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 $$\begin{array}{rl}{u}_{tt}(x,t)& =a^2{u}_{xx}(x,t),t>0\\ u(0,t)& =0,u(L,t)=0,t>0\\ u(x,0)& =f(x),u_t(x,0)=g(x),0\le x\le L\end{array}$$  where  $f(x)$ is the initial position function and  $g(x)$ is the initial velocity.  We assume $u(x,t)=X(x)T(t)$, then translate the problem into solving $$\begin{array}{rl}X”(x)+\lambda X(x)& =0,X(0)=X(L)=0\\ T”(t)+a^2\lambda T(t)& =0.\end{array}$$ The only non-trivial solutions of  $X^{″}(x)+\lambda X(x)=0,X(0)=X(L)=0$ are  ${\lambda }_n=(\frac{n\pi }{L}{)}^2$ and  $X_n=c_n\text{sin}(\frac{n\pi }{L}x)$. Hence we need to solve  $T^{″}(t)+a^2(\frac{n\pi }{L}{)}^{2}T(t)=0.$ 

Case (1):  $g(x)=0$.} Then  $u_t(x,0)=g(x)=0=X(x)T^{\prime }(0)$ implies  $T^{\prime }(0)=0$. The solution of  $T^{″}(t)+a^2(\frac{n\pi }{L}{)}^{2}T(t)=0$ with  $T^{\prime }(0)=0$ will be  $T_n(t)=k_n\text{cos}(\frac{an\pi }{L}t)$. We let $$u_n(x,t)=\text{cos}(\frac{an\pi }{L}t)\text{sin}(\frac{n\pi }{L}x).$$ and write $$u(x,t)=\sum _{n=1}^{\mathrm{\infty }}{d}_n{u}_n(x,t)=\sum _{n=1}^{\mathrm{\infty }}{d}_n\text{cos}(\frac{an\pi }{L}t)\text{sin}(\frac{n\pi }{L}x).$$ We use the initial condition  $u(x,0)=f(x)$ to solve for  $d_n$. $$u(x,0)=f(x)=\sum _{n=1}^{\mathrm{\infty }}{d}_n{u}_n(x,0)=\sum _{n=1}^{\mathrm{\infty }}{d}_n\text{sin}(\frac{n\pi }{L}x).$$ We see that  $f(x)$ is a sine Fourier series and hence  $d_n=\frac{2}{L}{\int }_0^{L}f(x)\text{sin}(\frac{n\pi }{L}x)dx$. 

Example 1: Suppose the displacement  $u(x,t)$ of a piece of flexible string is given by the initial-boundary value problem $$\begin{array}{rl}{u}_{tt}(x,t)& =9u_{xx}(x,t),t>0\\ u(0,t)& =0,u(\pi ,t)=0,t>0\\ u(x,0)& =10\text{sin}(2x)-20\text{sin}(4x),u_t(x,0)=0,0\le x\le \pi .\end{array}$$

(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,  $x=\frac{\pi }{2}$. 

(c) State the general form of its solution.

Exercise 1: Suppose the displacement  $u(x,t)$ of a piece of flexible string is given by the initial-boundary value

problem $$\begin{array}{rl}{u}_{tt}(x,t)& =16u_{xx}(x,t),t>0\\ u(0,t)& =0,u(6,t)=0,t>0\\ u(x,0)& =10\text{sin}(\pi x)-30\text{sin}(2\pi x),u_t(x,0)=0,0\le x\le 6.\end{array}$$

(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,  $x=3$. 

(c) State the general form of its solution.

Case (2):  $f(x)=0$.} Then  $u(x,0)=f(x)=0=X(x)T(0)$ implies  $T(0)=0$. The solution of  $T^{″}(t)+a^2(\frac{n\pi }{L}{)}^{2}T(t)=0$ with  $T(0)=0$ will be  $T_n(t)=k_n\text{sin}(\frac{an\pi }{L}t)$. We let  $$u_n(x,t)=\text{sin}(\frac{an\pi }{L}t)\text{sin}(\frac{n\pi }{L}x)$$ and write  $$u(x,t)=\sum _{n=1}^{\mathrm{\infty }}{d}_n{u}_n(x,t)=\sum _{n=1}^{\mathrm{\infty }}{d}_n\text{sin}(\frac{an\pi }{L}t)\text{sin}(\frac{n\pi }{L}x).$$ We use the initial condition  $u_t(x,0)=g(x)$ to solve for  $d_n$. $$u_t(x,0)=g(x)=\sum _{n=1}^{\mathrm{\infty }}{d}_n\frac{n\pi a}{L}\text{sin}(\frac{n\pi }{L}x).$$ We see that  $g(x)$ is a sine Fourier series and hence  $d_n\frac{n\pi a}{L}=\frac{2}{L}{\int }_0^{L}g(x)\text{sin}(\frac{n\pi }{L}x)dx$, i.e.  $d_n=\frac{2}{n\pi a}{\int }_0^{L}g(x)\text{sin}(\frac{n\pi }{L}x)dx$. 

