Home/Mathematics/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: \[ \left(\frac{\partial u}{\partial t}\right)(x, 0)=\left\{\begin{array}{cc} \alpha & 0<x \leq \frac{1}{2} \\ 0 & \frac{1}{2}<x \leq 1 \end{array}\right. \] 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]

Question Waiting for answers 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: \[ \left(\frac{\partial u}{\partial t}\right)(x, 0)=\left\{\begin{array}{cc} \alpha & 0<x \leq \frac{1}{2} \\ 0 & \frac{1}{2}<x \leq 1 \end{array}\right. \] 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]

QDW20Z The Asker · Advanced Mathematics

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]
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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]
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width: 0;}.MJX-TEX{font-family: MJXZERO, MJXTEX;}@font-face{font-family: MJXZERO; src: url("https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Zero.woff") format("woff");}@font-face{font-family: MJXTEX; src: url("https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Main-Regular.woff") format("woff");}mjx-c.mjx-c78::before{padding: 0.431em 0.528em 0 0; content: "x";}mjx-c.mjx-c3D::before{padding: 0.583em 0.778em 0.082em 0; content: "=";}mjx-c.mjx-c30::before{padding: 0.666em 0.5em 0.022em 0; content: "0";}mjx-c.mjx-c31::before{padding: 0.666em 0.5em 0 0; content: "1";}mjx-c.mjx-c75::before{padding: 0.442em 0.556em 0.011em 0; content: "u";}mjx-c.mjx-c28::before{padding: 0.75em 0.389em 0.25em 0; content: "(";}mjx-c.mjx-c2C::before{padding: 0.121em 0.278em 0.194em 0; content: ",";}mjx-c.mjx-c74::before{padding: 0.615em 0.389em 0.01em 0; content: "t";}mjx-c.mjx-c29::before{padding: 0.75em 0.389em 0.25em 0; content: ")";}mjx-c.mjx-c22C5::before{padding: 0.31em 0.278em 0 0; content: "\22C5";}mjx-c.mjx-c46::before{padding: 0.68em 0.653em 0 0; content: "F";}mjx-c.mjx-c61::before{padding: 0.448em 0.5em 0.011em 0; content: "a";}mjx-c.mjx-c6E::before{padding: 0.442em 0.556em 0 0; content: "n";}mjx-c.mjx-c64::before{padding: 0.694em 0.556em 0.011em 0; content: "d";}mjx-c.mjx-cA0::before{padding: 0 0.25em 0 0; content: "\A0";}(b) The fixed-end boundary conditions at x=0 andx=1 for the displacementu(x,t) are:u(0,t)=0and u(1,t)=0&#8901;The boundary conditions for the functionF(x)are:F(0)=0and&#160;F(1)=0&#8901;Explanation:Please refer to solution in this step.Step3/11mjx-container[jax="CHTML"]{line-height: 0;}mjx-container [space="2"]{margin-left: .167em;}mjx-container [space="4"]{margin-left: .278em;}mjx-mstyle{display: inline-block;}mjx-assistive-mml{position: absolute !important; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); padding: 1px 0px 0px 0px !important; border: 0px !important; display: block !important; width: auto !important; overflow: hidden !important; 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content: "(";}mjx-c.mjx-c2202::before{padding: 0.715em 0.566em 0.022em 0; content: "\2202";}mjx-c.mjx-c74::before{padding: 0.615em 0.389em 0.01em 0; content: "t";}mjx-c.mjx-c75::before{padding: 0.442em 0.556em 0.011em 0; content: "u";}mjx-c.mjx-c29::before{padding: 0.75em 0.389em 0.25em 0; content: ")";}mjx-c.mjx-c78::before{padding: 0.431em 0.528em 0 0; content: "x";}mjx-c.mjx-c2C::before{padding: 0.121em 0.278em 0.194em 0; content: ",";}mjx-c.mjx-c30::before{padding: 0.666em 0.5em 0.022em 0; content: "0";}mjx-c.mjx-c3D::before{padding: 0.583em 0.778em 0.082em 0; content: "=";}mjx-c.mjx-c22C5::before{padding: 0.31em 0.278em 0 0; content: "\22C5";}(c) The physical process behind the mathematical relation given by (&#8706;t&#8706;u)(x,0)is the initial velocity of the string. 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content: "c";}mjx-c.mjx-c1D434.TEX-I::before{padding: 0.716em 0.75em 0 0; content: "A";}mjx-c.mjx-c32::before{padding: 0.666em 0.5em 0 0; content: "2";}mjx-c.mjx-c2202::before{padding: 0.715em 0.566em 0.022em 0; content: "\2202";}mjx-c.mjx-c31::before{padding: 0.666em 0.5em 0 0; content: "1";}mjx-c.mjx-cA0::before{padding: 0 0.25em 0 0; content: "\A0";}mjx-c.mjx-c2F::before{padding: 0.75em 0.5em 0.25em 0; content: "/";}mjx-c.mjx-c2212::before{padding: 0.583em 0.778em 0.082em 0; content: "\2212";}mjx-c.mjx-c1D458.TEX-I::before{padding: 0.694em 0.521em 0.011em 0; content: "k";}mjx-c.mjx-c61::before{padding: 0.448em 0.5em 0.011em 0; content: "a";}mjx-c.mjx-c6E::before{padding: 0.442em 0.556em 0 0; content: "n";}mjx-c.mjx-c64::before{padding: 0.694em 0.556em 0.011em 0; content: "d";}mjx-c.mjx-c6B::before{padding: 0.694em 0.528em 0 0; content: "k";}(d) Using the method of separation of variables, we can assume that u(x,t)=F(x)G(t)&#8901; Substituting this into the wave equation, we get:&#178;&#178;F(x)G(t)=cA2&#8901;&#8706;x&#178;F(x)&#8706;t&#178;G(t)Dividing both sides by F(x)G(t), we get:&#178;&#178;1=cA2&#8901;&#8706;x&#178;F(x)&#160;/&#160;F(x)&#8706;t&#178;G(t)&#160;/&#160;G(t)This leads to two constant-coefficient ordinary differential equations:&#178;&#178;&#8706;x&#178;F(x)=&#8722;kF(x)&#160;and&#8706;t&#178;G(t)=kcA2A222G(t)Explanation:Please refer to solution in this step.Step5/11mjx-container[jax="CHTML"]{line-height: 0;}mjx-container [space="4"]{margin-left: .278em;}mjx-mstyle{display: inline-block;}mjx-assistive-mml{position: absolute !important; 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src: url("https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Zero.woff") format("woff");}@font-face{font-family: MJXTEX; src: url("https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Main-Regular.woff") format("woff");}mjx-c.mjx-c6B::before{padding: 0.694em 0.528em 0 0; content: "k";}mjx-c.mjx-c3D::before{padding: 0.583em 0.778em 0.082em 0; content: "=";}mjx-c.mjx-c30::before{padding: 0.666em 0.5em 0.022em 0; content: "0";}mjx-c.mjx-c2C::before{padding: 0.121em 0.278em 0.194em 0; content: ",";}mjx-c.mjx-c2202::before{padding: 0.715em 0.566em 0.022em 0; content: "\2202";}mjx-c.mjx-c78::before{padding: 0.431em 0.528em 0 0; content: "x";}mjx-c.mjx-c46::before{padding: 0.68em 0.653em 0 0; content: "F";}mjx-c.mjx-c28::before{padding: 0.75em 0.389em 0.25em 0; content: "(";}mjx-c.mjx-c29::before{padding: 0.75em 0.389em 0.25em 0; content: ")";}mjx-c.mjx-c22C5::before{padding: 0.31em 0.278em 0 0; content: "\22C5";}(e). Given a separation constant k=0, the general solution of the problem is the set of all functions that satisfy the equation &#178;&#8706;x&#178;F(x)=0&#8901;Explanation:This means that any function that is linear in x is a solution. However, since the boundary conditions are F(0) = 0 and F(1) = 0, the only solution is the trivial solution F(x) = 0.Step6/11mjx-container[jax="CHTML"]{line-height: 0;}mjx-container [space="4"]{margin-left: .278em;}mjx-mtext{display: inline-block; text-align: left;}mjx-mstyle{display: inline-block;}mjx-assistive-mml{position: absolute !important; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); padding: 1px 0px 0px 0px !important; border: 0px !important; display: block !important; width: auto !important; overflow: hidden !important; -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none;}mjx-math{display: inline-block; text-align: left; line-height: 0; text-indent: 0; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; border-collapse: collapse; word-wrap: normal; word-spacing: normal; white-space: nowrap; direction: ltr; padding: 1px 0;}mjx-mi{display: inline-block; text-align: left;}mjx-c{display: inline-block;}mjx-utext{display: inline-block; padding: .75em 0 .2em 0;}mjx-mo{display: inline-block; text-align: left;}mjx-mn{display: inline-block; text-align: left;}mjx-mspace{display: inline-block; text-align: left;}mjx-c::before{display: block; width: 0;}.MJX-TEX{font-family: MJXZERO, MJXTEX;}.TEX-I{font-family: MJXZERO, MJXTEX-I;}@font-face{font-family: MJXZERO; src: url("https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Zero.woff") format("woff");}@font-face{font-family: MJXTEX; src: url("https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Main-Regular.woff") format("woff");}@font-face{font-family: MJXTEX-I; src: url("https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Math-Italic.woff") format("woff");}mjx-c.mjx-c6B::before{padding: 0.694em 0.528em 0 0; content: "k";}mjx-c.mjx-c3E::before{padding: 0.54em 0.778em 0.04em 0; content: "&gt;";}mjx-c.mjx-c30::before{padding: 0.666em 0.5em 0.022em 0; content: "0";}mjx-c.mjx-c2C::before{padding: 0.121em 0.278em 0.194em 0; content: ",";}mjx-c.mjx-c2202::before{padding: 0.715em 0.566em 0.022em 0; content: "\2202";}mjx-c.mjx-c78::before{padding: 0.431em 0.528em 0 0; content: "x";}mjx-c.mjx-c46::before{padding: 0.68em 0.653em 0 0; content: "F";}mjx-c.mjx-c28::before{padding: 0.75em 0.389em 0.25em 0; content: "(";}mjx-c.mjx-c29::before{padding: 0.75em 0.389em 0.25em 0; content: ")";}mjx-c.mjx-c3D::before{padding: 0.583em 0.778em 0.082em 0; content: "=";}mjx-c.mjx-c2212::before{padding: 0.583em 0.778em 0.082em 0; content: "\2212";}mjx-c.mjx-c1D458.TEX-I::before{padding: 0.694em 0.521em 0.011em 0; content: "k";}mjx-c.mjx-c22C5::before{padding: 0.31em 0.278em 0 0; content: "\22C5";}mjx-c.mjx-c73::before{padding: 0.448em 0.394em 0.011em 0; content: "s";}mjx-c.mjx-c69::before{padding: 0.669em 0.278em 0 0; content: "i";}mjx-c.mjx-c6E::before{padding: 0.442em 0.556em 0 0; content: "n";}mjx-c.mjx-cA0::before{padding: 0 0.25em 0 0; content: "\A0";}mjx-c.mjx-c61::before{padding: 0.448em 0.5em 0.011em 0; content: "a";}mjx-c.mjx-c64::before{padding: 0.694em 0.556em 0.011em 0; content: "d";}mjx-c.mjx-c63::before{padding: 0.448em 0.444em 0.011em 0; content: "c";}mjx-c.mjx-c6F::before{padding: 0.448em 0.5em 0.01em 0; content: "o";}(f). Given a separation constantk&gt;0,the general solution of the problem is the set of all functions that satisfy the equation &#178;&#8706;x&#178;F(x)=&#8722;kF(x)&#8901;The solution is a linear combination of &#960;&#960;sin(k&#960;x)&#160;and&#160;cos(k&#960;x)that satisfy the boundary conditions F(0) = 0 and F(1) = 0.Explanation:Please refer to solution in this step.Step7/11mjx-container[jax="CHTML"]{line-height: 0;}mjx-container [space="4"]{margin-left: .278em;}mjx-container [size="s"]{font-size: 70.7%;}mjx-block{display: block;}mjx-mstyle{display: inline-block;}mjx-mphantom{visibility: hidden;}mjx-assistive-mml{position: absolute !important; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); padding: 1px 0px 0px 0px !important; border: 0px !important; display: block !important; width: auto !important; overflow: hidden !important; -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none;}mjx-math{display: inline-block; text-align: left; line-height: 0; text-indent: 0; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; border-collapse: collapse; word-wrap: normal; wo ... 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