Wave equation

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Wave equation 1D fixed endpoints.gif
A pulse traveling through a string with fixed endpoints as modeled by the wave equation.
Spherical wave2.gif
Spherical waves coming from a point source.
2D Wave Function resize.gif
A solution to the 2D wave equation

The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.

Partial differential equation differential equation that contains unknown multivariable functions and their partial derivatives

In mathematics, a partial differential equation (PDE) is a differential equation that contains beforehand unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.

Wave oscillation that travels through space and matter

In physics, a wave is a disturbance that transfers energy through matter or space, with little or no associated mass transport. Waves consist of oscillations or vibrations of a physical medium or a field, around relatively fixed locations. From the perspective of mathematics, waves, as functions of time and space, are a class of signals.

Classical physics refers to theories of physics that predate modern, more complete, or more widely applicable theories

Classical physics refers to theories of physics that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the previous theories, or new theories based on the older paradigm, will often be referred to as belonging to the realm of "classical physics".

Contents

Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange. [1] [2] [3] [4] In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation. [5]

Musical instrument History and classification

A musical instrument is an instrument created or adapted to make musical sounds. In principle, any object that produces sound can be considered a musical instrument—it is through purpose that the object becomes a musical instrument. The history of musical instruments dates to the beginnings of human culture. Early musical instruments may have been used for ritual, such as a trumpet to signal success on the hunt, or a drum in a religious ceremony. Cultures eventually developed composition and performance of melodies for entertainment. Musical instruments evolved in step with changing applications.

Jean le Rond dAlembert French philosopher

Jean-Baptiste le Rond d'Alembert was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was co-editor with Denis Diderot of the Encyclopédie. D'Alembert's formula for obtaining solutions to the wave equation is named after him. The wave equation is sometimes referred to as d'Alembert's equation.

Leonhard Euler Swiss mathematician

Leonhard Euler was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.

Introduction

The wave equation is a hyperbolic partial differential equation. It typically concerns a time variable t, one or more spatial variables x1, x2, , xn, and a scalar function u = u (x1, x2, , xn; t), whose values could model, for example, the mechanical displacement of a wave. The wave equation for u is

Scalar (mathematics) real numbers in the context of linear algebra

A scalar is an element of a field which is used to define a vector space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector.

Displacement (vector) vector that is the shortest distance from the initial to the final position of a point P

A displacement is a vector whose length is the shortest distance from the initial to the final position of a point P. It quantifies both the distance and direction of an imaginary motion along a straight line from the initial position to the final position of the point. A displacement may be identified with the translation that maps the initial position to the final position.

where ∇2 is the (spatial) Laplacian and c is a constant.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, 2, or Δ. The Laplacian Δf(p) of a function f at a point p, is the rate at which the average value of f over spheres centered at p deviates from f(p) as the radius of the sphere grows. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems such as cylindrical and spherical coordinates, the Laplacian also has a useful form.

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression; it is usually a number, but may be any expression. In the latter case, the variables appearing in the coefficients are often called parameters, and must be clearly distinguished from the other variables.

Solutions of this equation describe propagation of disturbances out from the region at a fixed speed in one or in all spatial directions, as do physical waves from plane or localized sources; the constant c is identified with the propagation speed of the wave. This equation is linear. Therefore, the sum of any two solutions is again a solution: in physics this property is called the superposition principle.

Superposition principle fundamental physics principle stating that physical solutions of linear systems are linear

The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input A produces response X and input B produces response Y then input produces response.

The wave equation alone does not specify a physical solution; a unique solution is usually obtained by setting a problem with further conditions, such as initial conditions, which prescribe the amplitude and phase of the wave. Another important class of problems occurs in enclosed spaces specified by boundary conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments.

The wave equation, and modifications of it, are also found in elasticity, quantum mechanics, plasma physics and general relativity.

Wave equation in one space dimension

French scientist Jean-Baptiste le Rond d'Alembert (b. 1717) discovered the wave equation in one space dimension. Alembert.jpg
French scientist Jean-Baptiste le Rond d'Alembert (b. 1717) discovered the wave equation in one space dimension.

The wave equation in one space dimension can be written as follows:

.

