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{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 292 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 293 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 294 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 295 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 296 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 297 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 298 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 299 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 300 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 301 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Unit 5: Boundary Value Pro blems for Partial Differential Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Chapter 24: Wave Equation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Section 2 4.5: longitudinal vibrations in an elastic rod" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Copyright" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Copyright * 2001 by Addison Wesley Longman, Inc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 302 "All rights reserved. No part of th is publication may be reproduced, stored in a retrieval system, or tra nsmitted, in any form or by any means, electronic, mechanical, photoco pying, recording, or otherwise, without the prior written permission o f the publisher. Printed in the United States of America." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Intializa tions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "with(plots):\ninterface(showassumed=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introductio n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "A rod of length " }{TEXT 294 1 "L" }{TEXT -1 13 " and density " } {XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 134 " (mass per unit volume) \+ is deformed by pulling or pushing in opposite directions at its ends. \+ The rod is said to stretch of compress " }{TEXT 295 8 "linearly" } {TEXT -1 186 " along its axis if the restorative forces induced in the rod obey Hooke's law. In effect, the rod behaves like a spring which is both extensible and compressible. Such a rod is called " }{TEXT 296 7 "elastic" }{TEXT -1 58 " if after a deformation, it returns to i ts original state." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 282 "Under compression, the distance between contiguous cross -sections in the rod decreases, while under expansion (rerefaction), i t increases. Compression or expansion experienced by neighboring plan e cross-sections can propagate through the rod as a wave which obeys t he wave equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[tt]=c^2*u[xx]" "6#/&%\"uG6#%#ttG*&%\" cG\"\"#&F%6#%#xxG\"\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Since the disturbance travels along the axis of the \+ rod, the wave is said to be " }{TEXT 297 12 "longitudinal" }{TEXT -1 330 ". Thus, a steel rod, hammered on its end, will support longitudi nal waves of compression and rarefaction traveling the length of the r od. Similarly, the rubber engine mounts supporting the engine in an a utomobile, when subjected to shock, experience the same passage of dis tortions progressing as plane waves through the medium." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 "Although the longit udinal vibrations in a rod are governed by the same wave equation as t he vibrating string, the meaning of the displacement " }{XPPEDIT 18 0 "u(x,t)" "6#-%\"uG6$%\"xG%\"tG" }{TEXT -1 43 " is different, and subtl y so. In the rod, " }{XPPEDIT 18 0 "u(x,t)" "6#-%\"uG6$%\"xG%\"tG" } {TEXT -1 36 " measures the displacement, at time " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 38 ", of the plane in the rod that was at " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 259 " initially. Since these di splacements are relative to where the plane section was initially, the re is an additivity which makes the complete motion much harder to fat hom than the vibrations in the finite string where the displacement wa s directly observable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "In addition, the free endpoint condition whereby an end is free to move is formalized in terms of the " }{TEXT 298 6 "str ain" }{TEXT -1 287 ", a new concept for the reader not familier with t he theory of elasticity. Hence, our approach will be to state an appr opriate boundary value problem, nurture intuition by illustrating the \+ solution, and finally, develop the theory and derive the governing par tial differential equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 12 "Example 24.6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "Problem Statement" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "Formulate and sol ve an appropriate boundary value problem for the longitudinal elastic \+ vibrations in a rod of length " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 50 " whose left and right ends are both free. Assume " }{XPPEDIT 18 0 "c=1" "6#/%\"cG\"\"\"" }{TEXT -1 117 " as the wave speed, and uni formly stretch the rod by one unit, releasing it without imparting any initial velocity. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Problem Formulation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "If the rod is placed between " }{XPPEDIT 18 0 " x=0" "6#/%\"xG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=Pi" "6#/%\" xG%#PiG" }{TEXT -1 7 " on an " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 28 "-axis, and if a copy of the " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 112 "-axis is etched on the rod at equilibrium, then the desired bo undary value problem consists of the wave equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 278 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[tt] =u[xx]" "6#/&%\"uG6#%#ttG&F%6#%#xxG" }}{PARA 0 "" 0 "" {TEXT -1 23 "th e boundary conditions" }}{PARA 279 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[x](0,t)=0" "6#/-&%\"uG6#%\"xG6$\"\"!%\"tGF*" }}{PARA 280 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[x](Pi,t)=0" "6#/-&%\"uG6#%\"xG6$%# PiG%\"tG\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 26 "and the initial conditio ns" }}{PARA 281 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,0)=f(x)" "6 #/-%\"uG6$%\"xG\"\"!-%\"fG6#F'" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x/Pi " "6#*&%\"xG\"\"\"%#PiG!\"\"" }}{PARA 282 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "u[t](x,0)=0" "6#/-&%\"uG6#%\"tG6$%\"xG\"\"!F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "u(x,t)" "6#-%\"uG6$%\"xG%\"tG" }{TEXT -1 36 " measures the displacement, at time " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 55 ", of the plane section in the rod that was at location " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 216 " initially. The endpoint c onditions are derivatives because, as we will see later whenwe derive \+ the governing equations, \"free ends\" mean no forces act on the ends. In fact, the elastic force on a face at location " }{TEXT 299 1 "x" }{TEXT -1 13 " is given by " }{XPPEDIT 18 0 "u[x](x,t)" "6#-&%\"uG6#% \"xG6$F'%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 114 "To determine the initial distribution of displacements in the rod, we must first obtain the constant value of \+ the " }{TEXT 300 6 "strain" }{TEXT -1 93 ", defined as the change in l ength, per unit length. Thus, we compute the constant strain as " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 283 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Delta*L/L=([Pi+1]-[Pi])/Pi" "6#/*(%&DeltaG\"\"\"%\"LGF& F'!\"\"*&,&7#,&%#PiGF&F&F&F&7#F-F(F&F-F(" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/Pi" "6#*&\"\"\"F$%#PiG!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "The initial distributi on of displacements is then found by accumulating the strain along the rod, obtaining" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 284 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,0) = Int(1/Pi,s=0..x)" "6#/-%\"uG6$ %\"xG\"\"!-%$IntG6$*&\"\"\"F-%#PiG!\"\"/%\"sG;F(F'" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x/Pi" "6#*&%\"xG\"\"\"%#PiG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Finally, the initial velo city, namely, " }{XPPEDIT 18 0 "u[t](x,0)" "6#-&%\"uG6#%\"tG6$%\"xG\" \"!" }{TEXT -1 10 ", is zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 177 "This boundary value problem is new becau se we have not yet solved the wave equation with homogeneous Neumann c onditions (as derivative conditions at the endpoints are called.) " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Solution by Separation of Variables" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Assume " }{XPPEDIT 18 0 "u(x,t)=X(x)*T(t)" "6#/-%\"uG6$%\"xG%\"tG*&-%\"XG6#F'\"\"\"-%\"TG 6#F(F-" }{TEXT -1 10 ", that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "U := X(x)*T(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "as the separated form for the solution. As in Sections 24.1 a nd 24.2, the wave equation then yields the two ODEs" }}{PARA 0 "" 0 " " {TEXT -1 70 "Apply this separation assumption to the endpoint condit ions, obtaining" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 285 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`X''`(X)-lambda*X(x)=0" "6#/,&-%$X''G6# %\"XG\"\"\"*&%'lambdaGF)-F(6#%\"xGF)!\"\"\"\"!" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 286 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`T''`(t)-lambda*T(t)=0" "6#/,&-%$T''G6#%\"tG\"\"\"*&%'l ambdaGF)-%\"TG6#F(F)!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "However, applying the sep aration assumption to the endpoint conditions now gives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 287 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " u[x](0,t)=0" "6#/-&%\"uG6#%\"xG6$\"\"!%\"tGF*" }{TEXT -1 6 " => " } {XPPEDIT 18 0 "`X'`(0)*T(t)=0" "6#/*&-%#X'G6#\"\"!