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The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.
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]}
The wave equation is a hyperbolic partial differential equation. It typically concerns a time variable t, one or more spatial variables x_{1}, x_{2}, …, x_{n}, and a scalar function u = u (x_{1}, x_{2}, …, x_{n}; t), whose values could model the displacement of a wave. The wave equation for u is
where ∇^{2} is the (spatial) Laplacian and where c is a fixed constant.
Solutions of this equation that are initially zero outside some restricted region propagate out from the region at a fixed speed in all spatial directions, as do physical waves from a localized disturbance; the constant c is identified with the propagation speed of the wave. This equation is linear, as the sum of any two solutions is again a solution: in physics this property is called the superposition principle.
The equation alone does not specify a solution; a unique solution is usually obtained by setting a problem with further conditions, such as initial conditions, which prescribe the value and velocity of the wave. Another important class of problems specifies boundary conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments.
The wave equation, and also modifications of it, are found in elastic physics, quantum mechanics, plasma physics and general relativity, for example.
The wave equation in one space dimension can be written like this:
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 "y" may represent a second space dimension, as in the case of a string that is located in the x-y plane.
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).
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. stress) that is traveling in an elastic material. The forces exerted on the mass m at the location x+h are:
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.
We want to find solutions to u_{tt}−Δu = 0 for u : R^{n} × (0, ∞) → R with u(x, 0) = g(x) and u_{t}(x, 0) = h(x). See Evans for more details.
Assume n ≥ 3 is an odd integer and g ∈ C^{m+1}(R^{n}), h ∈ C^{m}(R^{n}) for m = (n+1)/2. Let \gamma_n = 1\cdot 3 \cdot 5 \cdot .. \cdot (n-2) and let
then
Assume n ≥ 2 is an even integer and g ∈ C^{m+1}(R^{n}), h ∈ C^{m}(R^{n}), for m = (n+2)/2. Let \gamma_n = 2 \cdot 4 \cdot .. \cdot n and let
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 u_{t} can be obtained from expansion of these functions in the appropriate trigonometric series.
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 u_{i}, 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
\frac{u_{i+1}-u_i}{\Delta x}\ f
(1)
and the vertical component of the force towards point i−1 is
\frac{u_{i-1}-u_i}{\Delta x}\ f
(2)
Taking the sum of these two forces and dividing with the mass m one gets for the vertical motion:
\ddot u_i=\left(\frac{f}{m\ \Delta x} \right) \left(u_{i+1} + u_{i-1}\ -\ 2u_i\right)
(3)
As the mass density is
this can be written
\ddot u_i=\left(\frac{f}{\rho\ {\Delta x}^2} \right) \left(u_{i+1} + u_{i-1}\ -\ 2u_i\right)
(4)
The wave equation is obtained by letting Δx → 0 in which case u_{i}(t) takes the form u(x, t) where u(x, t) is continuous function of two variables, \ddot u_i takes the form \partial^2 u \over \partial t^2 and
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 8-th order multistep method the 6 states displayed in figure 2 are found:
The red curve is the initial state at time zero at which the string is "let free" in a predefined shape ^{[10]} with all \dot u_i=0. The blue curve is the state at time \frac{L}{c}\ 0.25, i.e. after a time that corresponds to the time a wave that is moving with the nominal wave velocity c=\sqrt{\frac{f}{\rho}} would need for one fourth of the length of the string.
Figure 3 displays the shape of the string at the times \frac{L}{c}\ k\ 0.05\ \ k=6,\cdots ,11. The wave travels in direction right with the speed c=\sqrt{\frac{f}{\rho}} 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).
Figure 4 displays the shape of the string at the times \frac{L}{c}\ k\ 0.05\ \ k=12,\cdots ,17. 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 \frac{L}{c}\ k\ 0.05\ \ k=18,\cdots ,23 when the direction of motion is reversed. The red, green and blue curves are the states at the times \frac{L}{c}\ k\ 0.05\ \ k=18,\cdots ,20 while the 3 black curves correspond to the states at times \frac{L}{c}\ k\ 0.05\ \ k=21,\cdots ,23 with the wave starting to move back towards left.
Figure 6 and figure 7 finally display the shape of the string at the times \frac{L}{c}\ k\ 0.05\ \ k=24,\cdots ,29 and \frac{L}{c}\ k\ 0.05\ \ k=30,\cdots ,35. 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
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.
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 (x_{i}, t_{i}), the value of u(x_{i}, t_{i}) depends only on the values of f(x_{i}+ct_{i}) and f(x_{i}−ct_{i}) and the values of the function g(x) between (x_{i}−ct_{i}) and (x_{i}+ct_{i}). 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 (x_{i}, t_{i}) as R_{C}. 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, namingly x_{i}±ct_{i}, 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(x_{i}, t_{i}) 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 (x_{i}, t_{i}) 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.
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.
To model dispersive wave phenomena, those in which the speed of wave propagation varies with the frequency of the wave, the constant c is replaced by the phase velocity:
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.
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