Plot the solution, you can use software like Matlab to do this.
Class: Partial Differential Equations I
16.6 Characteristics and d'Alembert's Solution
In this section, we will derive d'Alembert's solution of a wave problem on the real line. We will denote partial derivatives by subscripts, u_t = u, and u_x = ∂u/∂x. The problem we will solve is:
∂²u/∂t² = c²∂²u/∂x² for x > 0, t > 0
and
u(x,0) = f(x), ∂u/∂t(x,0) = g(x) for x.
We are using u(x, t) for the position function of the wave. A graph of the wave's profile at time t is the graph of y = u(x, t) in the x, y-plane for that value of t.
This initial-boundary value problem is called the Cauchy problem for the wave equation. The lines x - ct = k and x + ct = k in the x, t-plane are called characteristics of the wave equation. These are straight lines of slope 1/c and 1/c in the x, t-plane. Exploiting these characteristics, make the change of variables:
ξ = x - ct, η = x + ct.
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3. Solve the wave equation on the real line for the given initial position f and initial velocity g, first by separation of variables and the Fourier integral, and then by using d'Alembert's solution. Plot your solution on a computer, compare the separation of variables solution to the d'Alembert's solution. Use, for example, the sample python codes, waveeqn1.py or waveeqn2clamped.py, as starting points for plotting the solution.
∂²u/∂t² = c²∂²u/∂x² for x < 0
(9)
∂u/∂x(x,0) = f(x), ∂u/∂t(x,0) = g(x)
(10)
c = 1, g(x) = 0
(11)
f(x) = 2 - |x| if |x| > 2
(12)