\"\"\"-%\"TG6#%\"tGF )F(" }{TEXT -1 6 " => " }{XPPEDIT 18 0 "`X'`(0)=0" "6#/-%#X'G6#\"\"! F'" }{TEXT -1 1 " " }}{PARA 288 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[x](Pi,t)=0" "6#/-&%\"uG6#%\"xG6$%#PiG%\"tG\"\"!" }{TEXT -1 6 " => \+ " }{XPPEDIT 18 0 "`X'`(Pi)*T(t)=0" "6#/*&-%#X'G6#%#PiG\"\"\"-%\"TG6#% \"tGF)\"\"!" }{TEXT -1 6 " => " }{XPPEDIT 18 0 "`X'`(Pi)=0" "6#/-%#X 'G6#%#PiG\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 78 "In addition, applying the separation assu mption to the initial velocity yields" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 289 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[t](x,0)=0" "6#/ -&%\"uG6#%\"tG6$%\"xG\"\"!F+" }{TEXT -1 6 " => " }{XPPEDIT 18 0 "X(x )*`T'`(0)=0" "6#/*&-%\"XG6#%\"xG\"\"\"-%#T'G6#\"\"!F)F-" }{TEXT -1 6 " => " }{XPPEDIT 18 0 "`T'`(0)=0" "6#/-%#T'G6#\"\"!F'" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "a re sult we obtained each time we solved the wave equation with a similarl y vanishing initial velocity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 44 "We therefore face the Sturm-Liouville sys tem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 290 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`X''`(x)-lambda*X(x)=0" "6#/,&-%$X''G6#%\"xG\"\"\"*& %'lambdaGF)-%\"XG6#F(F)!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 291 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`X'`(0)=`X'`(Pi)" "6#/-%#X'G6#\"\"!-F %6#%#PiG" }{TEXT -1 5 " = 0 " }}{PARA 0 "" 0 "" {TEXT -1 38 "and the m odified initial value problem" }}{PARA 292 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`T''`(t)-lambda*T(t)=0" "6#/,&-%$T''G6#%\"tG\"\"\"*&%'l ambdaGF)-%\"TG6#F(F)!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 293 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`T'`(0)=0" "6#/-%#T'G6#\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "From Section 16.1, the Sturm-Liouville BVP for " }{XPPEDIT 18 0 "X (x)" "6#-%\"XG6#%\"xG" }{TEXT -1 21 " has the eigenvalues " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "-n^2" "6#,$ *$%\"nG\"\"#!\"\"" }{TEXT -1 2 ", " }{TEXT 256 1 "n" }{TEXT -1 15 " = \+ 0, 1, 2, ..." }}{PARA 0 "" 0 "" {TEXT -1 36 "and the corresponding eig enfunctions" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "X[n]=A [n]*cos(n*x)" "6#/&%\"XG6#%\"nG*&&%\"AG6#F'\"\"\"-%$cosG6#*&F'F,%\"xGF ,F," }{TEXT -1 2 ", " }{TEXT 257 1 "n" }{TEXT -1 15 " = 0, 1, 2, ..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The cor responding solutions for " }{XPPEDIT 18 0 "T(t)" "6#-%\"TG6#%\"tG" } {TEXT -1 9 " are then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "T[n]=B[n]*cos(n*t)" "6#/&%\"TG6#% \"nG*&&%\"BG6#F'\"\"\"-%$cosG6#*&F'F,%\"tGF,F," }{TEXT -1 2 ", " } {TEXT 258 1 "n" }{TEXT -1 15 " = 0, 1, 2, ..." }}{PARA 0 "" 0 "" {TEXT -1 36 "A typical eigensolution is therefore" }}{PARA 261 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "u[n](x,t)=a[n]*cos(n*x)*cos(n*t)" "6# /-&%\"uG6#%\"nG6$%\"xG%\"tG*(&%\"aG6#F(\"\"\"-%$cosG6#*&F(F0F*F0F0-F26 #*&F(F0F+F0F0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }{XPPEDIT 18 0 "u(x,t)" "6#-%\"uG6$%\"xG%\"tG" } {TEXT -1 68 " is a linear combination of all such eigensolutions. Eva luating at " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!" }{TEXT -1 71 " tells us the coefficients in the sum are the Fourier coefficients for " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 ", so we have the \+ solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,t) = a[0]/2" "6#/-%\"uG6$%\"xG%\"tG*&&%\" aG6#\"\"!\"\"\"\"\"#!\"\"" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "Sum(a[n]* cos(n*x)*cos(n*t),n=1..infinity)" "6#-%$SumG6$*(&%\"aG6#%\"nG\"\"\"-%$ cosG6#*&F*F+%\"xGF+F+-F-6#*&F*F+%\"tGF+F+/F*;F+%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[n]=2/Pi" "6#/&%\"aG6#%\"nG*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(f(x)*cos(n*x),x=0..Pi)" "6#-%$IntG6$*&-%\"fG6#% \"xG\"\"\"-%$cosG6#*&%\"nGF+F*F+F+/F*;\"\"!%#PiG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "f(x) =x/Pi" "6#/-%\"fG6#%\"xG*&F'\"\"\"%#PiG!\"\"" }{TEXT -1 14 ", that is, for" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f := x/Pi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "the Fourier coefficients a re " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 294 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "a[0]=2/Pi" "6#/&%\"aG6#\"\"!*&\"\"#\"\"\"%#PiG!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x/Pi,x=0..Pi)=1" "6#/-%$IntG6$*& %\"xG\"\"\"%#PiG!\"\"/F(;\"\"!F*F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 295 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[ n]=2/Pi" "6#/&%\"aG6#%\"nG*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(``(x/Pi)*cos(n*x),x=0..Pi)=2" "6#/-%$IntG6$*&-%!G6# *&%\"xG\"\"\"%#PiG!\"\"F--%$cosG6#*&%\"nGF-F,F-F-/F,;\"\"!F.