In 1747, d'Alembert derived the first partial differential equation (PDE for short) in the history of mathematics, namely the wave equation.
This section provides an introduction to one-dimensional wave equations and corresponding initial value problems.
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The wave equation is a typical example of more general class of partial differential equations called hyperbolic equations. They occur in classical physics, geology, acoustics, electromagnetics, and fluid dynamics. Wave equations usually describe wave propagations in different media.
Historically, the problem of a vibrating string such as that of a musical instrument was first studied by the French mathematician, mechanical physicist, philosopher, and music theorist Jean le Rond
d'Alembert.
Because he was the first who found a solution of one-dimensional wave equation in 1746, the latter is usually referred to as d'Alembert's equation.
Many others contributed to study of the wave equation, among first of them we mention Leonhard Euler (who discovered the wave equation in three space dimensions), Daniel Bernoulli ( the Euler–Bernoulli beam equation), and Joseph-Louis
Lagrange (classical and celestial mechanics).
The wave equation for real-valued function \( u(x_1, x_2, \ldots , x_n , t) \) of n spatial variables and a time variable t is
\begin{equation} \label{EqWave.1}
\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u , \qquad \mbox{or} \qquad \square u =0 ,
\end{equation}
where c is a positive constant (having dimensions of speed) and
\begin{equation} \label{EqWave.2}
\nabla^2 u \equiv \Delta u = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2} \qquad\mbox{and} \qquad \square u \equiv \square_c u = \frac{\partial^2 u}{\partial t^2} - c^2 \Delta u .
\end{equation}
Suppose we have a medium whose displacement may be described by a scalar function u(x,t), where \( {\bf x} \in \mathbb{R}^n , \quad t\in\mathbb{R} . \) Suppose that the system is conservative and it has the Lagrangian \( {\cal L} = \mbox{K} - \Pi , \) where the kinetic energy K and potential energy Π of the medium are
\[
\mbox{K} \left( u_t \right) = \frac{1}{2} \int \rho\,u_t\,{\text d}{\bf x} , \qquad \Pi \left( u \right) = \frac{1}{2} \int k \left\vert \nabla u \right\vert^2 {\text d}{\bf x} .
\]
Here ρ(x) is a mass-density and k(x) is a stiffness, both assumed positive, and \( u_t = \partial u/\partial t . \) The corresponding action becomes
\[
S \left( u \right) = \int {\text d}t \, {\cal L} \left( u, u_t \right) = \int {\text d}t \int {\text d}{\bf x} \,\frac{1}{2} \left\{ \rho\,u_t^2 - k\left\vert \nabla u \right\vert^2 \right\} .
\]
Note that the kinetic and potential energies and the Lagrangian are functions of the spacial field and velocity at each fixed time, whereas the action is a functional of the space-time field u(x, t), obtained by integrating the Lagrangian with respect to time.
The Euler--Lagrange equation is satisfied by a stationary point (which is a function u(x, t)) of this action becomes
If ρ and k are constants, then we get the wave equation
\[
\square_c u \equiv u_{tt} - c^2 \Delta u =0 \qquad\mbox{or}\qquad \frac{\partial^2 u}{\partial t^2} - c^2 \nabla^2 u =0 .
\]
The action functional for the wave equation is not positive definite. Therefore, we cannot expect a solution of the wave equation to be a minimizer of the action, in general, only a critical point. ■
We derive the wave equation in one space dimension that models the
transverse vibrations of an elastic string. If such string is placed
horizontally between end points x=0 and x=ℓ, it can
freely vibrate within a vertical plane. Generally speaking it is not
true; however, if displacements u(x,t) are small, we can assume
that spring motion occur only within a plane perpendicular to its
equilibrium horizontal position.
■
Perhaps the easiest case is observed with the investigation of
mechanical vibrations. Suppose that an elastic string of length ℓ
is tightly stretched between two supports at the same horizontal
level, which we identify with x-axis. Then its end points may
be taken as x=0 and x=ℓ. The elastic string may be
thought of as a guitar or violin string, a guy wire, or possibly an
electric power line. The positions of points on the string can be
described by the displacement, which we denote by u(x,t), from
the equilibrium horizontal position. If damping effects, such as air
resistance, are neglected, and if the magnitude of the motion is not
too large, then the displacement function satisfies the partial
differential equation (called one dimensional wave equation)
in the domain 0 < x < ℓ 0 < t
< ∞. The constant coefficient c² is given by
\[
c^2 = T/\rho ,
\]
where T is the tension (force) in the string, and ρ is the
mass per unit length of the string material (density). To describe the
motion of the string completely, we need to impose some auxiliary
conditions. Of these, we need to specify the initial displacement and
its initial velocity
where d and v are known functions. If we consider a
ideal (and not realistic) case that the string has an infinite length,
we arrive at so called the initial value problem:
which can be integrated directly. This leads to the conclusion that a
solution of the wave equation u_{tt}
- c²u_{xx} = 0 is the sum
\[
u(x,t) = f(x+ct) + g(x-ct)
\]
of two functions f(ξ) and g(ξ) of one
variable. This formula represents a superposition of two waves, one
traveling to the right and another traveling to the left, each with
velocity x. However, in practice, traveling waves are excited
by the initial disturbance
where d(x) is the initial displacement (initial configuration)
and v(x) is the initial velocity of the string. Upon
substituting the general solution into the initial condition, we get
two equations
where the given continuous functions d(x) and v(x) are assumed to be zero outside some disk. We define the energy function as a function of time variable t:
Hence, the nergy function E(t) is a constant, so E(t) = E(0).