This equation is typically described as having only one space dimension x, because the only other independent variable is the time t. Nevertheless, the dependent variable u may represent a second space dimension, if, for example, the displacement u takes place in y-direction, as in the case of a string that is located in the x–y plane.

Derivation of the wave equation

The wave equation in one space dimension can be derived in a variety of different physical settings. Most famously, it can be derived for the case of a string that is vibrating in a two-dimensional plane, with each of its elements being pulled in opposite directions by the force of tension. [6]

Another physical setting for derivation of the wave equation in one space dimension utilizes Hooke's Law. In the theory of elasticity, Hooke's Law is an approximation for certain materials, stating that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress).

From Hooke's law

The wave equation in the one-dimensional case can be derived from Hooke's Law in the following way: Imagine an array of little weights of mass m interconnected with massless springs of length h. The springs have a spring constant of k:

Array of masses.svg

Here the dependent variable u(x) measures the distance from the equilibrium of the mass situated at x, so that u(x) essentially measures the magnitude of a disturbance (i.e. strain) that is traveling in an elastic material. The forces exerted on the mass m at the location x + h are:

The equation of motion for the weight at the location x + h is given by equating these two forces:

where the time-dependence of u(x) has been made explicit.

If the array of weights consists of N weights spaced evenly over the length L = Nh of total mass M = Nm, and the total spring constant of the array K = k/N we can write the above equation as:

Taking the limit N → ∞, h → 0 and assuming smoothness one gets:

which is from the definition of a second derivative. KL2/M is the square of the propagation speed in this particular case.

1-d standing wave as a superposition of two waves traveling in opposite directions 1d wave equation animation.gif
1-d standing wave as a superposition of two waves traveling in opposite directions

Stress pulse in a bar

In the case of a stress pulse propagating through a beam the beam acts much like an infinite number of springs in series and can be taken as an extension of the equation derived for Hooke's law. A beam of constant cross-section made from a linear elastic material has a stiffness K given by

Where A is the cross-sectional area and E is the Young's modulus of the material. The wave equation becomes

AL is equal to the volume of the beam and therefore

where is the density of the material. The wave equation reduces to

The speed of a stress wave in a bar is therefore .

General solution

Algebraic approach

The one-dimensional wave equation is unusual for a partial differential equation in that a relatively simple general solution may be found. Defining new variables: [7]

changes the wave equation into

which leads to the general solution

or equivalently:

In other words, solutions of the 1D wave equation are sums of a right traveling function F and a left traveling function G. "Traveling" means that the shape of these individual arbitrary functions with respect to x stays constant, however the functions are translated left and right with time at the speed c. This was derived by Jean le Rond d'Alembert. [8]

Another way to arrive at this result is to note that the wave equation may be "factored":

As a result, if we define v thus,

then

From this, v must have the form G(x + ct), and from this the correct form of the full solution u can be deduced. [9]

For an initial value problem, the arbitrary functions F and G can be determined to satisfy initial conditions:

The result is d'Alembert's formula:

In the classical sense if f(x) ∈ Ck and g(x) ∈ Ck−1 then u(t, x) ∈ Ck. However, the waveforms F and G may also be generalized functions, such as the delta-function. In that case, the solution may be interpreted as an impulse that travels to the right or the left.

The basic wave equation is a linear differential equation and so it will adhere to the superposition principle. This means that the net displacement caused by two or more waves is the sum of the displacements which would have been caused by each wave individually. In addition, the behavior of a wave can be analyzed by breaking up the wave into components, e.g. the Fourier transform breaks up a wave into sinusoidal components.

Plane wave eigenmodes

Another way to solve for the solutions to the one-dimensional wave equation is to first analyze its frequency eigenmodes. A so-called eigenmode is a solution that oscillates in time with a well-defined constant angular frequency , with which the temporal part of the wave function for such eigenmode takes a specific form . The rest of the wave function is then only dependent on the spatial variable , hence amounting to separation of variables. Now writing the wave function as

we can obtain an ordinary differential equation for the spatial part

Therefore:

which is precisely an eigenvalue equation for , hence the name eigenmode. It has the well-known plane wave solutions

with wave number .

