Section 6.8 The wave equation
Objective:
1. The solution set of a wave equation (vibration of a string) with different initial conditions
The wave equation problem is \begin{align*} u_{tt}(x,t) & =a^{2}u_{xx}(x,t),t \gt 0\\ u(0,t) & =0,u(L,t)=0,t \gt 0\\ u(x,0) & =f(x),u_{t}(x,0)=g(x),0\leq x\leq L \end{align*} where [latex]f(x)[/latex] is the initial position function and [latex]g(x)[/latex] is the initial velocity. We assume [latex]u(x,t)=X(x)T(t)[/latex], then translate the problem into solving \begin{align*} X”(x)+\lambda X(x) & =0,X(0)=X(L)=0\\ T”(t)+a^{2}\lambda T(t) & =0. \end{align*} The only non-trivial solutions of [latex]X''(x)+\lambda X(x)=0,X(0)=X(L)=0[/latex] are [latex]\lambda_{n}=(\frac{n\pi}{L})^{2}[/latex] and [latex]X_{n}=c_{n}\text{sin}(\frac{n\pi}{L}x)[/latex]. Hence we need to solve [latex]T''(t)+a^{2}(\frac{n\pi}{L})^{2}T(t)=0.[/latex]
Case (1): [latex]g(x)=0[/latex].} Then [latex]u_{t}(x,0)=g(x)=0=X(x)T'(0)[/latex] implies [latex]T'(0)=0[/latex]. The solution of [latex]T''(t)+a^{2}(\frac{n\pi}{L})^{2}T(t)=0[/latex] with [latex]T'(0)=0[/latex] will be [latex]T_{n}(t)=k_{n}\text{cos}(\frac{an\pi}{L}t)[/latex]. We let \[ u_n(x,t)=\text{cos}(\frac{an\pi}{L}t)\text{sin}(\frac{n\pi}{L}x). \] and write \[ u(x,t)=\sum_{n=1}^{\infty}d_{n}u_{n}(x,t)=\sum_{n=1}^{\infty}d_{n}\text{cos}(\frac{an\pi}{L}t)\text{sin}(\frac{n\pi}{L}x). \] We use the initial condition [latex]u(x,0)=f(x)[/latex] to solve for [latex]d_{n}[/latex]. \[ u(x,0)=f(x)=\sum_{n=1}^{\infty}d_{n}u_{n}(x,0)=\sum_{n=1}^{\infty}d_{n}\text{sin}(\frac{n\pi}{L}x). \] We see that [latex]f(x)[/latex] is a sine Fourier series and hence [latex]d_{n}=\frac{2}{L}\int_{0}^{L}f(x)\text{sin}(\frac{n\pi}{L}x)dx[/latex].
Example 1: Suppose the displacement [latex]u(x,t)[/latex] of a piece of flexible string is given by the initial-boundary value problem \begin{align*} u_{tt}(x,t) & =9u_{xx}(x,t),t \gt 0\\ u(0,t) & =0,u(\pi,t)=0,t \gt 0\\ u(x,0) & =10\text{sin}(2x)-20\text{sin}(4x),u_{t}(x,0)=0,0\leq x\leq\pi. \end{align*}
(a) What is the physical meaning of its boundary conditions?
(b) What are the initial displacement and initial velocity of the string at the mid- point, [latex]x=\frac{\pi}{2}[/latex].
(c) State the general form of its solution.
Exercise 1: Suppose the displacement [latex]u(x,t)[/latex] of a piece of flexible string is given by the initial-boundary value
problem \begin{align*} u_{tt}(x,t) & =16u_{xx}(x,t),t \gt 0\\ u(0,t) & =0,u(6,t)=0,t \gt 0\\ u(x,0) & =10\text{sin}(\pi x)-30\text{sin}(2\pi x),u_{t}(x,0)=0,0\leq x\leq6. \end{align*}
(a) What is the physical meaning of its boundary conditions?
(b) What are the initial displacement and initial velocity of the string at the mid- point, [latex]x=3[/latex].
(c) State the general form of its solution.
Case (2): [latex]f(x)=0[/latex].} Then [latex]u(x,0)=f(x)=0=X(x)T(0)[/latex] implies [latex]T(0)=0[/latex]. The solution of [latex]T''(t)+a^{2}(\frac{n\pi}{L})^{2}T(t)=0[/latex] with [latex]T(0)=0[/latex] will be [latex]T_{n}(t)=k_{n}\text{sin}(\frac{an\pi}{L}t)[/latex]. We let \[ u_{n}(x,t)=\text{sin}(\frac{an\pi}{L}t)\text{sin}(\frac{n\pi}{L}x) \] and write \[ u(x,t)=\sum_{n=1}^{\infty}d_{n}u_{n}(x,t)=\sum_{n=1}^{\infty}d_{n}\text{sin}(\frac{an\pi}{L}t)\text{sin}(\frac{n\pi}{L}x).\] We use the initial condition [latex]u_{t}(x,0)=g(x)[/latex] to solve for [latex]d_{n}[/latex]. \[ u_{t}(x,0)=g(x)=\sum_{n=1}^{\infty}d_{n}\frac{n\pi a}{L}\text{sin}(\frac{n\pi}{L}x). \] We see that [latex]g(x)[/latex] is a sine Fourier series and hence [latex]d_{n}\frac{n\pi a}{L}=\frac{2}{L}\int_{0}^{L}g(x)\text{sin}(\frac{n\pi}{L}x)dx[/latex], i.e. [latex]d_{n}=\frac{2}{n\pi a}\int_{0}^{L}g(x)\text{sin}(\frac{n\pi}{L}x)dx[/latex].
Example 2: Suppose the displacement [latex]u(x,t)[/latex] of a piece of flexible string is given by the initial-boundary value problem \begin{align*} u_{tt}(x,t) & =49u_{xx}(x,t),t \gt 0\\ u(0,t) & =0,u(2,t)=0,t \gt 0\\ u(x,0) & =0,u_{t}(x,0)=20-x^{2},0\leq x\leq2. \end{align*}
(a) What is the physical meaning of its boundary conditions?
(b) What are the initial displacement and initial velocity of the string at the mid- point, [latex]x=1[/latex].
(c) State the general form of its solution.
Exercise 2: Suppose the displacement [latex]u(x,t)[/latex] of a piece of flexible string is given by the initial-boundary value
problem \begin{align*} u_{tt}(x,t) & =9u_{xx}(x,t),t \gt 0\\ u(0,t) & =0,u(4,t)=0,t \gt 0\\ u(x,0) & =0,u_{t}(x,0)=10+2x,0\leq x\leq4. \end{align*}
(a) What is the physical meaning of its boundary conditions?
(b) What are the initial displacement and initial velocity of the string at the mid- point, [latex]x=2[/latex].
(c) State the general form of its solution.
Case (3): [latex]f(x)\neq0[/latex] and [latex]g(x)\neq0[/latex].} We let [latex]v(x,t)[/latex] and [latex]w(x,t)[/latex] be the functions satisfy the boundary problems \begin{align*} v_{tt}(x,t) & =a^{2}v_{xx}(x,t),t \gt 0\\ v(0,t) & =0,v(L,t)=0,t \gt 0\\ v(x,0) & =f(x),v_{t}(x,0)=0,0\leq x\leq L \end{align*} with the solution, \[ v(x,t)=\sum_{n=1}^{\infty}d_{n}\text{cos}(\frac{an\pi}{L}t)\text{sin}(\frac{n\pi}{L}x),\text{ and }d_{n}=\frac{2}{L}\int_{0}^{L}f(x)\text{sin}(\frac{n\pi}{L}x)dx.\] \begin{align*} w_{tt}(x,t) & =a^{2}w_{xx}(x,t),t \gt 0\\ w(0,t) & =0,w(L,t)=0,t \gt 0\\ w(x,0) & =0,w_{t}(x,0)=g(x),0\leq x\leq L \end{align*} with the solution, \[ w(x,t)=\sum_{n=1}^{\infty}e_{n}\text{sin}(\frac{an\pi}{L}t)\text{sin}(\frac{n\pi}{L}x),\text{ and }e_{n}=\frac{2}{n\pi a}\int_{0}^{L}g(x)\text{sin}(\frac{n\pi}{L}x)dx.\] Then [latex]u(x,t)=v(x,t)+w(x,t)[/latex] satisfy \begin{align*} u_{tt}(x,t) & =a^{2}u_{xx}(x,t),t \gt 0\\ u(0,t) & =0,u(L,t)=0,t \gt 0\\ u(x,0) & =f(x),u_{t}(x,0)=g(x),0\leq x\leq L \end{align*} with solution \begin{align*} u(x,t) & =v(x,t)+w(x,t)=\sum_{n=1}^{\infty}d_{n}\text{cos}(\frac{an\pi}{L}t)\text{sin}(\frac{n\pi}{L}x)+\sum_{n=1}^{\infty}e_{n}\text{sin}(\frac{an\pi}{L}t)\text{sin}(\frac{n\pi}{L}x) \end{align*} and [latex]d_{n}=\frac{2}{L}\int_{0}^{L}f(x)\text{sin}(\frac{n\pi}{L}x)dx[/latex], [latex]e_{n}=\frac{2}{n\pi a}\int_{0}^{L}g(x)\text{sin}(\frac{n\pi}{L}x)dx.[/latex]
Example 3: Suppose the displacement [latex]u(x,t)[/latex] of a piece of flexible string is given by the initial-boundary value problem \begin{align*} u_{tt}(x,t) & =16u_{xx}(x,t),t \gt 0\\ u(0,t) & =0,u(2\pi,t)=0,t \gt 0\\ u(x,0) & =20\text{cos}(2x),u_{t}(x,0)=20-x^{2},0\leq x\leq2\pi. \end{align*}
(a) What is the physical meaning of its boundary conditions?
(b) What are the initial displacement and initial velocity of the string at the mid- point, [latex]x=\pi[/latex].
(c) State the general form of its solution.
Exercise 3: Suppose the displacement [latex]u(x,t)[/latex] of a piece of flexible string is given by the initial-boundary value problem \begin{align*} u_{tt}(x,t) & =4u_{xx}(x,t),t \gt 0\\ u(0,t) & =0,u(10,t)=0,t \gt 0\\ u(x,0) & =20+10\text{cos}(\pi x)-30\text{cos}(2\pi x),u_{t}(x,0)=10+2x,0\leq x\leq10. \end{align*}
(a) What is the physical meaning of its boundary conditions?
(b) What are the initial displacement and initial velocity of the string at the mid- point, [latex]x=5[/latex].
(c) State the general form of its solution.