The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. This article focuses on waves in classical physics. Quantum physics uses an operator-based wave equation often as a relativistic wave equation.
Introduction The wave equation is a hyperbolic 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 ( x , y , z , t ) {\displaystyle u=u(x,y,z,t)} of a time variable t {\displaystyle t} (a variable representing time) and one or more spatial variables x , y , z {\displaystyle x,y,z} (variables representing a position in a space under discussion). At the same time, 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 {\displaystyle x} component for the x axis) of a vector wave without sources of waves in the considered domain (i.e., space and time). For example, in the Cartesian coordinate system, for ( E x , E y , E z ) {\displaystyle (E_{x},E_{y},E_{z})} as the representation of an electric vector field wave E → {\displaystyle {\vec {E}}} in the absence of wave sources, each coordinate axis component E i , i = x , y , z , {\displaystyle E_{i},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. The scalar wave equation is
where
c {\displaystyle c} is a fixed non-negative real coefficient representing the propagation speed of the wave
u {\displaystyle u} is a scalar field representing the displacement or, more generally, the conserved quantity (e.g. pressure or density)
x , y , {\displaystyle x,y,} and z {\displaystyle z} are the three spatial coordinates and t {\displaystyle t} being the time coordinate. The equation states that, at any given point, the second derivative of u {\displaystyle u} with respect to time is proportional to the sum of the second derivatives of u {\displaystyle u} with respect to space, with the constant of proportionality being the square of the speed of the wave. Using notations from vector calculus, the wave equation can be written compactly as
u t t = c 2 Δ u , {\displaystyle u_{tt}=c^{2}\Delta u,}
or
◻ u = 0 , {\displaystyle \Box u=0,} where the double subscript denotes the second-order partial derivative with respect to time, Δ {\displaystyle \Delta } is the Laplace operator and ◻ {\displaystyle \Box } the d'Alembert operator, defined as:
u t t = ∂ 2 u ∂ t 2 , Δ = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 , ◻ = 1 c 2 ∂ 2 ∂ t 2 − Δ . {\displaystyle u_{tt}={\frac {\partial ^{2}u}{\partial t^{2}}},\qquad \Delta ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}},\qquad \Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\Delta .}
A solution to this (two-way) wave equation can be quite complicated. Still, it can be analyzed as a linear combination of simple solutions that are sinusoidal plane waves with various directions of propagation and wavelengths but all with the same propagation speed c {\displaystyle 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.
Wave equation in one space dimension
The wave equation in one spatial dimension can be written as follows:
∂ 2 u ∂ t 2 = c 2 ∂ 2 u ∂ x 2 . {\displaystyle {\frac {\partial ^{2}u}{\partial t^{2}}}=c^{2}{\frac {\partial ^{2}u}{\partial x^{2}}}.} This equation is typically described as having only one spatial dimension x {\displaystyle x} , because the only other independent variable is the time t {\displaystyle t} .
Derivation
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).
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 {\displaystyle m} interconnected with massless springs of length h {\displaystyle h} . The springs have a spring constant of k {\displaystyle k} :
Here the dependent variable u ( x ) {\displaystyle u(x)} measures the horizontal displacement from equilibrium of the mass situated at x {\displaystyle x} , so that u ( x ) {\displaystyle 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 {\displaystyle m} at the location x + h {\displaystyle x+h} is:
F Hooke = F x + 2 h − F x = k [ u ( x + 2 h , t ) − u ( x + h , t ) ] − k [ u ( x + h , t ) − u ( x , t ) ] . {\displaystyle {\begin{aligned}F_{\text{Hooke}}&=F_{x+2h}-F_{x}=k[u(x+2h,t)-u(x+h,t)]-k[u(x+h,t)-u(x,t)].\end{aligned}}}
By equating the latter equation with
F Newton = m a ( t ) = m ∂ 2 ∂ t 2 u ( x + h , t ) , {\displaystyle {\begin{aligned}F_{\text{Newton}}&=m\,a(t)=m\,{\frac {\partial ^{2}}{\partial t^{2}}}u(x+h,t),\end{aligned}}}
the equation of motion for the weight at the location x + h {\displaystyle x+h} is obtained:
∂ 2 ∂ t 2 u ( x + h , t ) = k m [ u ( x + 2 h , t ) − u ( x + h , t ) − u ( x + h , t ) + u ( x , t ) ] . {\displaystyle {\frac {\partial ^{2}}{\partial t^{2}}}u(x+h,t)={\frac {k}{m}}[u(x+2h,t)-u(x+h,t)-u(x+h,t)+u(x,t)].}
If the array of weights consists of N {\displaystyle N} weights spaced evenly over the length L = N h {\displaystyle L=Nh} of total mass M = N m {\displaystyle M=Nm} , and the total spring constant of the array K = k / N {\displaystyle K=k/N} , we can write the above equation as
∂ 2 ∂ t 2 u ( x + h , t ) = K L 2 M [ u ( x + 2 h , t ) − 2 u ( x + h , t ) + u ( x , t ) ] h 2 . {\displaystyle {\frac {\partial ^{2}}{\partial t^{2}}}u(x+h,t)={\frac {KL^{2}}{M}}{\frac {[u(x+2h,t)-2u(x+h,t)+u(x,t)]}{h^{2}}}.}
Taking the limit N → ∞ , h → 0 {\displaystyle N\rightarrow \infty ,h\rightarrow 0} and assuming smoothness, one gets
∂ 2 u ( x , t ) ∂ t 2 = K L 2 M ∂ 2 u ( x , t ) ∂ x 2 , {\displaystyle {\frac {\partial ^{2}u(x,t)}{\partial t^{2}}}={\frac {KL^{2}}{M}}{\frac {\partial ^{2}u(x,t)}{\partial x^{2}}},}
which is from the definition of a second derivative. K L 2 / M {\displaystyle KL^{2}/M} is the square of the propagation speed in this particular case.
Stress pulse in a bar 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 {\displaystyle K} given by
K = E A L , {\displaystyle K={\frac {EA}{L}},}
where A {\displaystyle A} is the cross-sectional area, and E {\displaystyle E} is the Young's modulus of the material. The wave equation becomes
∂ 2 u ( x , t ) ∂ t 2 = E A L M ∂ 2 u ( x , t ) ∂ x 2 . {\displaystyle {\frac {\partial ^{2}u(x,t)}{\partial t^{2}}}={\frac {EAL}{M}}{\frac {\partial ^{2}u(x,t)}{\partial x^{2}}}.}
A L {\displaystyle AL} is equal to the volume of the bar, and therefore
A L M = 1 ρ , {\displaystyle {\frac {AL}{M}}={\frac {1}{\rho }},}
where ρ {\displaystyle \rho } is the density of the material. The wave equation reduces to
∂ 2 u ( x , t ) ∂ t 2 = E ρ ∂ 2 u ( x , t ) ∂ x 2 . {\displaystyle {\frac {\partial ^{2}u(x,t)}{\partial t^{2}}}={\frac {E}{\rho }}{\frac {\partial ^{2}u(x,t)}{\partial x^{2}}}.}
The speed of a stress wave in a bar is therefore E / ρ {\displaystyle {\sqrt {E/\rho }}} .
General solution
Algebraic approach For the one-dimensional wave equation a relatively simple general solution may be found. Defining new variables
ξ = x − c t , η = x + c t {\displaystyle {\begin{aligned}\xi &=x-ct,\\\eta &=x+ct\end{aligned}}}
changes the wave equation into
∂ 2 u ∂ ξ ∂ η ( x , t ) = 0 , {\displaystyle {\frac {\partial ^{2}u}{\partial \xi \partial \eta }}(x,t)=0,}
which leads to the general solution
u ( x , t ) = F ( ξ ) + G ( η ) = F ( x − c t ) + G ( x + c t ) . {\displaystyle u(x,t)=F(\xi )+G(\eta )=F(x-ct)+G(x+ct).}
In other words, the solution is the sum of a right-traveling function F {\displaystyle F} and a left-traveling function G {\displaystyle 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 {\displaystyle c} . This was derived by Jean le Rond d'Alembert. Another way to arrive at this result is to factor the wave equation using two first-order differential operators:
[ ∂ ∂ t − c ∂ ∂ x ] [ ∂ ∂ t + c ∂ ∂ x ] u = 0. {\displaystyle \left[{\frac {\partial }{\partial t}}-c{\frac {\partial }{\partial x}}\right]\left[{\frac {\partial }{\partial t}}+c{\frac {\partial }{\partial x}}\right]u=0.}
Then, for our original equation, we can define
v ≡ ∂ u ∂ t + c ∂ u ∂ x , {\displaystyle v\equiv {\frac {\partial u}{\partial t}}+c{\frac {\partial u}{\partial x}},}
and find that we must have
∂ v ∂ t − c ∂ v ∂ x = 0. {\displaystyle {\frac {\partial v}{\partial t}}-c{\frac {\partial v}{\partial x}}=0.}
This advection equation can be solved by interpreting it as telling us that the directional derivative of v {\displaystyle v} in the ( 1 , − c ) {\displaystyle (1,-c)} direction is 0. This means that the value of v {\displaystyle v} is constant on characteristic lines of the form x + ct = x0, and thus that v {\displaystyle v} must depend only on x + ct, that is, have the form H(x + ct). Then, to solve the first (inhomogenous) equation relating v {\displaystyle 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
[ ∂ ∂ t + c ∂ ∂ x ] G ( x + c t ) = H ( x + c t ) . {\displaystyle \left[{\frac {\partial }{\partial t}}+c{\frac {\partial }{\partial x}}\right]G(x+ct)=H(x+ct).}
Expanding out the left side, rearranging terms, then using the change of variables s = x + ct simplifies the equation to
G ′ ( s ) = H ( s ) 2 c . {\displaystyle G'(s)={\frac {H(s)}{2c}}.}
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). For an initial-value problem, the arbitrary functions F and G can be determined to satisfy initial conditions:
u ( x , 0 ) = f ( x ) , {\displaystyle u(x,0)=f(x),}
u t ( x , 0 ) = g ( x ) . {\displaystyle u_{t}(x,0)=g(x).}
The result is d'Alembert's formula:
u ( x , t ) = f ( x − c t ) + f ( x + c t ) 2 + 1 2 c ∫ x − c t x + c t g ( s ) d s . {\displaystyle u(x,t)={\frac {f(x-ct)+f(x+ct)}{2}}+{\frac {1}{2c}}\int _{x-ct}^{x+ct}g(s)\,ds.}
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 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:
u ω ( x , t ) = e − i ω t f ( x ) . {\displaystyle u_{\omega }(x,t)=e^{-i\omega t}f(x).}
This produces an ordinary differential equation for the spatial part f(x):
∂ 2 u ω ∂ t 2 = ∂ 2 ∂ t 2 ( e − i ω t f ( x ) ) = − ω 2 e − i ω t f ( x ) = c 2 ∂ 2 ∂ x 2 ( e − i ω t f ( x ) ) . {\displaystyle {\frac {\partial ^{2}u_{\omega }}{\partial t^{2}}}={\frac {\partial ^{2}}{\partial t^{2}}}\left(e^{-i\omega t}f(x)\right)=-\omega ^{2}e^{-i\omega t}f(x)=c^{2}{\frac {\partial ^{2}}{\partial x^{2}}}\left(e^{-i\omega t}f(x)\right).}
Therefore,
d 2 d x 2 f ( x ) = − ( ω c ) 2 f ( x ) , {\displaystyle {\frac {d^{2}}{dx^{2}}}f(x)=-\left({\frac {\omega }{c}}\right)^{2}f(x),}
which is precisely an eigenvalue equation for f(x), hence the name eigenmode. Known as the Helmholtz equation, it has the well-known plane-wave solutions
f ( x ) = A e ± i k x , {\displaystyle f(x)=Ae^{\pm ikx},}
with wave number k = ω/c. The total wave function for this eigenmode is then the linear combination
u ω ( x , t ) = e − i ω t ( A e − i k x + B e i k x ) = A e − i ( k x + ω t ) + B e i ( k x − ω t ) , {\displaystyle u_{\omega }(x,t)=e^{-i\omega t}\left(Ae^{-ikx}+Be^{ikx}\right)=Ae^{-i(kx+\omega t)}+Be^{i(kx-\omega t)},}
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 e − i ω t , {\displaystyle e^{-i\omega t},} so that a full solution can be decomposed into an eigenmode expansion:
u ( x , t ) = ∫ − ∞ ∞ s ( ω ) u ω ( x , t ) d ω , {\displaystyle u(x,t)=\int _{-\infty }^{\infty }s(\omega )u_{\omega }(x,t)\,d\omega ,}
or in terms of the plane waves,
u ( x , t ) = ∫ − ∞ ∞ s + ( ω ) e − i ( k x + ω t ) d ω + ∫ − ∞ ∞ s − ( ω ) e i ( k x − ω t ) d ω = ∫ − ∞ ∞ s + ( ω ) e − i k ( x + c t ) d ω