The total wave function for this eigenmode is then the linear combination

where complex numbers depend in general on any initial and boundary conditions of the problem.

Eigenmodes are useful in constructing a full solution to the wave equation, because each of them evolves in time trivially with the phase factor . so that a full solution can be decomposed into an eigenmode expansion

or in terms of the plane waves,

which is exactly in the same form as in the algebraic approach. Functions are known as the Fourier component and are determined by initial and boundary conditions. This is a so-called frequency-domain method, alternative to direct time-domain propagations, such as FDTD method, of the wave packet , which is complete for representing waves in absence of time dilations. Completeness of the Fourier expansion for representing waves in the presence of time dilations has been challenged by chirp wave solutions allowing for time variation of . [10] The chirp wave solutions seem particularly implied by very large but previously inexplicable radar residuals in the flyby anomaly, and differ from the sinusoidal solutions in being receivable at any distance only at proportionally shifted frequencies and time dilations, corresponding to past chirp states of the source.

Scalar wave equation in three space dimensions

Swiss mathematician and physicist Leonhard Euler (b. 1707) discovered the wave equation in three space dimensions. Leonhard Euler 2.jpg
Swiss mathematician and physicist Leonhard Euler (b. 1707) discovered the wave equation in three space dimensions.

A solution of the initial-value problem for the wave equation in three space dimensions can be obtained from the corresponding solution for a spherical wave. The result can then be also used to obtain the same solution in two space dimensions.

Spherical waves

The wave equation can be solved using the technique of separation of variables. To obtain a solution with constant frequencies, let us first Fourier transform the wave equation in time as

So we get,

This is the Helmholtz equation and can be solved using separation of variables. If spherical coordinates are used to describe a problem, then the solution to the angular part of the Helmholtz equation is given by spherical harmonics and the radial equation now becomes [11]

Here and the complete solution is now given by

where and are the spherical Hankel functions. To gain a better understanding of the nature of these spherical waves, let us go back and look at the case when . In this case, there is no angular dependence and the amplitude depends only on the radial distance i.e. . In this case, the wave equation reduces to

This equation can be rewritten as

where the quantity satisfies the one-dimensional wave equation. Therefore, there are solutions in the form

where F and G are general solutions to the one-dimensional wave equation, and can be interpreted as respectively an outgoing or incoming spherical wave. Such waves are generated by a point source, and they make possible sharp signals whose form is altered only by a decrease in amplitude as r increases (see an illustration of a spherical wave on the top right). Such waves exist only in cases of space with odd dimensions.[ citation needed ]

For physical examples of non-spherical wave solutions to the 3D wave equation that do possess angular dependence, see dipole radiation.

Monochromatic spherical wave

Cut-away of spherical wavefronts, with a wavelength of 10 units, propagating from a point source. Spherical Wave.gif
Cut-away of spherical wavefronts, with a wavelength of 10 units, propagating from a point source.

Although the word "monochromatic" is not exactly accurate since it refers to light or electromagnetic radiation with well-defined frequency, the spirit is to discover the eigenmode of the wave equation in three dimensions. Following the derivation in the previous section on Plane wave eigenmodes, if we again restrict our solutions to spherical waves that oscillate in time with well-defined constant angular frequency , then the transformed function has simply plane wave solutions,

,

or

.

From this we can observe that the peak intensity of the spherical wave oscillation, characterized as the squared wave amplitude

.

drops at the rate proportional to , an example of the inverse-square law.

Solution of a general initial-value problem

The wave equation is linear in u and it is left unaltered by translations in space and time. Therefore, we can generate a great variety of solutions by translating and summing spherical waves. Let φ(ξ, η, ζ) be an arbitrary function of three independent variables, and let the spherical wave form F be a delta function: that is, let F be a weak limit of continuous functions whose integral is unity, but whose support (the region where the function is non-zero) shrinks to the origin. Let a family of spherical waves have center at (ξ, η, ζ), and let r be the radial distance from that point. Thus

If u is a superposition of such waves with weighting function φ, then

the denominator 4πc is a convenience.

From the definition of the delta function, u may also be written as

where α, β, and γ are coordinates on the unit sphere S, and ω is the area element on S. This result has the interpretation that u(t, x) is t times the mean value of φ on a sphere of radius ct centered at x:

It follows that

The mean value is an even function of t, and hence if

then

These formulas provide the solution for the initial-value problem for the wave equation. They show that the solution at a given point P, given (t, x, y, z) depends only on the data on the sphere of radius ct that is intersected by the light cone drawn backwards from P. It does not depend upon data on the interior of this sphere. Thus the interior of the sphere is a lacuna for the solution. This phenomenon is called Huygens' principle . It is true for odd numbers of space dimension, where for one dimension the integration is performed over the boundary of an interval with respect to the Dirac measure. It is not satisfied in even space dimensions. The phenomenon of lacunas has been extensively investigated in Atiyah, Bott and Gårding (1970, 1973).

Scalar wave equation in two space dimensions

In two space dimensions, the wave equation is

We can use the three-dimensional theory to solve this problem if we regard u as a function in three dimensions that is independent of the third dimension. If

then the three-dimensional solution formula becomes

where α and β are the first two coordinates on the unit sphere, and is the area element on the sphere. This integral may be rewritten as a double integral over the disc D with center (x,y) and radius ct:

It is apparent that the solution at (t, x, y) depends not only on the data on the light cone where

but also on data that are interior to that cone.

Scalar wave equation in general dimension and Kirchhoff's formulae

We want to find solutions to utt − Δu = 0 for u : Rn × (0, ∞) → R with u(x, 0) = g(x) and ut(x, 0) = h(x). See Evans for more details.

Odd dimensions

Assume n ≥ 3 is an odd integer and gCm+1(Rn), hCm(Rn) for m = (n + 1)/2. Let and let

then

uC2(Rn × [0, ∞))
utt − Δu = 0 in Rn × (0, ∞)

Even dimensions

Assume n ≥ 2 is an even integer and gCm+1(Rn), hCm(Rn), for m = (n + 2)/2. Let and let

then

uC2(Rn × [0, ∞))
utt − Δu = 0 in Rn × (0, ∞)

Problems with boundaries

One space dimension

The Sturm–Liouville formulation

A flexible string that is stretched between two points x = 0 and x = L satisfies the wave equation for t > 0 and 0 < x < L. On the boundary points, u may satisfy a variety of boundary conditions. A general form that is appropriate for applications is

where a and b are non-negative. The case where u is required to vanish at an endpoint is the limit of this condition when the respective a or b approaches infinity. The method of separation of variables consists in looking for solutions of this problem in the special form

A consequence is that

The eigenvalue λ must be determined so that there is a non-trivial solution of the boundary-value problem

This is a special case of the general problem of Sturm–Liouville theory. If a and b are positive, the eigenvalues are all positive, and the solutions are trigonometric functions. A solution that satisfies square-integrable initial conditions for u and ut can be obtained from expansion of these functions in the appropriate trigonometric series.

Investigation by numerical methods

Approximating the continuous string with a finite number of equidistant mass points one gets the following physical model:

Figure 1: Three consecutive mass points of the discrete model for a string String wave 0.svg
Figure 1: Three consecutive mass points of the discrete model for a string

If each mass point has the mass m, the tension of the string is f, the separation between the mass points is Δx and ui, i = 1, ..., n are the offset of these n points from their equilibrium points (i.e. their position on a straight line between the two attachment points of the string) the vertical component of the force towards point i + 1 is

 

 

 

 

(1)

and the vertical component of the force towards point i − 1 is

 

 

 

 

(2)

Taking the sum of these two forces and dividing with the mass m one gets for the vertical motion:

 

 

 

 

(3)

As the mass density is

this can be written

 

 

 

 

(4)

The wave equation is obtained by letting Δx → 0 in which case ui(t) takes the form u(x, t) where u(x, t) is continuous function of two variables, takes the form and

But the discrete formulation ( 3 ) of the equation of state with a finite number of mass point is just the suitable one for a numerical propagation of the string motion. The boundary condition

where L is the length of the string takes in the discrete formulation the form that for the outermost points u1 and un the equation of motion are

 

 

 

 

(5)

and

 

 

 

 

(6)

while for 1 < i < n

 

 

 

 

(7)

where

If the string is approximated with 100 discrete mass points one gets the 100 coupled second order differential equations ( 5 ), ( 6 ) and ( 7 ) or equivalently 200 coupled first order differential equations.

