One of the simplest and most classical model formed by a hyperbolic equation is the model governing the motion of the waves of a vibrating string. The equation is called the wave equation and we look
(x,t) t
x x−vt=
ξ η
x+vt=
x−vt x+vt
t
x x−vt=
ξ η
x+vt=
x−vt x+vt
(x,t)
ξ
A B C
D
FIGURE 2.8: Domain of dependence for the wave equation initial value problem (left). The propa-gation of signals from pointξ occurs with speed at most λ, i.e., within the shaded region on the left panel.
at the following initial value problem:
utt− λ2uxx = 0, (2.18a)
u(x, 0) = h(x), (2.18b)
ut(x, 0) = g(x), (2.18c)
where we note that, being (2.18) of second order, we need two initial conditions. We can consider two approaches for finding the solution of the above equation, given the initial conditions. As seen in section 2, this equation can be written as a two-by-two system of linear first order PDEs given by (2.4). In this section, however, we will try to solve this equation without recasting it as a system of fist order PDEs. This approach follows a more standard path-line, similar to the original solution due to d’Alembert, that makes use of the principle of separation of variables.
It is easy to verify that if we write:
x + λt = ξ x− λt = η, (2.19)
equation (2.18) transforms into the simpler equation:
uξη = 0.
This equation states that uξis independent of η, i.e., it is a function of ξ: uξ = f (ξ). After integration, we have that
u(ξ, η) = Z
f (ζ)dζ + G(η) = F (ξ) + G(η),
i.e., u(ξ, η) is the sum of two functions of ξ and η, respectively. Using (2.19), we have the solution in the original variables:
u(x, t) = F (x + λt) + G(x− λt). (2.20)
The form of this solution shows a superposition of two waves, solutions of vt− λvx = 0 and wt+ λwx = 0, respectively. This corresponds to the fact that the differential operator can be factored into the product of two operators: If we impose the initial conditions (2.18b) and (2.18c) we obtain:
u(x, 0) = F (x) + G(x) = h(x) (2.21)
ut(x, 0) = λF0(x)− λG0(x) = g(x). (2.22)
Differentiating the first equation and solving the ensuing linear system for F (x) and G(x) we obtain:
F0(x) = λh0(x) + g(x) The above general solution to the wave equation is determined uniquely by the values of h and g for t = 0, that is from the initial data (2.18b) and (2.18c). The fact that we have x± λt as argument of the initial functions tells us that the domain of dependence of u(x, t) is the triangular region shown in Figure 2.8. In other words, given a point (x, t), the solution u at that point depends only upon the values of u in the domain of dependence.
t
x λ < 0
t
x λ > 0
FIGURE 2.9: Characteristic lines for the scalar advection equation on a semi-infinite domain in the case λ < 0 (left) and λ > 0 (right). The circle and square dots indicate the points where initial conditions and boundary conditions, respectively, need to be imposed.
The scalar advection equation. Let us start with the simplest equation on a semi-infinite domain:
ut+ λux = 0, 0 < x <∞,
u(x, 0) = f (x), 0 < t <∞.
Figure 2.9 show on the characteristic lines in the case λ < 0 and λ > 0 in the left and right graphs, respectively. In the first case, the solution is uniquely determined by the initial data at t = 0. In the second case, conditions at x = 0 for all the times determine the solution. These are called boundary conditions, and not initial conditions, for obvious reasons. The set of all boundary and initial data are called “Cauchy” data, as they define the Cauchy problem.
Let us now turn our attention to the same problem but defined on the closed interval I = [0, 1]. By the same reasoning as above, if λ < 0 we would have to specify boundary conditions at x = 1. In the case λ > 0 boundary conditions would have to be specified at x = 0. We stress here that we cannot specify BCs at the wrong boundary, as this will not give a well-posed problem. If we look at the graph of u(x, t) at fixed times as a function of x, we realize immediately that the correct boundary condition is given at the point where the wave enters the domain, and no boundary conditions can be give at outgoing boundaries.
Remark 2.9. First, we note that the solution u(x, t) of the form (2.20) satisfies the following rela-tionship:
u(x, t)− u(x + λζ, t + ζ) − u(x − λϑ, t + ϑ) + u(x + λζ − λϑ, t + ζ + ϑ) = 0.
This equation implies that for any parallelogram built on the characteristic lines (e.g., ABCD in Figure 2.8, right panel), we have that the sum of the solution at opposite vertices is a constant:
u(A) + u(C) = u(B) + u(D). (2.24)
We will be using consistently this observation in the following paragraphs to obtain trajectories (set of characteristic lines) of the wave equations in a compact domain.
The wave equation. We want to look now at an Initial-Boundary Value problem for the wave equa-tion. In this case, our equation is specified in a compact domain Ω ∈ Rd, identified in our case by an interval. We can think of Ω = I = [0, 1]. In this case, the solution of our wave equation must be restricted within this interval, and we have to specify additional conditions besides the initial ones, called boundary conditions. However, as we will see, we cannot specify boundary conditions at will.
We can state our initial-boundary value problem as:
Problem 2.10. Find u(x, t) such that:
utt− λ2uxx = 0,
u(x, 0) = h(x); ut(x, 0) = g(x), u(0, t) = α(t); u(1, t) = β(t).
This problem corresponds to restricting all the characteristic lines seen above to the interval [0, 1].
However, it is intuitive that if we do this we need to specify certain conditions at x = 0 and x = 1 for all the times. These boundary conditions cannot be arbitrary. For example, at x = 0 we cannot specify conditions along the characteristic line x + λt, and vice-versa, at x = 1 we cannot specify conditions along the characteristic line x−λt. Moreover, continuity of the solution and continuous dependence of the solution to the initial data (i.e., well-posedness) require that we specify these BCs imposing some compatibility conditions. To see this we work out the solution to our IBV problem 2.10 constructing the characteristic lines and using Remark 2.9 in the ensuing regions.
In the plane x − t the characteristic lines and the domain of dependence can be described as in Figure 2.10. Looking separately at the solution in the different regions, we see that in region I the solution is determined by the initial data at t = 0, and its expression is given by (2.23). In a point of region II, say point A = (x, t), the solution can be evaluated by (2.24) by noting that in B the solution is given by the left boundary condition, and in C and D by the initial condition. We note that we need to construct the parallelogram using the characteristic lines. Following this method we can determine the solution at every point of the domain.
If the initial data h(x) and g(x) together with the boundary conditions α(t) and β(t) are arbitrarily, the solution may not be smooth and can have jumps across the characteristic lines. If we want a smooth (C2) solution we need to require that:
α(0) = h(0); α0(0) = g(0); α00(0) = λ2h00(0);
β(0) = h(1); β0(0) = g0(1); β00(0) = λ2h00(1). (2.25) Looking at Figure 2.10 it is intuitive that the solution along the characteristic lines x− λt = C is determined either by the initial data or by the boundary condition on the left. The boundary condition
t
(t)α(t) β
x t
0 1
A B C D
A’
B’
C’
D’
I
II III
IV
V VI
VII
u=h(x); u =g(x)
FIGURE2.10: Characteristic lines for the wave equation defined on a compact domain Ω = [0, 1].
on the right does not affect the solution u(x, t) at any time. The opposite is true for the characteristic lines x + λt = C. This suggests that we cannot specify boundary conditions everywhere in the domain, as is done for elliptic equations. From a physical point of view, we may argue that we need to specify conditions only at the inflow boundaries, in our case x = 0 for the wave moving to the right corresponding to the characteristic lines x− λt = C and x = 1 for the characteristic line x + λt = C.
The type of boundary conditions that we can specify are either Dirichlet (specified value of u) on an inflow, or a reflection boundary at the outflow. No other types of BCs can be specified. In any case, compatibility conditions between BCs and ICs need to be determined to ensure smooth solutions.
Example 2.11. Given the following IBV:
wtt = wxx 0≤ x ≤ 1; t > 0;
w(x, 0) = φ(x) t = 0;
wt(x, 0) = ψ(x) t = 0;
wx = αLwt x = 0;
wx = αRwt x = 1.
Find the values of αLand αRso that the problem is well-posed.
Rewrite the wave equation as a system of first-order equations:
Matrix A can be diagonalized as:
Λ = U AUT; Λ = −1 0
Thus, the original wave equation has been transformed into a system of two separate advection equa-tions. Looking at Example 2.3, we see that vI is a wave propagating with celerity λ = −λ1 = 1 while vII has λ =−λ2 =−1 and the characteristic directions are the eigenvectors r1 = (−1, 1)T and r2 = (1, 1)T. Thus we recover the solution developed in Section 2.10.
Since we have a wave traveling from left to right and a wave traveling from right to left we need a boundary condition on the left and on the right boundaries, respectively. Intuitively, these BCs will relate vI with vII through the proper well-posed boundary conditions, at x = 0 and x = 1, respectively. To see this, we note that:
v = U r; r = UTv.
Using the initial conditions, we have:
wx(x, 0) = φ0(x) and wt(x, 0) = ψ(x), yielding for x = 0 and x = 1:
αL= φ0(0)
ψ(0) and αR = φ0(1) ψ(1).