\"\"#" } {TEXT -1 1 " " }{XPPEDIT 18 0 "((-1)^n-1)/Pi^2/n^2" "6#*(,&),$\"\"\"! \"\"%\"nGF'F'F(F'*$%#PiG\"\"#F(*$F)F,F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "which we obtain in M aple as follows. First, obtain " }{XPPEDIT 18 0 "a[0]" "6#&%\"aG6#\" \"!" }{TEXT -1 27 " with the separate integral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "q0 := (2/Pi )*Int(f,x=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "whose value is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a0 := value (q0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "The general Fourier coefficient is given by" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "qa := (2/Pi)*Int(f*cos(n*x),x=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Tell Maple that " }{TEXT 259 1 "n" }{TEXT -1 18 " is an integer via" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "assu me(n,integer);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "and evaluate this defining integra l, obtaining" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "a := simplify(value(qa));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "The pa rtial sum " }}{PARA 296 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[20](x ,t)=1/2" "6#/-&%\"uG6#\"#?6$%\"xG%\"tG*&\"\"\"F-\"\"#!\"\"" }{TEXT -1 6 " + 2 " }{XPPEDIT 18 0 "Sum(``(((-1)^n-1)/Pi^2/n^2)*cos(n*x)*cos(n* t),n=1..20)" "6#-%$SumG6$*(-%!G6#*(,&),$\"\"\"!\"\"%\"nGF.F.F/F.*$%#Pi G\"\"#F/*$F0F3F/F.-%$cosG6#*&F0F.%\"xGF.F.-F66#*&F0F.%\"tGF.F./F0;F.\" #?" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "u20 := a0/2 + sum(a*cos(n*x)*cos(n*t),n=1..20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "is graph ed as a surface in Figure 24.18, below." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "plot3d(u20, x=0..Pi, \+ t=0..4*Pi, axes=frame, color=black, labels=[x,t,u], labelfont=[TIMES,I TALIC,12], scaling=constrained, tickmarks=[3,6,2], orientation=[-45,50 ]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 157 "The wave-like properties of the solution are appa rent, but the detailed interpretation of the meaning of these waves re quires more investigation. Remember, " }{XPPEDIT 18 0 "u(x,t)" "6#-% \"uG6$%\"xG%\"tG" }{TEXT -1 33 " gives the displacement, at time " } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 35 ", of the plane section that \+ was at " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 150 " initially. What we want to see is a representation of the physical motion of the elas tic rod, and unlike the transverse waves on the finite string, " } {XPPEDIT 18 0 "u(x,t)" "6#-%\"uG6$%\"xG%\"tG" }{TEXT -1 47 " does not \+ directly show these physical motions." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "An animation of the solution " } {XPPEDIT 18 0 "u(x,t)" "6#-%\"uG6$%\"xG%\"tG" }{TEXT -1 152 " is a dyn amic interpretation of the solution surface, but is still not a repres entation of the physical motions in the rod. It shows the displacemen ts " }{XPPEDIT 18 0 "u(x,t)" "6#-%\"uG6$%\"xG%\"tG" }{TEXT -1 38 " whi ch each face that was at location " }{XPPEDIT 18 0 "x" "6#%\"xG" } {TEXT -1 29 " is now experiencing at time " }{XPPEDIT 18 0 "t" "6#%\"t G" }{TEXT -1 294 ". It tells us that whatever is happening is happeni ng linearly, and it takes place from the endpoints in, and back, with \+ a wave-front like a shock wave. Several frames from this animation ap pear in Figure 24.19, but the full animation is generated in Maple by \+ executing the following command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "p1 := animate(u20, x=0...Pi , t=0..4*Pi, frames=80, labels=[x,u], color=black, labelfont=[TIMES,IT ALIC,12], xtickmarks=3, ytickmarks=[0,1], scaling=constrained):\np1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 152 "What does this animation mean? What is it telling us \+ about the longitudinal vibrations in the rod? For openers, the animat ion shows the displacements " }{XPPEDIT 18 0 "u(x,t)" "6#-%\"uG6$%\"xG %\"tG" }{TEXT -1 38 " which each face that was at location " }{TEXT 261 1 "x" }{TEXT -1 29 " is now experiencing at time " }{TEXT 262 1 "t " }{TEXT -1 112 ". Unfortunately, it is still hard from this animatio n to comprehend the actual motions taking place in the rod." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 163 "However, it do es tell us that whatever is happening is happening linearly, and it ta kes place from the endpoints in, and back, with a wave-front like a sh ock wave." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 211 "A more revealing animation simulates the actual rod by imagining \+ a set of eleven equispaced scratch marks on the rod, and following the motion of these scratch marks in real time. Figure 24.20 shows the b ar at " }{XPPEDIT 18 0 "t=3.7" "6#/%\"tG$\"#P!\"\"" }{TEXT -1 69 " but the full animation is generated by the following Maple commands." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 269 "uu := unapply(u20, x, t):\nF1 := (X,T) -> [[X+uu(X,T),0],[X+uu( X,T),1]]:\nF2 := T -> \{seq(F1(k*Pi/10,T), k = 0..10)\}:\nF3 := T -> p lot(F2(T), color=black):\np2 := display([seq(F3(k*Pi/20), k = 0..79)], insequence=true, scaling=constrained, ytickmarks=2, labels=[x,``]):\n p2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 726 "The uniformly stretched rod relaxes to its natura l length, with the collapse occurring from the \"outside, in.