Propagating these up to the times

using an 8th order multistep method the 6 states displayed in figure 2 are found:

Figure 2: The string at 6 consecutive epochs, the first (red) corresponding to the initial time with the string in rest String wave 1.svg
Figure 2: The string at 6 consecutive epochs, the first (red) corresponding to the initial time with the string in rest
Figure 3: The string at 6 consecutive epochs String wave 2.svg
Figure 3: The string at 6 consecutive epochs
Figure 4: The string at 6 consecutive epochs String wave 3.svg
Figure 4: The string at 6 consecutive epochs
Figure 5: The string at 6 consecutive epochs String wave 4.svg
Figure 5: The string at 6 consecutive epochs
Figure 6: The string at 6 consecutive epochs String wave 5.svg
Figure 6: The string at 6 consecutive epochs
Figure 7: The string at 6 consecutive epochs String wave 6.svg
Figure 7: The string at 6 consecutive epochs

The red curve is the initial state at time zero at which the string is "let free" in a predefined shape [12] with all . The blue curve is the state at time i.e. after a time that corresponds to the time a wave that is moving with the nominal wave velocity would need for one fourth of the length of the string.

Figure 3 displays the shape of the string at the times . The wave travels in direction right with the speed without being actively constraint by the boundary conditions at the two extremes of the string. The shape of the wave is constant, i.e. the curve is indeed of the form f(xct).

Figure 4 displays the shape of the string at the times . The constraint on the right extreme starts to interfere with the motion preventing the wave to raise the end of the string.

Figure 5 displays the shape of the string at the times when the direction of motion is reversed. The red, green and blue curves are the states at the times while the 3 black curves correspond to the states at times with the wave starting to move back towards left.

Figure 6 and figure 7 finally display the shape of the string at the times and . The wave now travels towards left and the constraints at the end points are not active any more. When finally the other extreme of the string the direction will again be reversed in a way similar to what is displayed in figure 6.

Several space dimensions

A solution of the wave equation in two dimensions with a zero-displacement boundary condition along the entire outer edge. Drum vibration mode12.gif
A solution of the wave equation in two dimensions with a zero-displacement boundary condition along the entire outer edge.

The one-dimensional initial-boundary value theory may be extended to an arbitrary number of space dimensions. Consider a domain D in m-dimensional x space, with boundary B. Then the wave equation is to be satisfied if x is in D and t > 0. On the boundary of D, the solution u shall satisfy

where n is the unit outward normal to B, and a is a non-negative function defined on B. The case where u vanishes on B is a limiting case for a approaching infinity. The initial conditions are

where f and g are defined in D. This problem may be solved by expanding f and g in the eigenfunctions of the Laplacian in D, which satisfy the boundary conditions. Thus the eigenfunction v satisfies

in D, and

on B.

In the case of two space dimensions, the eigenfunctions may be interpreted as the modes of vibration of a drumhead stretched over the boundary B. If B is a circle, then these eigenfunctions have an angular component that is a trigonometric function of the polar angle θ, multiplied by a Bessel function (of integer order) of the radial component. Further details are in Helmholtz equation.

If the boundary is a sphere in three space dimensions, the angular components of the eigenfunctions are spherical harmonics, and the radial components are Bessel functions of half-integer order.

Inhomogeneous wave equation in one dimension

The inhomogeneous wave equation in one dimension is the following:

with initial conditions given by

The function s(x, t) is often called the source function because in practice it describes the effects of the sources of waves on the medium carrying them. Physical examples of source functions include the force driving a wave on a string, or the charge or current density in the Lorenz gauge of electromagnetism.

One method to solve the initial value problem (with the initial values as posed above) is to take advantage of a special property of the wave equation in an odd number of space dimensions, namely that its solutions respect causality. That is, for any point (xi, ti), the value of u(xi, ti) depends only on the values of f(xi + cti) and f(xicti) and the values of the function g(x) between (xicti) and (xi + cti). This can be seen in d'Alembert's formula, stated above, where these quantities are the only ones that show up in it. Physically, if the maximum propagation speed is c, then no part of the wave that can't propagate to a given point by a given time can affect the amplitude at the same point and time.

In terms of finding a solution, this causality property means that for any given point on the line being considered, the only area that needs to be considered is the area encompassing all the points that could causally affect the point being considered. Denote the area that casually affects point (xi, ti) as RC. Suppose we integrate the inhomogeneous wave equation over this region.

To simplify this greatly, we can use Green's theorem to simplify the left side to get the following:

The left side is now the sum of three line integrals along the bounds of the causality region. These turn out to be fairly easy to compute

In the above, the term to be integrated with respect to time disappears because the time interval involved is zero, thus dt = 0.

For the other two sides of the region, it is worth noting that x ± ct is a constant, namely xi ± cti, where the sign is chosen appropriately. Using this, we can get the relation dx ± cdt = 0, again choosing the right sign:

And similarly for the final boundary segment:

Adding the three results together and putting them back in the original integral:

Solving for u(xi, ti) we arrive at

In the last equation of the sequence, the bounds of the integral over the source function have been made explicit. Looking at this solution, which is valid for all choices (xi, ti) compatible with the wave equation, it is clear that the first two terms are simply d'Alembert's formula, as stated above as the solution of the homogeneous wave equation in one dimension. The difference is in the third term, the integral over the source.

Other coordinate systems

In three dimensions, the wave equation, when written in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the Mathieu differential equation.

Further generalizations

Elastic waves

The elastic wave equation in three dimensions describes the propagation of waves in an isotropic homogeneous elastic medium. Most solid materials are elastic, so this equation describes such phenomena as seismic waves in the Earth and ultrasonic waves used to detect flaws in materials. While linear, this equation has a more complex form than the equations given above, as it must account for both longitudinal and transverse motion:

where:

Note that in this equation, both force and displacement are vector quantities. Thus, this equation is sometimes known as the vector wave equation. As an aid to understanding, the reader will observe that if f and ∇ ⋅ u are set to zero, this becomes (effectively) Maxwell's equation for the propagation of the electric field E, which has only transverse waves.

Dispersion relation

In dispersive wave phenomena, the speed of wave propagation varies with the wavelength of the wave, which is reflected by a dispersion relation

where ω is the angular frequency and k is the wavevector describing plane wave solutions. For light waves, the dispersion relation is ω = ±c|k|, but in general, the constant speed c gets replaced by a variable phase velocity:

See also

Notes

  1. Cannon, John T.; Dostrovsky, Sigalia (1981). The evolution of dynamics, vibration theory from 1687 to 1742. Studies in the History of Mathematics and Physical Sciences. 6. New York: Springer-Verlag. pp. ix + 184 pp. ISBN   978-0-3879-0626-3.GRAY, JW (July 1983). "BOOK REVIEWS". Bulletin (New Series) of the American Mathematical Society. 9 (1). (retrieved 13 Nov 2012).
  2. Gerard F Wheeler. The Vibrating String Controversy, (retrieved 13 Nov 2012). Am. J. Phys., 1987, v55, n1, p33–37.
  3. For a special collection of the 9 groundbreaking papers by the three authors, see First Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. – the controversy about vibrating strings (retrieved 13 Nov 2012). Herman HJ Lynge and Son.
  4. For de Lagrange's contributions to the acoustic wave equation, one can consult Acoustics: An Introduction to Its Physical Principles and Applications Allan D. Pierce, Acoustical Soc of America, 1989; page 18 (retrieved 9 Dec 2012).
  5. 1 2 3 Speiser, David. Discovering the Principles of Mechanics 1600–1800 , p. 191 (Basel: Birkhäuser, 2008).
  6. Tipler, Paul and Mosca, Gene. Physics for Scientists and Engineers, Volume 1: Mechanics, Oscillations and Waves; Thermodynamics , pp. 470–471 (Macmillan, 2004).
  7. Eric W. Weisstein. "d'Alembert's Solution". MathWorld . Retrieved 2009-01-21.
  8. D'Alembert (1747) "Recherches sur la courbe que forme une corde tenduë mise en vibration" (Researches on the curve that a tense cord forms [when] set into vibration), Histoire de l'académie royale des sciences et belles lettres de Berlin, vol. 3, pages 214–219.
  9. http://math.arizona.edu/~kglasner/math456/linearwave.pdf.
  10. V Guruprasad (2015), "Observational evidence for travelling wave modes bearing distance proportional shifts", EPL , 110 (5): 54001, arXiv: 1507.08222 , Bibcode:2015EL....11054001G, doi:10.1209/0295-5075/110/54001
  11. John David Jackson, Classical Electrodynamics, 3rd Edition, Wiley, page 425. ISBN   978-0-471-30932-1
  12. The initial state for "Investigation by numerical methods" is set with quadratic splines as follows:
    for
    for
    for
    with

