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.[2]

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. strain) that is traveling in an elastic material. The forces exerted on the mass m at the location x + h are:

$${\displaystyle {\begin{aligned}F_{\text{Newton}}&=m\,a(t)=m\,{\frac {\partial ^{2}}{\partial t^{2}}}u(x+h,t)\\F_{\text{Hooke}}&=F_{x+2h}-F_{x}=k\left[{u(x+2h,t)-u(x+h,t)}\right]-k[u(x+h,t)-u(x,t)]\end{aligned}}}$$

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

$${\displaystyle {\partial ^{2} \over \partial t^{2}}u(x+h,t)={k \over m}[u(x+2h,t)-u(x+h,t)-u(x+h,t)+u(x,t)]}$$

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:

$${\displaystyle {\partial ^{2} \over \partial t^{2}}u(x+h,t)={KL^{2} \over M}{u(x+2h,t)-2u(x+h,t)+u(x,t) \over h^{2}}.}$$

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

$${\displaystyle {\frac {\partial ^{2}u(x,t)}{\partial t^{2}}}={\frac {KL^{2}}{M}}{\frac {\partial ^{2}u(x,t)}{\partial x^{2}}},}$$

which is from the definition of a second derivative. 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

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

$${\displaystyle K={\frac {EA}{L}},}$$

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

$${\displaystyle {\frac {\partial ^{2}u(x,t)}{\partial t^{2}}}={\frac {EAL}{M}}{\frac {\partial ^{2}u(x,t)}{\partial x^{2}}}.}$$

AL is equal to the volume of the bar and therefore

$${\displaystyle {\frac {AL}{M}}={\frac {1}{\rho }},}$$

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

$${\displaystyle {\frac {\partial ^{2}u(x,t)}{\partial t^{2}}}={\frac {E}{\rho }}{\frac {\partial ^{2}u(x,t)}{\partial x^{2}}}.}$$

The speed of a stress wave in a bar is therefore √E/ρ.

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:[3]

$${\displaystyle {\begin{aligned}\xi &=x-ct\\\eta &=x+ct\end{aligned}}}$$

changes the wave equation into

$${\displaystyle {\frac {\partial ^{2}u}{\partial \xi \partial \eta }}=0,}$$

which leads to the general solution

$${\displaystyle u(\xi ,\eta )=F(\xi )+G(\eta )}$$

or equivalently:

$${\displaystyle u(x,t)=F(x-ct)+G(x+ct).}$$

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.[4]

Another way to arrive at this result is to factor the wave equation into two one-way wave equations:

$${\displaystyle \left[{\frac {\partial }{\partial t}}-c{\frac {\partial }{\partial x}}\right]\left[{\frac {\partial }{\partial t}}+c{\frac {\partial }{\partial x}}\right]u=0.}$$

i.e.

$${\displaystyle {\frac {\partial u}{\partial t}}-c{\frac {\partial u}{\partial x}}=0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\frac {\partial u}{\partial t}}+c{\frac {\partial u}{\partial x}}=0}$$

As a result, if we define v thus,

$${\displaystyle {\frac {\partial u}{\partial t}}+c{\frac {\partial u}{\partial x}}=v,}$$

then

$${\displaystyle {\frac {\partial v}{\partial t}}-c{\frac {\partial v}{\partial x}}=0.}$$

From this, v must have the form G(x + ct), and from this the correct form of the full solution u can be deduced.[5] 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:

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

$${\displaystyle u_{t}(x,0)=g(x).}$$

The result is d'Alembert's formula:

$${\displaystyle u(x,t)={\frac {f(x-ct)+f(x+ct)}{2}}+{\frac {1}{2c}}\int _{x-ct}^{x+ct}g(s)\,ds}$$

In the classical sense if f(x) ∈ C^k and g(x) ∈ C^k−1 then u(t, x) ∈ C^k. 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:

$${\displaystyle u_{\omega }(x,t)=e^{-i\omega t}f(x).}$$

This produces an ordinary differential equation for the spatial part 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:

$${\displaystyle {\frac {d^{2}}{dx^{2}}}f(x)=-\left({\frac {\omega }{c}}\right)^{2}f(x),}$$

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

$${\displaystyle f(x)=Ae^{\pm ikx},}$$

with wave number k = ω/c.

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

$${\displaystyle u_{\omega }(x,t)=e^{-i\omega t}\left(Ae^{-ikx}+Be^{ikx}\right)=Ae^{-i(kx+\omega t)}+Be^{i(kx-\omega t)},}$$

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 ${\displaystyle e^{-i\omega t}}$. so that a full solution can be decomposed into an eigenmode expansion

$${\displaystyle u(x,t)=\int _{-\infty }^{\infty }s(\omega )u_{\omega }(x,t)\,d\omega }$$

or in terms of the plane waves,

$${\displaystyle {\begin{aligned}u(x,t)&=\int _{-\infty }^{\infty }s_{+}(\omega )e^{-i(kx+\omega t)}\,d\omega +\int _{-\infty }^{\infty }s_{-}(\omega )e^{i(kx-\omega t)}\,d\omega \\&=\int _{-\infty }^{\infty }s_{+}(\omega )e^{-ik(x+ct)}\,d\omega +\int _{-\infty }^{\infty }s_{-}(\omega )e^{ik(x-ct)}\,d\omega \\&=F(x-ct)+G(x+ct)\end{aligned}}}$$

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 packet u(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.[6] 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.

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

$${\displaystyle \Psi (\mathbf {r} ,t)=\int _{-\infty }^{\infty }\Psi (\mathbf {r} ,\omega )e^{-i\omega t}\,d\omega .}$$

So we get,

$${\displaystyle \left(\nabla ^{2}+{\frac {\omega ^{2}}{c^{2}}}\right)\Psi (\mathbf {r} ,\omega )=0.}$$

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[9]

$${\displaystyle \left[{\frac {d^{2}}{dr^{2}}}+{\frac {2}{r}}{\frac {d}{dr}}+k^{2}-{\frac {l(l+1)}{r^{2}}}\right]f_{l}(r)=0}$$

Here k ≡ ω/c and the complete solution is now given by

$${\displaystyle \Psi (\mathbf {r} ,\omega )=\sum _{lm}\left[A_{lm}^{(1)}h_{l}^{(1)}(kr)+A_{lm}^{(2)}h_{l}^{(2)}(kr)\right]Y_{lm}(\theta ,\phi ),}$$

where h⁽¹⁾
_l(kr) and h⁽²⁾
_l(kr) 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 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

$${\displaystyle {\begin{aligned}&{}\left(\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\right)\Psi (\mathbf {r} ,t)=0\\\rightarrow &{}\left({\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial }{\partial r}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\right)u(r,t)=0\end{aligned}}}$$

This equation can be rewritten as

$${\displaystyle {\frac {\partial ^{2}(ru)}{\partial t^{2}}}-c^{2}{\frac {\partial ^{2}(ru)}{\partial r^{2}}}=0;}$$

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

$${\displaystyle u(r,t)={\frac {1}{r}}F(r-ct)+{\frac {1}{r}}G(r+ct),}$$

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

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 ω, then the transformed function ru(r, t) has simply plane wave solutions,

$${\displaystyle ru(r,t)=Ae^{i(\omega t\pm kr)},}$$

or

$${\displaystyle u(r,t)={\frac {A}{r}}e^{i\left(\omega t\pm kr\right)}.}$$

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

$${\displaystyle I=|u(r,t)|^{2}={\frac {|A|^{2}}{r^{2}}}.}$$

drops at the rate proportional to 1/r², an example of the inverse-square law.

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

$${\displaystyle r^{2}=(x-\xi )^{2}+(y-\eta )^{2}+(z-\zeta )^{2}.}$$

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

$${\displaystyle u(t,x,y,z)={\frac {1}{4\pi c}}\iiint \varphi (\xi ,\eta ,\zeta ){\frac {\delta (r-ct)}{r}}\,d\xi \,d\eta \,d\zeta$$ ;}

the denominator 4πc is a convenience.

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

$${\displaystyle u(t,x,y,z)={\frac {t}{4\pi }}\iint _{S}\varphi (x+ct\alpha ,y+ct\beta ,z+ct\gamma )\,d\omega ,}$$

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:

$${\displaystyle u(t,x,y,z)=tM_{ct}[\phi ].}$$

It follows that

$${\displaystyle u(0,x,y,z)=0,\quad u_{t}(0,x,y,z)=\phi (x,y,z).}$$

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

$${\displaystyle v(t,x,y,z)={\frac {\partial }{\partial t}}\left(tM_{ct}[\psi ]\right),}$$

then

$${\displaystyle v(0,x,y,z)=\psi (x,y,z),\quad v_{t}(0,x,y,z)=0.}$$

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).