\" The c enter of the rod is the only part of the rod that does not move. Then , the collapse continues past the point of reaching natural length, an d the rod contracts by an amount equal to the original stretch. This \+ continuing contraction now takes place from \"inside, out\" as the cen ter of the rod first experiences the compression, with the outside of \+ the rod feeling the compression last. The rod then expands from this \+ state of full compression to natural length, from \"outside, in.\" Ha ving achieved natural length, the rod continues expanding to its initi al stretched state, this time from the \"inside, out.\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "In addition to this simulation of the rod itself, it is also instructive to examine an an imation of " }{XPPEDIT 18 0 "u[x](x,t)" "6#-&%\"uG6#%\"xG6$F'%\"tG" } {TEXT -1 57 ", which is the localized (per-face) value of the strain, \+ " }{XPPEDIT 18 0 "Delta*L/L" "6#*(%&DeltaG\"\"\"%\"LGF%F&!\"\"" } {TEXT -1 117 ". This animation shows more clearly how the compression and rarefaction waves travel with a \"front\" through the rod." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "To comput e the derivative " }{XPPEDIT 18 0 "u[x](x,t)" "6#-&%\"uG6#%\"xG6$F'%\" tG" }{TEXT -1 69 " we have to differentiate the infinite sum represent ing the solution " }{XPPEDIT 18 0 "u(x,t)" "6#-%\"uG6$%\"xG%\"tG" } {TEXT -1 166 ". The derivative of a Fourier series converges even mor e slowly than the original series. Where we obtained acceptable resul ts with a twenty-term approximation for " }{XPPEDIT 18 0 "u(x,t)" "6#- %\"uG6$%\"xG%\"tG" }{TEXT -1 128 " itself, it will take 100 terms for \+ the derived series to converge. This translates into a modest increas e in computation time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Figure 24.21 shows, at times " }{XPPEDIT 18 0 "t=1" "6#/%\"tG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t=4" "6#/%\"tG\" \"%" }{TEXT -1 111 ", the strain in the rod. However, executing the f ollowing Maple commands will generate the complete animation." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "u100 := a0/2 + sum(a*cos(n*x)*cos(n*t),n=1..100):\nux100 := dif f(u100,x):\np3 := animate(ux100, x=0..Pi, t=0..4*Pi, frames=80, color= black, labels=[x,strain], xtickmarks=3, ytickmarks=[-.3,0,.3]):\np3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 506 "The animation shows, as a function of time, the changi ng strain in the rod. The strain moves longitudinally through the rod , as a shock wave. It starts out uniform across the rod, and drops to zero with a shock-front. When the strain is completely gone, the rod has reached its natural length. As the rod then enters a compressive mode, the strain becomes negative, and the region of compression expa nds from the center out to the ends. At this juncture the strain grap h is a constant negative amount." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 226 "The compression then starts to vanish fr om the ends inwards until the rod again reaches its natural length. T his corresponds to the vanishing of the strain. As the rod enters an \+ expansion phase, the strain appears above the " }{TEXT 260 1 "x" } {TEXT -1 141 "-axis, growing, with a shock-front, outwards from the ce nter. The expansion continues until the rod returns to its original s tretched state." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 231 "Finally, we place side-by-side the three animations of d isplacement, physical motion, and strain. Run continuously, and slowl y enough, the resulting animation allows the three facets of the solut ion to be compared and coordinated." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "display(array([p||(1..3)]) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Theory and Derivation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "A rod which obeys a line ar law of elasticity is said to obey " }{TEXT 301 11 "Hooke's law" } {TEXT -1 134 ", wherein forces are proportional to displacement. This linear relationship between force and displacement, expressed by the \+ formula " }{XPPEDIT 18 0 "F=k*x" "6#/%\"FG*&%\"kG\"\"\"%\"xGF'" } {TEXT -1 197 ", is a staple of elementary calculus and general physics , especially in any discussion of the behavior of a spring. In fact, \+ the linear spring was an essential element in Chapters 5 and 6. Thus, " }{XPPEDIT 18 0 "F" "6#%\"FG" }{TEXT -1 61 ", the force required to \+ stretch a spring, is proportional to " }{XPPEDIT 18 0 "x" "6#%\"xG" } {TEXT -1 63 ", the amount of the stretch. The constant of proportiona lity, " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 32 ", is called the spr ing constant." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "In the context of elasticity, Hooke's law is sometimes st ated as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 270 6 "stress" }{TEXT -1 24 " is proportional to the " }{TEXT 271 6 "s train" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 " where stress is the force per unit cross-sectional area and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 297 "" 0 "" {TEXT 263 6 "strain" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "Delta*L/L" "6#*(%&DeltaG\"\"\"%\"LGF% F&!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "is defined as the elongation per unit length. In the context of e lasticity, Hooke's law is then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 265 "" 0 "" {TEXT 266 6 "stress" }{TEXT -1 3 " = " }{TEXT 303 1 "E" }{TEXT -1 1 " " }{TEXT 267 6 "strain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 302 1 "E" }{TEXT -1 61 " is Young's modulus, the constant of proportionality between " } {TEXT 268 6 "stress" }{TEXT -1 5 " and " }{TEXT 269 6 "strain" }{TEXT -1 43 ". This gives Young's modulus the units of " }}{PARA 266 "" 0 " " {TEXT -1 5 " E ~ " }{XPPEDIT 18 0 "stress/strain" "6#*&%'stressG\"\" \"%'strainG!\"\"" }{TEXT -1 3 " ~ " }{XPPEDIT 18 0 "``(force/area)/``( length/length)" "6#*&-%!G6#*&%&forceG\"\"\"%%areaG!\"\"F)-F%6#*&%'leng thGF)F/F+F+" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "force/area" "6#*&%&for ceG\"\"\"%%areaG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "In fact, starting with " }{XPPEDIT 18 0 "F=k*x" "6#/% \"FG*&%\"kG\"\"\"%\"xGF'" }{TEXT -1 26 ", and dividing through by " } {TEXT 304 1 "A" }{TEXT -1 35 ", the cross-sectional area, we have" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 274 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "F/A=k/A" "6#/*&%\"FG\"\"\"%\"AG!\"\"*&%\"kGF&F'F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "x" "6#%\"xG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The displacement " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 19 " is the elongation " }{XPPEDIT 18 0 " Delta" "6#%&DeltaG" }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 40 " from t he definition of strain, namely, " }{XPPEDIT 18 0 "strain=Delta*L/L" " 6#/%'strainG*(%&DeltaG\"\"\"%\"LGF'F(!\"\"" }{TEXT -1 32 ". Hence, th e spring law becomes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 275 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "stress=F/A" "6#/%'stressG*&%\"FG\" \"\"%\"AG!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "k/A" "6#*&%\"kG\"\" \"%\"AG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Delta" "6#%&DeltaG" } {XPPEDIT 18 0 "L" "6#%\"LG" }}{PARA 276 "" 0 "" {TEXT -1 28 " \+ = " }{XPPEDIT 18 0 "k/A" "6#*&%\"kG\"\"\"%\"AG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L*strain" "6#*&%\"LG\"\"\"%'strainGF% " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 277 "" 0 "" {TEXT -1 24 " \+ = " }{XPPEDIT 18 0 "E*strain" "6#*&%\"EG\"\"\"%'str ainGF%" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " where " }{XPPEDIT 18 0 "E" "6#%\"EG" }{TEXT -1 22 ", Young's modulus, \+ is " }{XPPEDIT 18 0 "k*L/A" "6#*(%\"kG\"\"\"%\"LGF%%\"AG!\"\"" }{TEXT -1 59 ". Hence, the two statements of Hooke's law are equivalent." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 365 "Since lo ngitudinal waves travel in the direction of the rod's axis, we want a \+ coordinate system along that axis. However, we must assume a linear e lasticity which keeps plane sections both plane and parallel. Thus, f orces within the rod obey the linear Hooke's law and preserve the orie ntation of all plane sections which are perpendicular to the axis of t he rod." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Put the rod (whose equilibrium length is " }{XPPEDIT 18 0 "L" "6#% \"LG" }{TEXT -1 205 ") on a ruler, aligning the left end of the rod wi th the left end of the ruler. At equilibrium, paint a copy of the rul er's coordinates on the rod. For clarity, take this linear coodinate \+ system to be an " }{TEXT 264 1 "x" }{TEXT -1 19 "-coordinate system." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 311 "Identi fy a face (i.e., a plane section) within the rod by the coordinate it \+ had at equilibrium. If the equilibrium coordinates are painted on the rod, then these coordinates travel with the rod should the rod be str etched or compressed. In addition, leave the ruler on the table so th e original meaning of an " }{TEXT 265 1 "x" }{TEXT -1 25 "-coordinate \+ is preserved." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Consider " }{TEXT 305 1 "R" }{TEXT -1 58 ", a segment of t he rod bounded at equilibrium by faces at " }{TEXT 293 1 "x" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x+dx" "6#,&%\"xG\"\"\"%#dxGF%" }{TEXT -1 27 ". This segment has length " }{XPPEDIT 18 0 "dx" "6#%#dxG" }{TEXT -1 52 " and is shown in the following figure, FIgure 24.22." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 621 " p4 := plot([10+2*cos(t),3+3*sin(t),t=0..2*Pi],color=black):\np5 := plo t([2*cos(t),3+3*sin(t),t=Pi/2..3*Pi/2],color=black):\np6 := plot([3+2* cos(t),3+3*sin(t),t=Pi/2..3*Pi/2],color=black):\np7 := plot([5+2*cos(t ),3+3*sin(t),t=Pi/2..3*Pi/2],color=black):\np8 := plot([5+2*cos(t),3+3 *sin(t),t=-Pi/2..Pi],color=black, linestyle=2):\np9 := plot(6,x=0..10, color=black):\np10 := plot(0,x=0..12,color=black):\np11 := textplot(\{ [2,-.5,`x`], [5,-.5,`x+dx`], [12.5,0,`x`], [10,-.5,`L`]\}, font=[TIMES ,ITALIC,12]):\np12 := textplot([2.4,3,`R`], font=[TIMES,ITALIC,12]):\n display([p||(4..12)],scaling=constrained,view=[-2..13,-1..6],axes=none );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u(x,t)" "6#-%\"uG6$%\"xG%\"tG " }{TEXT -1 46 " denote the displacement experienced, at time " } {TEXT 272 1 "t" }{TEXT -1 52 ", by the face, which at equilibrium, was located at " }{TEXT 273 1 "x" }{TEXT -1 66 ". It helps to remember t hat this face still bears the coordinate " }{TEXT 274 1 "x" }{TEXT -1 59 ". The displacement undergone by this face is the quantity " } {XPPEDIT 18 0 "u(x,t)" "6#-%\"uG6$%\"xG%\"tG" }{TEXT -1 199 ", and is \+ measured along the axis of the rod. Unlike the vibrating string where the displacements are very obvious, the lateral displacements of face s in the rod are much more difficult to visualize." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "The partial differential equation governing longitudinal vibrations in the rod will be a conse quence of " }{TEXT 306 1 "F" }{TEXT -1 3 " = " }{TEXT 307 1 "m" } {TEXT -1 1 " " }{TEXT 308 1 "a" }{TEXT -1 62 ", Newton's second law of motion. It's not hard to guess that " }{TEXT 275 1 "a" }{TEXT -1 51 ", the acceleration of the face with the coordinate " }{TEXT 276 1 "x " }{TEXT -1 28 " painted on it, is given by " }{XPPEDIT 18 0 "a=u[tt]( x,t)" "6#/%\"aG-&%\"uG6#%#ttG6$%\"xG%\"tG" }{TEXT -1 15 ". Determinin g " }{TEXT 309 1 "F" }{TEXT -1 52 ", the net force acting on this face , is more subtle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "At time " }{TEXT 277 1 "t" }{TEXT -1 42 ", the left and ri ght faces of the segment " }{TEXT 310 1 "R" }{TEXT -1 32 " are now loc ated respectively at" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "X[left] := x+u(x,t);\nX[right] := (x+dx)+ u(x+dx,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "The length of the deformed segment is the n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Delta := X[right]-X[left];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 278 6 "change" }{TEXT -1 121 " in length of this segment is ther efore the relative displacement between the left and right faces of th e segment, namely" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 11 "Delta - dx;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "the relative displacement between the left and right faces of the segment. The st rain associated with the face which, at equilibrium, was at location \+ " }{TEXT 280 1 "x" }{TEXT -1 32 ", is given by the limiting ratio" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Limit((Delta-dx)/dx,dx=0)=Limit((u(x+dx,t)-u(x,t))/dx,d x=0)" "6#/-%&LimitG6$*&,&%&DeltaG\"\"\"%#dxG!\"\"F*F+F,/F+\"\"!-F%6$*& ,&-%\"uG6$,&%\"xGF*F+F*%\"tGF*-F46$F7F8F,F*F+F,/F+F." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 215 "Notice that strain, an extensive property, is ascribed to a point by this limit. This defin ition parallels the definition of pressure which is also an extensive \+ property, force per unit area, ascribed to a point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Thus, the strain on a \+ single face with coordinate " }{TEXT 279 1 "x" }{TEXT -1 17 " painted \+ on it is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "q3 := Limit((u(x+dx,t)-u(x,t))/dx,dx=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "so that the value of the strain is " }{XPPEDIT 18 0 "strain=u[x ](x,t)" "6#/%'strainG-&%\"uG6#%\"xG6$F)%\"tG" }{TEXT -1 10 ", that is, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "strain = convert(value(q3),diff);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "Th is is a fundamental result in the theory of elasticity. Getting this \+ right at the beginning is well worth the time invested in getting it. \+ Thus, at time " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 14 " and locat ion " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 40 ", the strain caused b y the displacement " }{XPPEDIT 18 0 "u(x,t)" "6#-%\"uG6$%\"xG%\"tG" } {TEXT -1 102 ", is positive if the applied force tends to increase the size of the region R, and negative otherwise." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The force on a plane sect ion can be obtained from the strain, using Hooke's law, " }{TEXT 281 6 "stress" }{TEXT -1 3 " = " }{TEXT 286 1 "E" }{TEXT -1 1 " " }{TEXT 282 6 "strain" }{TEXT -1 8 ". With " }{TEXT 287 1 "A" }{TEXT -1 48 " \+ as the cross-sectional area of the rod, we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "force /area" "6#*&%&forceG\"\"\"%%areaG!