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In physics, a wave packet is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere. Each component wave function, and hence the wave packet, are solutions of a wave equation. Depending on the wave equation, the wave packet's profile may remain constant or it may change (dispersion) while propagating.

Acoustic impedance and specific acoustic impedance are measures of the opposition that a system presents to the acoustic flow resulting of an acoustic pressure applied to the system. The SI unit of acoustic impedance is the pascal second per cubic metre or the rayl per square metre, while that of specific acoustic impedance is the pascal second per metre or the rayl. In this article the symbol rayl denotes the MKS rayl. There is a close analogy with electrical impedance, which measures the opposition that a system presents to the electrical flow resulting from an electrical voltage applied to the system.

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.

In mathematics, the Hamilton–Jacobi equation (HJE) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.

Helmholtz equation

In mathematics and physics, the Helmholtz equation, named for Hermann von Helmholtz, is the linear partial differential equation

Nonlinear Schrödinger equation

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose-Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.

In physics, the acoustic wave equation governs the propagation of acoustic waves through a material medium. The form of the equation is a second order partial differential equation. The equation describes the evolution of acoustic pressure or particle velocity u as a function of position x and time . A simplified form of the equation describes acoustic waves in only one spatial dimension, while a more general form describes waves in three dimensions.

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:

The telegrapher's equations are a pair of coupled, linear differential equations that describe the voltage and current on an electrical transmission line with distance and time. The equations come from Oliver Heaviside who in the 1880s developed the transmission line model, which is described in this article. The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can appear along the line. The theory applies to transmission lines of all frequencies including high-frequency transmission lines, audio frequency, low frequency and direct current.

The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics determined by Maxwell's equations, as an analogue for all types of particles.

Mild-slope equation The combined effects of diffraction and refraction for water waves propagating over variable depth and with lateral boundaries

In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.

Equatorial Rossby waves, often called planetary waves, are very long, low frequency waves found near the equator and are derived using the equatorial beta plane approximation.

In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation.

A Sverdrup wave is a wave in the ocean, which is affected by gravity and Earth's rotation.

In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems for the Laplacian in a bounded domain in the plane with smooth boundary. The methods use the theory of bounded operators on Hilbert space. They can be used to deduce regularity properties of solutions and to solve the corresponding eigenvalue problems.

SST turbulence model is a widely used and robust two-equation eddy-viscosity turbulence model used in Computational Fluid Dynamics. The model combines the k-omega turbulence model and K-epsilon turbulence model such that the k-omega is used in the inner region of the boundary layer and switches to the k-epsilon in the free shear flow.

Stokes problem

In fluid dynamics, Stokes problem also known as Stokes second problem or sometimes referred to as Stokes boundary layer or Oscillating boundary layer is a problem of determining the flow created by an oscillating solid surface, named after Sir George Stokes. This is considered as one of the simplest unsteady problem that have exact solution for the Navier-Stokes equations. In turbulent flow, this is still named a Stokes boundary layer, but now one has to rely on experiments, numerical simulations or approximate methods in order to obtain useful information on the flow.

In fluid dynamics, Beltrami flows are flows in which the vorticity vector and the velocity vector are parallel to each other. In other words, Beltrami flow is a flow where Lamb vector is zero. It is named after the Italian mathematician Eugenio Beltrami due to his derivation of the Beltrami vector field, while initial developments in fluid dynamics were done by the Russian scientist Ippolit S. Gromeka in 1881.

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