Assume n ≥ 3 is an odd integer and g ∈ C^m+1(Rⁿ), h ∈ C^m(Rⁿ) for m = (n + 1)/2. Let γ_n = 1 × 3 × 5 × ⋯ × (n − 2) and let

$${\displaystyle u(x,t)={\frac {1}{\gamma _{n}}}\left[\partial _{t}\left({\frac {1}{t}}\partial _{t}\right)^{\frac {n-3}{2}}\left(t^{n-2}{\frac {1}{|\partial B_{t}(x)|}}\int _{\partial B_{t}(x)}g\,dS\right)+\left({\frac {1}{t}}\partial _{t}\right)^{\frac {n-3}{2}}\left(t^{n-2}{\frac {1}{|\partial B_{t}(x)|}}\int _{\partial B_{t}(x)}h\,dS\right)\right]}$$

then

• u ∈ C²(Rⁿ × [0, ∞))
• u_tt − Δu = 0 in Rⁿ × (0, ∞)
• ${\displaystyle \lim _{(x,t)\to (x^{0},0)}u(x,t)=g(x^{0})}$
• ${\displaystyle \lim _{(x,t)\to (x^{0},0)}u_{t}(x,t)=h(x^{0})}$

Assume n ≥ 2 is an even integer and g ∈ C^m+1(Rⁿ), h ∈ C^m(Rⁿ), for m = (n + 2)/2. Let γ_n = 2 × 4 × ⋯ × n and let

$${\displaystyle u(x,t)={\frac {1}{\gamma _{n}}}\left[\partial _{t}\left({\frac {1}{t}}\partial _{t}\right)^{\frac {n-2}{2}}\left(t^{n}{\frac {1}{|B_{t}(x)|}}\int _{B_{t}(x)}{\frac {g}{(t^{2}-|y-x|^{2})^{\frac {1}{2}}}}dy\right)+\left({\frac {1}{t}}\partial _{t}\right)^{\frac {n-2}{2}}\left(t^{n}{\frac {1}{|B_{t}(x)|}}\int _{B_{t}(x)}{\frac {h}{(t^{2}-|y-x|^{2})^{\frac {1}{2}}}}dy\right)\right]}$$

then

• u ∈ C²(Rⁿ × [0, ∞))
• u_tt − Δu = 0 in Rⁿ × (0, ∞)
• ${\displaystyle \lim _{(x,t)\to (x^{0},0)}u(x,t)=g(x^{0})}$
• ${\displaystyle \lim _{(x,t)\to (x^{0},0)}u_{t}(x,t)=h(x^{0})}$

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

$${\displaystyle u^{inc}(x,t)=Ae^{i(k_{1}x-\omega t)};\ A\in \mathbb {C} }$$

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

$${\displaystyle u^{ref}(x,t)=Be^{i(-k_{1}x-\omega t)};\ u^{trans}(x,t)=Ce^{i(k_{2}x-\omega t)};\ B,C\in \mathbb {C} }$$

The continuity condition at the boundary is

$${\displaystyle u^{inc}(0,t)+u^{ref}(0,t)=u^{trans}(0,t);\ u_{x}^{inc}(0,t)+u_{x}^{ref}(0,t)=u_{x}^{trans}(0,t)}$$

This gives the equations

$${\displaystyle A+B=C;\ A-B={\frac {k_{2}}{k_{1}}}C={\frac {c_{1}}{c_{2}}}C}$$

And we have the reflectivity and transmissivity

$${\displaystyle {\frac {B}{A}}={\frac {c_{2}-c_{1}}{c_{2}+c_{1}}};\ {\frac {C}{A}}={\frac {2c_{2}}{c_{2}+c_{1}}}}$$

When c₂ < c₁, the reflected wave has a reflection phase change of 180°, since B/A < 0. The energy conservation can be verified by

$${\displaystyle {\frac {B^{2}}{c_{1}}}+{\frac {C^{2}}{c_{2}}}={\frac {A^{2}}{c_{1}}}}$$

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 doesn't move, whereas the limiting case of c₂ → ∞ corresponds to a "free end".

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

$${\displaystyle -u_{x}(t,0)+au(t,0)=0,}$$

$${\displaystyle u_{x}(t,L)+bu(t,L)=0,}$$

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

$${\displaystyle u(t,x)=T(t)v(x).}$$

A consequence is that

$${\displaystyle {\frac {T''}{c^{2}T}}={\frac {v''}{v}}=-\lambda .}$$

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

$${\displaystyle v''+\lambda v=0,}$$

$${\displaystyle -v'(0)+av(0)=0,\quad v'(L)+bv(L)=0.}$$

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:

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 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

$${\displaystyle {\frac {u_{i+1}-u_{i}}{\Delta x}}f}$$

(1)

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

$${\displaystyle {\frac {u_{i-1}-u_{i}}{\Delta x}}f}$$

(2)

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

$${\displaystyle {\ddot {u}}_{i}=\left({\frac {f}{m\Delta x}}\right)\left(u_{i+1}+u_{i-1}-2u_{i}\right)}$$

(3)

As the mass density is

$${\displaystyle \rho ={\frac {m}{\Delta x}}}$$

this can be written

$${\displaystyle {\ddot {u}}_{i}=\left({\frac {f}{\rho {\Delta x}^{2}}}\right)\left(u_{i+1}+u_{i-1}-2u_{i}\right)}$$

(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, ··u_i takes the form ∂²u/∂t² and

$${\displaystyle {\frac {u_{i+1}+u_{i-1}-2u_{i}}{{\Delta x}^{2}}}\to {\frac {\partial ^{2}u}{\partial x^{2}}}}$$

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

$${\displaystyle u(0,t)=u(L,t)=0}$$

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

$${\displaystyle {\ddot {u}}_{1}={\left({\frac {c}{\Delta x}}\right)}^{2}\left(u_{2}-2u_{1}\right)}$$

(5)

and

$${\displaystyle {\ddot {u}}_{n}={\left({\frac {c}{\Delta x}}\right)}^{2}\left(u_{n-1}-2u_{n}\right)}$$

(6)

while for 1 < i < n

$${\displaystyle {\ddot {u}}_{i}={\left({\frac {c}{\Delta x}}\right)}^{2}\left(u_{i+1}+u_{i-1}-2u_{i}\right)}$$

(7)

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

$${\displaystyle {\frac {L}{c}}k(0.05),\,k=0,\dots ,5}$$

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

Figure 3: The string at 6 consecutive epochs

Figure 4: The string at 6 consecutive epochs

Figure 5: The string at 6 consecutive epochs

Figure 6: The string at 6 consecutive epochs

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[10] with all ${\displaystyle {\dot {u}}_{i}=0}$. The blue curve is the state at time ${\displaystyle {\tfrac {L}{c}}(0.25),}$ i.e. after a time that corresponds to the time a wave that is moving with the nominal wave velocity c=√ f/ρ would need for one fourth of the length of the string.

Figure 3 displays the shape of the string at the times ${\displaystyle {\tfrac {L}{c}}k(0.05),\,k=6,\dots ,11}$. The wave travels in direction 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).

Figure 4 displays the shape of the string at the times ${\displaystyle {\tfrac {L}{c}}k(0.05),\,k=12,\dots ,17}$. 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 ${\displaystyle {\tfrac {L}{c}}k(0.05),\,k=18,\dots ,23}$ when the direction of motion is reversed. The red, green and blue curves are the states at the times ${\displaystyle {\tfrac {L}{c}}k(0.05),\,k=18,\dots ,20}$ while the 3 black curves correspond to the states at times ${\displaystyle {\tfrac {L}{c}}k(0.05),\,k=21,\dots ,23}$ with the wave starting to move back towards left.

Figure 6 and figure 7 finally display the shape of the string at the times ${\displaystyle {\tfrac {L}{c}}k(0.05),\,k=24,\dots ,29}$ and ${\displaystyle {\tfrac {L}{c}}k(0.05),\,k=30,\dots ,35}$. 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.

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

$${\displaystyle {\frac {\partial u}{\partial n}}+au=0,}$$

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

$${\displaystyle u(0,x)=f(x),\quad u_{t}(0,x)=g(x),}$$

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

$${\displaystyle \nabla \cdot \nabla v+\lambda v=0,}$$

in D, and

$${\displaystyle {\frac {\partial v}{\partial n}}+av=0,}$$

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 elastic wave equation (also known as the Navier–Cauchy 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:

$${\displaystyle \rho {\ddot {\mathbf {u} }}=\mathbf {f} +(\lambda +2\mu )\nabla (\nabla \cdot \mathbf {u} )-\mu \nabla \times (\nabla \times \mathbf {u} )}$$

where:

• 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),
• and 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.

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

$${\displaystyle \omega =\omega (\mathbf {k} ),}$$

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:

$${\displaystyle v_{\mathrm {p} }={\frac {\omega (k)}{k}}.}$$