Example 2: Suppose the displacement  $u(x,t)$ of a piece of flexible string is given by the initial-boundary value problem $$\begin{array}{rl}{u}_{tt}(x,t)& =49u_{xx}(x,t),t>0\\ u(0,t)& =0,u(2,t)=0,t>0\\ u(x,0)& =0,u_t(x,0)=20-x^2,0\le x\le 2.\end{array}$$

(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,  $x=1$. 

(c) State the general form of its solution.

Exercise 2: Suppose the displacement  $u(x,t)$ of a piece of flexible string is given by the initial-boundary value

problem $$\begin{array}{rl}{u}_{tt}(x,t)& =9u_{xx}(x,t),t>0\\ u(0,t)& =0,u(4,t)=0,t>0\\ u(x,0)& =0,u_t(x,0)=10+2x,0\le x\le 4.\end{array}$$

(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,  $x=2$. 

(c) State the general form of its solution.

Case (3):  $f(x)\ne 0$ and  $g(x)\ne 0$.} We let  $v(x,t)$ and  $w(x,t)$ be the functions satisfy the boundary problems $$\begin{array}{rl}{v}_{tt}(x,t)& =a^2{v}_{xx}(x,t),t>0\\ v(0,t)& =0,v(L,t)=0,t>0\\ v(x,0)& =f(x),v_t(x,0)=0,0\le x\le L\end{array}$$ with the solution, $$v(x,t)=\sum _{n=1}^{\mathrm{\infty }}{d}_n\text{cos}(\frac{an\pi }{L}t)\text{sin}(\frac{n\pi }{L}x),\text{ and }{d}_n=\frac{2}{L}{\int }_0^{L}f(x)\text{sin}(\frac{n\pi }{L}x)dx.$$ $$\begin{array}{rl}{w}_{tt}(x,t)& =a^2{w}_{xx}(x,t),t>0\\ w(0,t)& =0,w(L,t)=0,t>0\\ w(x,0)& =0,w_t(x,0)=g(x),0\le x\le L\end{array}$$ with the solution, $$w(x,t)=\sum _{n=1}^{\mathrm{\infty }}{e}_n\text{sin}(\frac{an\pi }{L}t)\text{sin}(\frac{n\pi }{L}x),\text{ and }{e}_n=\frac{2}{n\pi a}{\int }_0^{L}g(x)\text{sin}(\frac{n\pi }{L}x)dx.$$ Then  $u(x,t)=v(x,t)+w(x,t)$ satisfy $$\begin{array}{rl}{u}_{tt}(x,t)& =a^2{u}_{xx}(x,t),t>0\\ u(0,t)& =0,u(L,t)=0,t>0\\ u(x,0)& =f(x),u_t(x,0)=g(x),0\le x\le L\end{array}$$ with solution $$\begin{array}{rl}u(x,t)& =v(x,t)+w(x,t)=\sum _{n=1}^{\mathrm{\infty }}{d}_n\text{cos}(\frac{an\pi }{L}t)\text{sin}(\frac{n\pi }{L}x)+\sum _{n=1}^{\mathrm{\infty }}{e}_n\text{sin}(\frac{an\pi }{L}t)\text{sin}(\frac{n\pi }{L}x)\end{array}$$ and  $d_n=\frac{2}{L}{\int }_0^{L}f(x)\text{sin}(\frac{n\pi }{L}x)dx$, $e_n=\frac{2}{n\pi a}{\int }_0^{L}g(x)\text{sin}(\frac{n\pi }{L}x)dx.$

Example 3: Suppose the displacement  $u(x,t)$ of a piece of flexible string is given by the initial-boundary value problem $$\begin{array}{rl}{u}_{tt}(x,t)& =16u_{xx}(x,t),t>0\\ u(0,t)& =0,u(2\pi ,t)=0,t>0\\ u(x,0)& =20\text{cos}(2x),u_t(x,0)=20-x^2,0\le x\le 2\pi .\end{array}$$

(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,  $x=\pi$. 

(c) State the general form of its solution.

Exercise 3: Suppose the displacement  $u(x,t)$ of a piece of flexible string is given by the initial-boundary value problem $$\begin{array}{rl}{u}_{tt}(x,t)& =4u_{xx}(x,t),t>0\\ u(0,t)& =0,u(10,t)=0,t>0\\ u(x,0)& =20+10\text{cos}(\pi x)-30\text{cos}(2\pi x),u_t(x,0)=10+2x,0\le x\le 10.\end{array}$$

(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,  $x=5$. 

(c) State the general form of its solution.