\"\"" }{TEXT -1 3 " = " }{TEXT 288 7 "stress " }{TEXT -1 2 "= " }{TEXT 311 1 "E" }{TEXT -1 1 " " }{TEXT 283 6 "strain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 270 "" 0 "" {TEXT -1 1 " " }{TEXT 284 5 "force" }{TEXT -1 3 " = " }{TEXT 312 1 "E" }{TEXT -1 1 " " }{TEXT 313 1 "A" }{TEXT -1 1 " " }{TEXT 285 6 "strain" }{TEXT -1 3 " = " }{TEXT 314 1 "E" }{TEXT -1 1 " " }{TEXT 315 1 "A" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u[x](x,t)" "6#-&%\"uG6#%\"xG6$F'%\"tG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Thus, the elastic force on the \+ face whose equilibrium coordinate was " }{TEXT 289 1 "x" }{TEXT -1 13 ", is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 271 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x,t)=E*A*u[x](x,t)" "6#/-%\"FG6$%\"xG %\"tG*(%\"EG\"\"\"%\"AGF+-&%\"uG6#F'6$F'F(F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "To represent this force in Mapl e, define" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "F := (x,t) -> E*A*D[1](u)(x,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "so t hat" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "'F(x,t)' = convert(F(x,t),diff);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The \+ net force on segment " }{TEXT 316 1 "R" }{TEXT -1 73 " is given by the sum of the forces acting on the left and right faces of " }{TEXT 317 1 "R" }{TEXT -1 9 ", namely," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 298 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "E*A*u[x](x+dx,t)-E*A*u[x](x ,t)" "6#,&*(%\"EG\"\"\"%\"AGF&-&%\"uG6#%\"xG6$,&F,F&%#dxGF&%\"tGF&F&*( F%F&F'F&-&F*6#F,6$F,F0F&!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "which, is Maple, is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "`net force` := F(x+dx,t)-F(x ,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "The total mass of segment " }{TEXT 318 1 "R" } {TEXT -1 12 " is given by" }}{PARA 272 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "m = [rho*A*dx]" "6#/%\"mG7#*(%$rhoG\"\"\"%\"AGF(%#dxGF( " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "since the volume of the segment is " }{TEXT 319 1 "A" }{TEXT -1 46 ", the c ross-sectional area, times the length, " }{TEXT 290 2 "dx" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "T aking the acceleration of a face as " }{XPPEDIT 18 0 "u[tt](x,t)" "6#- &%\"uG6#%#ttG6$%\"xG%\"tG" }{TEXT -1 19 ", we can write the " }{TEXT 320 1 "i" }{TEXT -1 35 "-component of Newton's second law, " }{TEXT 321 1 "F" }{TEXT -1 3 " = " }{TEXT 322 1 "m" }{TEXT -1 1 " " }{TEXT 323 1 "a" }{TEXT -1 4 ", as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 299 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "E*A*u[x](x+dx,t)-E*A*u[x](x ,t)=rho*A*dx*u[tt](x,t)" "6#/,&*(%\"EG\"\"\"%\"AGF'-&%\"uG6#%\"xG6$,&F -F'%#dxGF'%\"tGF'F'*(F&F'F(F'-&F+6#F-6$F-F1F'!\"\"**%$rhoGF'F(F'F0F'-& F+6#%#ttG6$F-F1F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "that is, as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "q4 := `net force` = rho*A*dx*diff(u(x,t),t,t);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Divide by " }{XPPEDIT 18 0 "A*rho*dx" "6#*(%\"AG\"\"\"%$r hoGF%%#dxGF%" }{TEXT -1 7 " to get" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 300 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "E/rho" "6#*&%\"EG\"\" \"%$rhoG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "((u[x](x+dx)-u[x](x,t)) /dx=u[tt](x,t)" "6#/*&,&-&%\"uG6#%\"xG6#,&F*\"\"\"%#dxGF-F--&F(6#F*6$F *%\"tG!\"\"F-F.F4-&F(6#%#ttG6$F*F3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "q5 := simplify(q4/(A*rho*dx));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "which, in the limit as " }{XPPEDIT 18 0 "dx*``-``*`>`*0" "6#,&*&%#dxG\"\"\"%!GF&F&*(F'F&%\">GF&\"\"!F&!\"\"" }{TEXT -1 9 ", bec omes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 301 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "E/rho" "6#*&%\"EG\"\"\"%$rhoG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "u[xx](x,t)=u[tt](x,t)" "6#/-&%\"uG6#%#xxG6$%\"xG%\"t G-&F&6#%#ttG6$F*F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "q6 := convert(map(limit,q5,dx=0),diff);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Upon defining " }{XPPEDIT 18 0 "c^2=E/rho" "6#/*$%\"cG\"\"#*&% \"EG\"\"\"%$rhoG!\"\"" }{TEXT -1 45 " , the partial differential equat ion becomes " }{XPPEDIT 18 0 "u[tt]=c^2*u[xx]" "6#/&%\"uG6#%#ttG*&%\"c G\"\"#&F%6#%#xxG\"\"\"" }{TEXT -1 9 ", that is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "algsubs(E/r ho=c^2,q6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "With a slight rearrangement and the intr oduction of subscript notation for derivatives, this equation can be w ritten as" }}{PARA 273 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[tt] = \+ c^2*u[xx]" "6#/&%\"uG6#%#ttG*&%\"cG\"\"#&F%6#%#xxG\"\"\"" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "the very same wave equation we derived for the vibrating string. The difference here is that " }{XPPEDIT 18 0 "u(x,t)" "6#-%\"uG6$%\"xG%\"tG" }{TEXT -1 53 " \+ represents the displacement of the face that was at " }{TEXT 291 1 "x " }{TEXT -1 9 " at time " }{TEXT 292 1 "t" }{TEXT -1 62 ", and this di splacement is lateral, along the axis of the rod." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }