# Question Waiting for answersThe vibration of a uniform string is modelled by the wave equation $$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}$$, where $$u(x, t)$$ is the displacement of the string and $$c$$ is a non-zero constant, $$c=1$$. Assume that both ends of the string are fixed. The initial position of the string is set by $$u(x, 0)=x(1-x)$$. The string is also set into motion from its initial position with an initial velocity: $\left(\frac{\partial u}{\partial t}\right)(x, 0)=\left\{\begin{array}{cc} \alpha & 0<x \leq \frac{1}{2} \\ 0 & \frac{1}{2}<x \leq 1 \end{array}\right.$ where $$\alpha$$ is a small non-zero constant. Follow the steps below and use the method of separation of variables $$u(x, t)=F(x) G(t)$$ to find the displacement $$u(x, t)$$. (a) Plot the initial position of the string in MATLAB.[1 mark] (b) Write down the fixed-end boundary conditions at $$x=0$$ and $$x=1$$ for the displacement $$u(x, t)$$, and then deduce the boundary conditions for the function $$F(x)$$. [2 marks] (c) Explain in your own words what is the physical process behind the mathematical relation given by $$\left(\frac{\partial u}{\partial t}\right)(x, 0)$$. [1 mark] (d) Using the method of separation of variables, find two constant-coefficient ordinary differential equations. [2 marks] (e) Given a separation constant $$k=0$$, discuss the general solution of the problem. [3 marks] (f) Given a separation constant $$k>0$$, discuss the general solution of the problem. [3 marks] (g) Given a separation constant $$k=-p^{2}<0$$ i. Show all steps to find the eigen-function $$F_{n}(x)$$. ii. Given the eigen-values $$k_{n}$$, find $$G_{n}(t)$$. Hint: Use the notation $$C_{n}$$ and $$D_{n}$$ for the constants involved. iii. Write a general form of the $$u_{n}(x, t)$$ solution of the problem. iv. Use the superposition principle to obtain a general solution of the problem. v. Find $$C_{n}$$. vi. Find $$D_{n}$$. [10 marks] (h) Find the series form for the subsequent time-dependent displacement $$u(x, t)$$ of the uniform string and write down the first six non-zero terms of the solution. [3 marks]

Transcribed Image Text: The vibration of a uniform string is modelled by the wave equation $$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}$$, where $$u(x, t)$$ is the displacement of the string and $$c$$ is a non-zero constant, $$c=1$$. Assume that both ends of the string are fixed. The initial position of the string is set by $$u(x, 0)=x(1-x)$$. The string is also set into motion from its initial position with an initial velocity: \[ \left(\frac{\partial u}{\partial t}\right)(x, 0)=\left\{\begin{array}{cc} \alpha & 00 \), discuss the general solution of the problem. [3 marks] (g) Given a separation constant $$k=-p^{2}<0$$ i. Show all steps to find the eigen-function $$F_{n}(x)$$. ii. Given the eigen-values $$k_{n}$$, find $$G_{n}(t)$$. Hint: Use the notation $$C_{n}$$ and $$D_{n}$$ for the constants involved. iii. Write a general form of the $$u_{n}(x, t)$$ solution of the problem. iv. Use the superposition principle to obtain a general solution of the problem. v. Find $$C_{n}$$. vi. Find $$D_{n}$$. [10 marks] (h) Find the series form for the subsequent time-dependent displacement $$u(x, t)$$ of the uniform string and write down the first six non-zero terms of the solution. [3 marks]
Transcribed Image Text: The vibration of a uniform string is modelled by the wave equation $$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}$$, where $$u(x, t)$$ is the displacement of the string and $$c$$ is a non-zero constant, $$c=1$$. Assume that both ends of the string are fixed. The initial position of the string is set by $$u(x, 0)=x(1-x)$$. The string is also set into motion from its initial position with an initial velocity: \[ \left(\frac{\partial u}{\partial t}\right)(x, 0)=\left\{\begin{array}{cc} \alpha & 00 \), discuss the general solution of the problem. [3 marks] (g) Given a separation constant $$k=-p^{2}<0$$ i. Show all steps to find the eigen-function $$F_{n}(x)$$. ii. Given the eigen-values $$k_{n}$$, find $$G_{n}(t)$$. Hint: Use the notation $$C_{n}$$ and $$D_{n}$$ for the constants involved. iii. Write a general form of the $$u_{n}(x, t)$$ solution of the problem. iv. Use the superposition principle to obtain a general solution of the problem. v. Find $$C_{n}$$. vi. Find $$D_{n}$$. [10 marks] (h) Find the series form for the subsequent time-dependent displacement $$u(x, t)$$ of the uniform string and write down the first six non-zero terms of the solution. [3 marks]