Wave equation

The wave equation , also D'Alembert equation according to Jean-Baptiste le Rond d'Alembert , determines the propagation of waves such as sound or light . It is one of the hyperbolic differential equations .

If the medium or vacuum only conducts the wave and does not generate waves itself, it is more precisely the homogeneous wave equation , the linear partial differential equation of the second order

${\displaystyle {\frac {1}{c^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}-\sum _{i=1}^{n}{\frac {\partial ^{2}u}{\partial x_{i}^{2}}}=0}$

for a real function of spacetime. Here is the dimension of the room. The parameter is the speed of propagation of the wave, i.e. the speed of sound for sound (in a homogeneous and isotropic medium) and the speed of light for light. ${\displaystyle u(t,x_{1},\dots ,x_{n})}$ ${\displaystyle n}$ ${\displaystyle c}$

The differential operator of the wave equation is called the D'Alembert operator and is noted with the symbol . ${\displaystyle \Box }$

${\displaystyle \Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\sum _{i=1}^{n}{\frac {\partial ^{2}}{\partial x_{i}^{2}}}}$,

The solutions to the wave equation are called waves . Because the equation is linear, waves overlap without affecting each other. Since the coefficients of the wave equation do not depend on the place or the time, waves behave regardless of where or when and in which direction they are excited. Displaced, delayed or rotated waves are also solutions to the wave equation.

The inhomogeneous wave equation is understood to be the linearly inhomogeneous partial differential equation

${\displaystyle \Box u=v\ .}$

It describes the development of waves over time in a medium that generates waves itself. The inhomogeneity is also called the source of the wave . ${\displaystyle v}$ ${\displaystyle u}$

The wave equation in a spatial dimension

The D'Alembert operator in a spatial dimension

${\displaystyle \Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-{\frac {\partial ^{2}}{\partial x^{2}}}}$

breaks down into the product due to the theorem of black as in the binomial formula ${\displaystyle (a^{2}-b^{2})=(a-b)(a+b)}$

${\displaystyle \Box =\left({\frac {1}{c}}{\frac {\partial }{\partial t}}-{\frac {\partial }{\partial x}}\right)\left({\frac {1}{c}}{\frac {\partial }{\partial t}}+{\frac {\partial }{\partial x}}\right)}$.

Hence the wave equation has the general solution in one spatial dimension

${\displaystyle u\left(t,x\right)=f(x+ct)+g(x-ct)}$

with any doubly differentiable functions and . The first summand is a wave moving to the left and the second summand is a wave moving to the right with the same shape. The straight lines are the characteristics of the wave equation. ${\displaystyle f(x)}$ ${\displaystyle g(x)}$ ${\displaystyle f(x+ct)}$ ${\displaystyle g(x-ct)}$ ${\displaystyle x\pm ct={\text{konstant}}}$

Be

${\displaystyle \phi (x)=u(0,x)=f(x)+g(x)}$

the initial value and

${\displaystyle \psi (x)={\frac {1}{c}}{\frac {\partial u}{\partial t}}(0,x)=f'(x)-g'(x)}$

the initial time derivative of the wave. These functions of space are collectively called the initial values ​​of the wave.

The integration of the last equation gives

${\displaystyle f(x)-g(x)=\int _{x_{0}}^{x}\psi (\xi )\,\mathrm {d} \xi \ .}$

By dissolving one gets

${\displaystyle f(x)={\frac {1}{2}}\left(\phi (x)+\int _{x_{0}}^{x}\psi (\xi )\,\mathrm {d} \xi \right)\ ,}$

${\displaystyle g(x)={\frac {1}{2}}\left(\phi (x)-\int _{x_{0}}^{x}\psi (\xi )\,\mathrm {d} \xi \right)\ .}$

The solution of the wave equation is therefore expressed in terms of its initial values

${\displaystyle u(t,x)={\frac {1}{2}}\left(\phi (x+ct)+\phi (x-ct)+\int _{x-ct}^{x+ct}\psi (\xi )\,\mathrm {d} \xi \right)\ .}$

This is also known as D'Alembert's solution to the wave equation ( d'Alembert , 1740s).

The wave equation in three spatial dimensions

The general solution of the wave equation can be expressed as a linear combination of plane waves

${\displaystyle u({\vec {x}},t)=\int \mathrm {d} \omega \int \mathrm {d} ^{3}{\vec {k}}\,A(\omega ,k)e^{\mathrm {i} ({\vec {k}}\cdot {\vec {x}}-\omega t)}\delta (\omega -c|{\vec {k}}|)}$

write. The delta distribution ensures that the dispersion relation is preserved. Such a plane wave moves in the direction of . With the superposition of such solutions, however, it is not obvious how their initial values ​​are related to the later solution. ${\displaystyle \omega =c|{\vec {k}}|}$ ${\displaystyle {\vec {k}}}$

The general solution of the homogeneous wave equation can be represented in three spatial dimensions by means of the mean values ​​of the initial values. Let the function and its time derivative be given at the beginning by functions and , ${\displaystyle u(t,{\vec {x}})}$ ${\displaystyle t=0}$ ${\displaystyle \phi }$ ${\displaystyle \psi }$

${\displaystyle u(0,{\vec {x}})=\phi ({\vec {x}}),\quad {\frac {1}{c}}{\frac {\partial }{\partial t}}u(0,{\vec {x}})=\psi ({\vec {x}})\,,}$

then is the linear combination of means

${\displaystyle u(t,{\vec {x}})=ct\,M_{t,{\vec {x}}}[\psi ]+{\frac {1}{c}}{\frac {\partial }{\partial t}}(ct\,M_{t,{\vec {x}}}[\phi ])}$

the corresponding solution of the homogeneous wave equation. Here designated

${\displaystyle M_{t,{\vec {x}}}[\chi ]={\frac {1}{4\,\pi }}\int _{-1}^{1}\mathrm {d} \cos \theta \int _{0}^{2\pi }\mathrm {d} \varphi \,\chi ({\vec {x}}+ct{\vec {n}}(\theta ,\varphi ))\quad {\text{mit}}\quad {\vec {n}}(\theta ,\varphi )={\begin{pmatrix}\sin \theta \cos \varphi \\\sin \theta \sin \varphi \\\cos \theta \end{pmatrix}}}$

is the mean value of the function averaged over a spherical shell around the point with radius in particular ${\displaystyle \chi \,,}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle c|t|.}$ ${\displaystyle M_{0,{\vec {x}}}[\chi ]=\chi ({\vec {x}}).}$

As this representation of the solution indicated by the initial values, the solution depends continuously on the initial values, depending on the time at the place only on the initial values in the places from, from which one in runtime with speed can reach. It thus satisfies the Huygens principle . ${\displaystyle t}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle {\vec {y}}}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle |t|}$ ${\displaystyle c}$

This principle does not apply to one-dimensional systems and in straight spatial dimensions. There the solutions currently also depend on initial values ​​at closer points , from which one can reach with a lower speed. ${\displaystyle t}$ ${\displaystyle {\vec {y}}}$ ${\displaystyle {\vec {x}}}$

The solution of the inhomogeneous wave equation in three spatial dimensions

${\displaystyle u(t,{\vec {x}})=ct\,M_{t,{\vec {x}}}[\psi ]+{\frac {1}{c}}{\frac {\partial }{\partial t}}(ct\,M_{t,{\vec {x}}}[\phi ])+{\frac {1}{4\pi }}\int _{|{\vec {z}}|\leq c|t|}\mathrm {d} ^{3}{\vec {z}}\,{\frac {v(ct-\operatorname {sign} (t)|{\vec {z}}|,{\vec {x}}+{\vec {z}})}{|{\vec {z}}|}}}$

currently only depends on the inhomogeneity on the backward light cone of , at negative times only on the inhomogeneity on the forward light cone. The inhomogeneity and the initial values ​​affect the solution at the speed of light. ${\displaystyle {\vec {x}}}$ ${\displaystyle t>0}$ ${\displaystyle {\vec {x}}}$

Retarded potential

The retarded potential

${\displaystyle u_{\text{retardiert}}(t,{\vec {x}})={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}\mathrm {d} ^{3}{\vec {z}}\,{\frac {v(ct-|{\vec {z}}|,\,{\vec {x}}+{\vec {z}})}{|{\vec {z}}|}}}$

is a solution of the inhomogeneous wave equation, which assumes that the inhomogeneity on all backward light cones drops faster than it does. It is the wave that is completely created by the medium without a passing wave. ${\displaystyle v}$ ${\displaystyle 1/r^{2}}$

In electrodynamics, the continuity equation limits the inhomogeneity. Thus the charge density of a non-vanishing total charge can never disappear everywhere. In perturbation theory, inhomogeneities occur that do not decrease spatially quickly enough. Then the associated retarded integral diverges and has a so-called infrared divergence.

The somewhat more complex representation of the solution through its initial values ​​at finite time and through integrals over finite sections of the light cone is free of such infrared divergences.

Lorentz invariance of the D'Alembert operator

The D'Alembert operator is invariant under translations and Lorentz transformations in the sense that, when applied to Lorentz concatenated functions, it gives the same result as the Lorentz concatenated derived function ${\displaystyle \Box }$ ${\displaystyle \Lambda }$ ${\displaystyle f\circ \Lambda ^{-1}}$

${\displaystyle (\Box f)\circ \Lambda ^{-1}=\Box \,(f\circ \Lambda ^{-1})\ .}$

Correspondingly, the Laplace operator is invariant under translations and rotations.

The homogeneous wave equation is invariant even under conformal transformations, especially under stretching.

See also

• Klein-Gordon equation
• Standing wave

literature

• Richard Courant , David Hilbert : Methods of Mathematical Physics. Volume 2. Second edition. Springer Verlag, Berlin 1968 ( Heidelberger Taschenbücher 31, ISSN  0073-1684 ).
• Fritz John : Partial Differential Equations, 4th edition, Springer 1982

Web links

• Gernot Pfanner, The Wave Equation (PDF)
• Norbert Dragon, Geometry of Relativity (PDF; 2.5 MB) Chapter 5.5

Individual evidence

1. ^ Eric Weisstein, d'Alembert's solution, Mathworld