Section 11.3-Motion of a stringby Alain Goriely, goriely@math.arizona.edu, (http://www.math.arizona.edu/~goriely) Abstract: This section illustrates Section 11.3 in Kreyszig 's book (8th ed.)Application Areas/Subjects: Engineering, Applied MathematicsKeywords: String Motion, 1D-Wave equation See Also: Other Worksheets in the same package.Prerequisites: plotsNote: Send me an e-mail (comments-criticisms) if you use this worksheet.restart;assume(n,integer):with(plots): setoptions(thickness=2): #set the tickness of the lines in the plots
<Text-field layout="Heading 1" style="Heading 1"><Font family="Times New Roman">Introduction</Font></Text-field>Let NiMvJSJ1Ry1GJDYkJSJ4RyUidEc= be the vertical displacement of the string. Under suitable assumptions (see 11.2), the displacement obeys the following PDEs (the so-called 1D wave equation):NiMvLSUlZGlmZkc2JCUidUctJSIkRzYkJSJ0RyIiIyomJSJjR0YsLUYlNiRGJy1GKTYkJSJ4R0YsIiIiwhere NiMlImNH is the speed of the wave (NiMvKiQlImNHIiIjKiYlJHJob0ciIiIlIlRHISIi : density of the string/ tension in the string) The string is attached at NiMvJSJ4RyUiTEc=, that is we have the boundary conditions NiMvLSUidUc2JCIiISUidEdGJw== and NiMvLSUidUc2JCUiTEclInRHIiIh Here we explore different solutions for the string, starting with initial data. Look at the animations of the string!
<Text-field layout="Heading 1" style="Heading 1"><Font family="Times New Roman">Section 1: The fundamental modes</Font></Text-field>The following functions are the solutions for boundary conditions NiMvLSUidUc2JCIiISUidEdGJw== and NiMvLSUidUc2JCUiTEclInRHIiIhSu:=u=(A[n]*cos(lambda[n]*t)+B[n]*sin(lambda[n]*t))*sin(n*Pi*x/L);Slambda:=lambda[n]=c*n*Pi/L;Let us verify that this is indeed a solution of the equation:Diff(u,`\$`(t,2)) - c^2*Diff(u,`\$`(x,2))=eval(subs(Su,diff(u,`\$`(t,2)) - c^2*diff(u,`\$`(x,2))));We want to see what these modes look like: Start with n=1, the FUNDAMENTAL MODE: