOV method, which are more general than the the method used by d'Alembert [Krey]. We will consider the acoustic or sound wave as a small amplitude disturbance of ambient conditions where second order effects can be ignored.

We start with the Euler continuity and momentum equations.

### 1st Edition

We assume an inviscid dry gas situation where gravitational effects are negligible. This means that the third and fifth terms of equation 10 can be ignored. Thus, equations 9 and 10 reduce to. Waves in solids are more complex than acoustic waves in fluids. Starting with Newton's second Law we arrive at the vector wave equation [Elm, chapter 7]. Irrotational waves are of the longitudinal type, or P-waves. Solenoidal waves are of the transverse type, or S-waves.

For further discussion refer to [Cia]. The fundamental equations of electromagnetism are the Maxwell Equations , which in differential form and SI units, are usually written as:. They take the familiar form of linear wave equation 4. For further discussion refer to [Sha]. Nonlinear waves are described by nonlinear equations, and therefore the superposition principle does not generally apply.

This means that nonlinear wave equations are more difficult to analyze mathematically and that no general analytical method for their solution exists. Thus, unfortunately, each particular wave equation has to be treated individually. An example of solving the Korteweg-de Vries equation by direct integration is given below. Some advanced methods that have been used successfully to obtain closed-form solutions are listed in section Closed form PDE solution methods , and example solutions to well known evolution equations are given in section Nonlinear wave equation solutions.

There are no general methods guaranteed to find closed form solutions to non-linear PDEs. Nevertheless, some problems can yield to a trial-and-error approach. This hit-and-miss method seeks to deduce candidate solutions by looking for clues from the equation form, and then systematically investigating whether or not they satisfy the particular PDE.

If the form is close to one with an already known solution, this approach may yield useful results.

However, success is problematical and relies on the analyst having a keen insight into the problem. We list below, in alphabetical order, a non-exhaustive selection of advanced solution methods that can assist in determining closed form solutions to nonlinear wave equations. We will not discuss further these methods and refer the reader to the references given for details.

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## Trimester on nonlinear waves

All these methods are greatly enhanced by use of a symbolic computer program such as: Maple V, Mathematica, Macysma, etc. The following are examples of techniques that transform PDEs into ODEs which are subsequently solved to obtain traveling wave solutions to the original equations. Some example applications of these and other methods can be found in [Gri].

A non-exhaustive selection of well known 1D nonlinear wave equations and their closed-form solutions is given below. The closed form solutions are given by way of example only, as nonlinear wave equations often have many possible solutions. Examples from the biological and medical fields can be found in Murray [Mur] and Murray [Mur]. A useful on-line resource is the DispersiveWiki [Dis]. It is, historically, the most famous solitary wave equation and describes small amplitude, shallow water waves in a channel, where symbols have the following meaning:.

After re-scaling and translating the dependent and independent variables to eliminate the physical constants using the transformations [Abl] ,. The KdV equation was found to have solitary wave solutions [Lam] , which confirmed John Scott-Russell's account of the solitary wave phenomena [Sco] discovered during his experimental investigations into water flow in channels to determine the most efficient design for canal boats [Jag]. Subsequently, the KdV equation has been shown to model various other nonlinear wave phenomena found in the physical sciences.

John Scott-Russell, a Scottish engineer and naval architect, also described in poetic terms his first encounter with the solitary wave phenomena, thus:. An experimental apparatus for re-creating the phenomena observed by Scott-Russell have been built at Herriot-Watt University. Scott-Russell also coined the term solitary wave and conducted some of the first experiments to investigate another nonlinear wave phenomena, the Doppler effect , publishing an independent explanation of the theory in [Sco].

## Regularity results for nonlinear wave equations

It is interesting to note that, a KdV solitary wave in water that experiences a change in depth will retain its general shape. However, on encountering shallower water its velocity and height will increase and its width decrease; whereas, on encountering deeper water its velocity and height will decrease and its width increase [Joh, pp ].

A closed form single soliton solution to the KdV equation 28 can be found using direct integration as follows. Hence, the taller a wave the faster it travels. The KdV equation also admits many other solutions including multiple soliton solutions, see figure 15 , and cnoidal periodic solutions. Interestingly, the KdV equation is invariant under a Galilean transformation , i. Linear and nonlinear evolutionary wave problems can very often be solved by application of general numerical techniques such as: finite difference, finite volume, finite element, spectral, least squares, weighted residual e.

## Wave equation - Wikipedia

These methods, which can all handle various boundary conditions , stiff problems and may involve explicit or implicit calculations, are well documented in the literature and will not be discussed further here. For general texts refer to [Bur] , [Sch] , [Sch] , and for more detailed discussion refer to [Lev] , [Mor] , [Zie]. Some wave problems do, however, present significant problems when attempting to find a numerical solution.

In particular we highlight problems that include shocks, sharp fronts or large gradients in their solutions. Because these problems often involve inviscid conditions zero or vanishingly small viscosity , it is often only practical to obtain weak solutions. Such problems are likely to occur when there is a hyperbolic strongly convective component present. In these situations weak solutions provide useful information. Detailed discussion of this approach is beyond the scope of this article and readers are referred to [Wes, chapters 9 and 10] for further discussion.

General methods are often not adequate for accurate resolution of steep gradient phenomena; they usually introduce non-physical effects such as smearing of the solution or spurious oscillations. Since publication of Godunov's order barrier theorem , which proved that linear methods cannot provide non-oscillatory solutions higher than first order [God] , [God] , these difficulties have attracted a lot of attention and a number of techniques have been developed that largely overcome these problems.