A pulse traveling through a string with fixed endpoints as modeled by the wave equation
Spherical waves coming from a point source
A solution to the 2D wave equation
The (two-way) wave equation is a second-order partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in scalars by scalar functions u = u (x1, x2, ..., xn; t) of a time variable t (a variable representing time) and one or more spatial variables x1, x2, ..., xn (variables representing a position in a space under discussion), while there are vector wave equations describing waves in vectors such as waves for an electrical field, magnetic field, and magnetic vector potential and elastic waves. By comparison with vector wave equations, the scalar wave equation can be seen as a special case of the vector wave equations; in the Cartesian coordinate system, the scalar wave equation is the equation to be satisfied by each component (for each coordinate axis, such as the x component for the x axis) of a vector wave without sources of waves in the considered domain (i.e., a space and time). For example, in the Cartesian coordinate system, for as the representation of an electric vector field wave in the absence of wave sources, each coordinate axis component (i = x, y, z) must satisfy the scalar wave equation. Other scalar wave equation solutions u are for physical quantities in scalars such as pressure in a liquid or gas, or the displacement along some specific direction of particles of a vibrating solid away from their resting (equilibrium) positions.
u is the factor representing a displacement from rest situation – it could be gas pressure above or below normal, or the height of water in a pond above or below rest, or something else.
t represents time.
is a term for how the displacement accelerates, i.e. not the speed at which the displacement is changing, but in fact the rate at which that displacement's speed is itself changing – its acceleration.
x represents space or position.
is a term for how the displacement is varying at the point x in one of the dimensions (like one of the axes on a graph). It is not the rate at which the displacement is changing across space, but in fact the rate at which the change itself is changing across space – its second derivative. In other words, this term shows how the displacement's changes are squashed up in a tiny surrounding area.
The equation states that at any given instance, at any given point, the way the displacement accelerates is proportional to the way the displacement's changes are squashed up in the surrounding area. Or, in even simpler terms, the way displacements get pushed around is proportional to how pointy the displacement is, and conversely.
A solution of this (two-way) wave equation can be quite complicated, but it can be analyzed as a linear combination of simple solutions that are sinusoidalplane waves with various directions of propagation and wavelengths but all with the same propagation speed c. This analysis is possible because the wave equation is linear and homogeneous, so that any multiple of a solution is also a solution, and the sum of any two solutions is again a solution. This property is called the superposition principle in physics.
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 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 variableu 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 xy plane.
Derivation of the wave equationedit
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 vibrating in a two-dimensional plane, with each of its elements being pulled in opposite directions by the force of tension.
Another physical setting for derivation of the wave equation in one space dimension uses 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 lawedit
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:
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 resulting force exerted on the mass m at the location x + h is:
By equating the latter equation with
the equation of motion for the weight at the location x + h is obtained:
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.
Stress pulse in a baredit
In the case of a stress pulse propagating longitudinally through a bar, the bar 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 uniform bar, i.e. 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 bar, and therefore
where ρ is the density of the material. The wave equation reduces to
The speed of a stress wave in a bar is therefore .
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
changes the wave equation into
which leads to the general solution
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.
This one-way wave equation can be solved by interpreting it as telling us that the directional derivative of v in the (1, -c) direction is 0. This means that the value of v is constant on characteristic lines of the form x + ct = x0, and thus that v must depend only on x + ct, that is, have the form H(x + ct). Then, to solve the first (inhomogenous) one-way wave equation relating v to u, we can note that its homogenous solution must be a function of the form F(x - ct), by logic similar to the above. Guessing a particular solution of the form G(x + ct), we find that
Expanding out the left side, rearranging terms, then using the change of variables s = x + ct simplifies the equation to
This means we can find a particular solution G of the desired form by integration. Thus, we have again shown that u obeys u(x, t) = F(x - ct) + G(x + ct) via one-way wave equations. Note that since the one-way wave differential operators commute for analytic functions by Clairaut's theorem, we could have equivalently set up our system of one-way wave equations so that we solved them in the opposite order.
The usual second-order wave equation is sometimes called the "two-way wave equation" (superposition of two waves) to distinguish it from the first-order one-way wave equation describing the wave propagation of a single wave in a pre-defined direction.
For an initial-value problem, the arbitrary functions F and G can be determined to satisfy initial conditions:
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.
Another way to solve 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 ω, so that the temporal part of the wave function takes the form e−iωt = cos(ωt) − i sin(ωt), and the amplitude is a function f(x) of the spatial variable x, giving a separation of variables for the wave function:
The total wave function for this eigenmode is then the linear combination
where complex numbers A, B 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 s±(ω) 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 packetu(x, t), 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 ω. 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.
Vectorial wave equation in three space dimensionsedit
The vectorial wave equation (from which the scalar wave equation can be directly derived) can be obtained by applying a force equilibrium to an infinitesimal volume element. In a homogeneous continuum (cartesian coordinate ) with a constant modulus of elasticity a vectorial, elastic deflection causes the stress tensor . The local equilibrium of a) the tension force due to deflection and b) the inertial force caused by the local acceleration can be written as
By merging density and elasticity module the sound velocity results (material law). After insertion, follows the well-known governing wave equation for a homogeneous medium:
(Note: Instead of vectorial only scalar can be used, i.e. waves are travelling only along the axis, and the scalar wave equation follows as .)
The above vectorial partial differential equation of the 2nd order delivers two mutually independent solutions. From the quadratic velocity term can be seen that there are two waves travelling in opposite directions and are possible, hence results the designation “two-way wave equation”.
It can be shown for plane longitudinal wave propagation that the synthesis of two one-way wave equations leads to a general two-way wave equation. For special two-wave equation with the d'Alembert operator results:
For this simplifies to
Therefore, the vectorial 1st-order one-way wave equation with waves travelling in a pre-defined propagation direction results as
Scalar wave equation in three space dimensionsedit
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.
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
To gain a better understanding of the nature of these spherical waves, let us go back and look at the case when l = 0. In this case, there is no angular dependence, and the amplitude depends only on the radial distance i.e. Ψ(r, t) → u(r, t). In this case, the wave equation reduces to
This equation can be rewritten as
where the quantity ru 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 and incoming spherical waves. The outgoing wave can be 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.
For physical examples of solutions to the 3D wave equation that possess angular dependence, see dipole radiation.
Monochromatic spherical waveedit
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 ω, then the transformed function ru(r, t) has simply plane-wave solutions:
From this we can observe that the peak intensity of the spherical-wave oscillation, characterized as the squared wave amplitude
The wave equation is linear in u and 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
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 dimensionsedit
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 dω 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 formulaeedit
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.
Assume n ≥ 3 is an odd integer, and g ∈ Cm+1(Rn), h ∈ Cm(Rn) for m = (n + 1)/2. Let γn = 1 × 3 × 5 × ⋯ × (n − 2) and let
Assume n ≥ 2 is an even integer and g ∈ Cm+1(Rn), h ∈ Cm(Rn), for m = (n + 2)/2. Let γn = 2 × 4 × ⋯ × n and let
u ∈ C2(Rn × [0, ∞))
utt − Δu = 0 in Rn × (0, ∞)
Problems with boundariesedit
One space dimensionedit
Reflection and transmission at the boundary of two mediaedit
For an incident wave traveling from one medium (where the wave speed is c1) to another medium (where the wave speed is c2), one part of the wave will transmit into the second medium, while another part reflects back into the other direction and stays in the first medium. The amplitude of the transmitted wave and the reflected wave can be calculated by using the continuity condition at the boundary.
Consider the component of the incident wave with an angular frequency of ω, which has the waveform
At t = 0, the incident reaches the boundary between the two media at x = 0. Therefore, the corresponding reflected wave and the transmitted wave will have the waveforms
The continuity condition at the boundary is
This gives the equations
and we have the reflectivity and transmissivity
When c2 < c1, the reflected wave has a reflection phase change of 180°, since B/A < 0. The energy conservation can be verified by
The above discussion holds true for any component, regardless of its angular frequency of ω.
The limiting case of c2 = 0 corresponds to a "fixed end" that does not move, whereas the limiting case of c2 → ∞ corresponds to a "free end".
The Sturm–Liouville formulationedit
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 (i.e. "fixed end") 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 eigenvalueλ 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 methodsedit
Approximating the continuous string with a finite number of equidistant mass points, one gets the following physical model:
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
and the vertical component of the force towards point i − 1 is
Taking the sum of these two forces and dividing with the mass m, one gets for the vertical motion:
As the mass density is
this can be written
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, üi takes the form ∂2u/∂t2, 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 equations of motion are
while for 1 < i < n
where c = √f/ρ.
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 are found:
The red curve is the initial state at time zero, at which the string is "let free" in a predefined shape 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 c = √f/ρ would need for one fourth of the length of the string.
The wave travels towards right with the speed c = √f/ρ 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(x − ct).
The constraint on the right extreme starts to interfere with the motion, preventing the wave to raise the end of the string.
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 with the wave starting to move back towards left.
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 is reached, the direction will again be reversed in a way similar to what is displayed in the figure above for
Several space dimensionsedit
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
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 dimensionedit
The inhomogeneous wave equation in one dimension is
with initial conditions
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(xi − cti) and the values of the function g(x) between (xi − cti) 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 cannot 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 causally 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 gives
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.
Wave equation for inhomogeneous media, three-dimensional caseedit
For one-way wave propagation, i.e. wave are travelling in a pre-defined wave direction ( or ) in inhomogeneous media, wave propagation can also be calculated with a tensorial one-way wave equation (resulting from factorization of the vectorial two-way wave equation), and an analytical solution can be derived.
The elastic wave equation (also known as the Navier–Cauchy equation) in three dimensions describes the propagation of waves in an isotropichomogeneouselastic 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:
λ and μ are the so-called Lamé parameters describing the elastic properties of the medium,
ρ is the density,
f is the source function (driving force),
u is the displacement vector.
By using ∇ × (∇ × u) = ∇(∇ ⋅ u) − ∇ ⋅ ∇ u = ∇(∇ ⋅ u) − ∆u, the elastic wave equation can be rewritten into the more common form of the Navier–Cauchy equation.
Note that in the elastic wave 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.
^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, p. 214–219.
See also: D'Alembert (1747) "Suite des recherches sur la courbe que forme une corde tenduë mise en vibration" (Further 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, p. 220–249.
See also: D'Alembert (1750) "Addition au mémoire sur la courbe que forme une corde tenduë mise en vibration," Histoire de l'académie royale des sciences et belles lettres de Berlin, vol. 6, p. 355–360.
^"First and second order linear wave equations" (PDF). math.arizona.edu. Archived from the original (PDF) on 2017-12-15.
^Bschorr, Oskar; Raida, Hans-Joachim (April 2021). "Spherical One-Way Wave Equation". Acoustics. 3 (2): 309–315. doi:10.3390/acoustics3020021. Text was copied from this source, which is available under a Creative Commons Attribution 4.0 International License.