In particular, if u_{1} and u_{2} are two solutions of the initial value problem for the wave equation, then v =
u_{1} - u_{2} has homogeneous initial conditions, and so E(t) = E(0) = 0. Since the nonnegative energy function is a constant, it implies that v = 0. So the solution of such Cauchy problem is unique.
▣
Example: Dirichlet boundary conditions.
Let us consider the vibrations of an infinitely long string (0 < x < ∞) that is fixed at one end.
We first assume that the
boundary condition at left end x = 0 are of first type
(Dirichlet):
As usual, the dot indicates derivative with respect to time variable:
\( \dot{u} = \partial u/\partial t \) and
\( \ddot{u} = \partial^2 u/\partial t^2 . \)
To understand the solution, we assume temporally that input is a
snakey.
(A "snakey" is a slinky-like device that consists of a large concentration of small-diameter metal coils.) If an upward displaced pulse is introduced at the left end of the snakey, it will travel rightward across the snakey until it reaches the fixed end on the right side of the snakey. Upon reaching the fixed end, the single pulse will reflect and undergo inversion. That is, the upward displaced pulse will become a downward displaced pulse. Now suppose that a second upward displaced pulse is introduced into the snakey at the precise moment that the first crest undergoes its fixed end reflection. If this is done with perfect timing, a rightward moving, upward displaced pulse will meet up with a leftward moving, downward displaced pulse in the exact middle of the snakey. As the two pulses pass through each other, they will undergo destructive interference.
The animation below shows several snapshots of the meeting of the two
pulses at various stages in their interference.
is valid only for 0 < x < ∞ since the initial displacement d(x) and the initial velocity v(x) are defined only for positive inputs. Therefore, the d'Alembert formula provides a solution to the wave equation only when x ±ct > 0. Since all ingredients (x, c, t) are positive, the expression x + ct > 0 always true. If x - ct > 0, then the d'Alembert solution is still valid. However, for negative values, it is false.
Now suppose that x - ct ≤ 0. Using the traveling wave formula \( u(x,t) = F(x-ct) + G(x+ct) , \) we try to satisfy the Dirichlet boundary condition
\[
u(x,0) = F(-ct) + G(ct) = 0 .
\]
Let w = -ct, so F(w) = -G(-w) for w ≤ 0, which reveals how the function F(w) should be defined. Therefore,
We can solve the above initial value problem using sine Fourier transform. To do this, we multiply both sides of the wave equation by \( \sin (kx) \) and integrate with respect to variable x from zero to infinity. Then integration by parts leads to
When these functions are identically zero (so we have the homogeneous boundary conditions, f(x) ≡ 0 and g(t) ≡ 0), the given problem has infinite many solutions:
\[
u(x,t) = \begin{cases}
\psi \left( \frac{x+t}{2} \right) - \psi \left( \frac{x-t}{2} \right) , & \ \mbox{ if } x \ge t , \\
\psi \left( \frac{x+t}{2} \right) - \psi \left( \frac{t-x}{2} \right) , & \ \mbox{ if } t \ge x ,
\end{cases}
\]
for arbitrary smooth function ψ.
▣
Example: .
We again consider the wave equation in the wedge-shaped domain Ω = { (x, y) : 0 < x < ∞, 0 < t < x/2 }:
Wazwaz, A.-M., One Dimensional Wave Equation, Chapter 5 in Partial Differential Equations and Solitary Waves Theory, Nonlinear Physical Science. Springer, Berlin, Heidelberg, 2009.
Wazwaz, A.-M., Blow-up for solutions of some linear wave equations with mixed nonlinear boundary conditions, Applied Mathematics and Computation, 2001,Volume 123, Issue 1, pp. 133--140. https://doi.org/10.1016/S0096-3003(00)00069-2
Wazwaz, A.-M., A reliable technique for solving the wave equation in an infinite one-dimensional medium, Applied Mathematics and Computation, 1998, Volume 79, Issue 1, 1--7. https://doi.org/10.1016/S0096-3003(97)10037-6
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