EXERCISES WAVE EQUATION. In Problems 1 and 2 solve the heat equation (1) subject to the given conditions. Assume a rod of length L.

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1 .4 WAVE EQUATION 445 EXERCISES.3 In Problems and solve the heat eqation () sbject to the given conditions. Assme a rod of length.. (, t), (, t) (, ),, > >. (, t), (, t) (, ) ( ) 3. Find the temperatre (, t) in a rod of length if the initial temperatre is f () throghot and if the ends and are inslated. 4. Solve Problem 3 if and f (),,. 5. Sppose heat is lost from the lateral srface of a thin rod of length into a srronding medim at temperatre zero. If the linear law of heat transfer applies, then the heat eqation takes on the form k h,, t, h a constant. Find the temperatre (, t) if the initial temperatre is f () throghot and the ends and are inslated. See Figre.3.3. Inslated Inslated Heat transfer from lateral srface of the rod Answers to selected odd-nmbered problems begin on page ANS-. 6. Solve Problem 5 if the ends and are held at temperatre zero. Discssion Problems 7. Figre.3.(b) shows the graphs of (, t) for t 6 for, p, p6, p4, and p. Describe or sketch the graphs of (, t) on the same time interval bt for the fied vales 3p4, 5p6, p, and p. 8. Find the soltion of the bondary-vale problem given in () (3) when f() sin(5p). Compter ab Assignments 9. (a) Solve the heat eqation () sbject to (, t), (, t), t (, ).8, 5.8( ), 5. (b) Use the 3D-plot application of yor CAS to graph the partial sm S 5 (, t) consisting of the first five nonzero terms of the soltion in part (a) for, t. Assme that k.635. Eperiment with varios three-dimensional viewing perspectives of the srface (called the ViewPoint option in Mathematica). FIGURE.3.3 Rod losing heat in Problem 5.4 WAVE EQUATION REVIEW MATERIA Reread pages of Section.. INTRODUCTION We are now in a position to solve the bondary-vale problem () that was discssed in Section.. The vertical displacement (, t) of the vibrating string of length shown in Figre..(a) is determined from a,, t (, t), (, t), t (, ) f (), g(),. t () () (3)

2 446 CHAPTER BOUNDARY-VAUE PROBEMS IN RECTANGUAR COORDINATES SOUTION OF THE BVP With the sal assmption that (, t) X()T(t), separating variables in () gives so that X X (4) T a T. (5) As in the preceding section, the bondary conditions () translate into X() and X(). Eqation (4) along with these bondary conditions is the reglar Strm-ioville problem X X, X(), X(). (6) Of the sal three possibilities for the parameter, l, l a, and l a, only the last choice leads to nontrivial soltions. Corresponding to l a, a, the general soltion of (4) is X c cos a c sin a. X() and X() indicate that c and c sin a. The last eqation again implies that a np or a np. The eigenvales and corresponding eigenfnctions of (6) are l n n p and X() c sin n, n,, 3,.... The general soltion of the second-order eqation (5) is then T(t) c 3 cos t c 4 sin t. By rewriting c c 3 as A n and c c 4 as B n, soltions that satisfy both the wave eqation () and bondary conditions () are and (, t) A n cos. (8) n t B n sin t n sin Setting t in (8) and sing the initial condition (, ) f () gives (, ) f () A n sin n. n Since the last series is a half-range epansion for f in a sine series, we can write A n b n : A n f () sin. (9) n d To determine B n, we differentiate (8) with respect to t and then set t : n A n cos t B n sin t n sin n A n X X T a T sin t B n cos t n sin g() t B n n n sin. For this last series to be the half-range sine epansion of the initial velocity g on the interval, the total coefficient B n npa mst be given by the form b n in (5) of Section.3, that is, B n g() sin n d (7)

3 .4 WAVE EQUATION 447 from which we obtain B n g() sin n. () d The soltion of the bondary-vale problem () (3) consists of the series (8) with coefficients A n and B n defined by (9) and (), respectively. We note that when the string is released from rest, then g() for every in the interval [, ], and conseqently, B n. PUCKED STRING A special case of the bondary-vale problem in () (3) is the model of the plcked string. We can see the motion of the string by plotting the soltion or displacement (, t) for increasing vales of time t and sing the animation featre of a CAS. Some frames of a movie generated in this manner are given in Figre.4.; the initial shape of the string is given in Figre.4.(a). Yo are asked to emlate the reslts given in the figre plotting a seqence of partial sms of (8). See Problems 7 and in Eercises (a) t = initial shape (b) t =. (c) t = (d) t =. (e) t =.6 (f) t =.9 FIGURE.4. Frames of a CAS movie STANDING WAVES Recall from the derivation of the one-dimensional wave eqation in Section. that the constant a appearing in the soltion of the bondary-vale problem in (), (), and (3) is given by T>, where r is mass per nit length and T is the magnitde of the tension in the string. When T is large enogh, the vibrating string prodces a msical sond. This sond is the reslt of standing waves. The soltion (8) is a sperposition of prodct soltions called standing waves or normal modes: (, t) (, t) (, t) 3 (, t). In view of (6) and (7) of Section 5. the prodct soltions (7) can be written as n (, t) C n sin n t n sin, where C n A n B n and f n is defined by sin f n A n C n and cos f n B n C n. For n,, 3,... the standing waves are essentially the graphs of sin(np), with a time-varying amplitde given by C n sin t n. Alternatively, we see from () that at a fied vale of each prodct fnction n (, t) represents simple harmonic motion with amplitde C n sin(np) and freqency f n na. In other words, each point on a standing wave vibrates with a different amplitde bt with the same freqency. When n, (, t) C sin a t sin ()

4 448 CHAPTER BOUNDARY-VAUE PROBEMS IN RECTANGUAR COORDINATES (a) First standing wave Node (b) Second standing wave is called the first standing wave, the first normal mode, or the fndamental mode of vibration. The first three standing waves, or normal modes, are shown in Figre.4.. The dashed graphs represent the standing waves at varios vales of time. The points in the interval (, ), for which sin(np), correspond to points on a standing wave where there is no motion. These points are called nodes. For eample, in Figres.4.(b) and.4.(c) we see that the second standing wave has one node at and the third standing wave has two nodes at 3 and 3. In general, the nth normal mode of vibration has n nodes. The freqency f a T B Nodes 3 3 (c) Third standing wave FIGURE.4. waves First three standing of the first normal mode is called the fndamental freqency or first harmonic and is directly related to the pitch prodced by a stringed instrment. It is apparent that the greater the tension on the string, the higher the pitch of the sond. The freqencies f n of the other normal modes, which are integer mltiples of the fndamental freqency, are called overtones. The second harmonic is the first overtone, and so on. EXERCISES.4 Answers to selected odd-nmbered problems begin on page ANS-. In Problems 8 solve the wave eqation () sbject to the given conditions... (, t), (, t) (, ) ( ), 4 t (, t), (, t) (, ), ( ) 3. (, t), (, t) (, ), given in Figre.4.3, f() 7. (, t), (, t) 8. (, ) h, h,, (, ),, This problem cold describe the longitdinal displacement (, t) of a vibrating elastic bar. The bondary conditions at and are called free-end conditions. See Figre.4.4. /3 /3 (, t) FIGURE.4.3 Initial displacement in Problem 3 (, t), (, t) (, ) 6 ( ), (, t), (, t) (, ), sin 6. (, t), (, t) (, ). sin 3p, FIGURE.4.4 Vibrating elastic bar in Problem 8 9. A string is stretched and secred on the -ais at and p for t. If the transverse vibrations take place in a medim that imparts a resistance proportional to the instantaneos velocity, then the wave eqation takes on the form,, t.

5 .4 WAVE EQUATION 449 Find the displacement (, t) if the string starts from rest from the initial displacement f ().. Show that a soltion of the bondary-vale problem is (, t) 4,, t (, t), (, t), t (, ), > >,, k () k sin(k ) cos (k ) t. (k ). The transverse displacement (, t) of a vibrating beam of length is determined from a forth-order partial differential eqation a 4,, t. 4 If the beam is simply spported, as shown in Figre.4.5, the bondary and initial conditions are (, t), (, t), t,, t (, ) f (), g(),. Solve for (, t). [Hint: For convenience se l a 4 when separating variables.] FIGURE.4.5 Simply spported beam in Problem. If the ends of the beam in Problem are embedded at and, the bondary conditions become, for t, (, t), (, t),. (a) Show that the eigenvales of the problem are n n >, where n, n,, 3,..., are the positive roots of the eqation cosh cos. (b) Show graphically that the eqation in part (a) has an infinite nmber of roots. (c) Use a calclator or a CAS to find approimations to the first for eigenvales. Use for decimal places. 3. Consider the bondary-vale problem given in (), (), and (3) of this section. If g() for, show that the soltion of the problem can be written as (, t) [ f ( at) f ( at)]. [Hint: Use the identity sin cos sin( ) sin( ).] 4. The vertical displacement (, t) of an infinitely long string is determined from the initial-vale problem () This problem can be solved withot separating variables. (a) Show that the wave eqation can be pt into the form hj by means of the sbstittions j at and h at. (b) Integrate the partial differential eqation in part (a), first with respect to h and then with respect to j, to show that (, t) F( at) G( at), where F and G are arbitrary twice differentiable fnctions, is a soltion of the wave eqation. Use this soltion and the given initial conditions to show that and a,, t (, ) f (), g(). t F() f () G() f () a a g(s)ds c g(s)ds c, where is arbitrary and c is a constant of integration. (c) Use the reslts in part (b) to show that (, t). (3) aat [ f ( at) f ( at)] g(s) ds at Note that when the initial velocity g(), we obtain (, t) [ f ( at) f ( at)], This last soltion can be interpreted as a sperposition of two traveling waves, one moving to the right (that is, f ( at) ) and one moving to the.

6 45 CHAPTER BOUNDARY-VAUE PROBEMS IN RECTANGUAR COORDINATES left ( f ( at)). Both waves travel with speed a and have the same basic shape as the initial displacement f (). The form of (, t) given in (3) is called d Alembert s soltion. In Problems 5 8 se d Alembert s soltion (3) to solve the initial-vale problem in Problem 4 sbject to the given initial conditions. 5. f () sin, g() 6. f () sin, g() cos 7. f (), g() sin 8. f () e, g() Compter ab Assignments 9. (a) Use a CAS to plot d Alembert s soltion in Problem 8 on the interval [5, 5] at the times t, t, t, t 3, and t 4. Sperimpose the graphs on one coordinate system. Assme that a. (b) Use the 3D-plot application of yor CAS to plot d Alembert s soltion (, t) in Problem 8 for 5 5, t 4. Eperiment with varios three-dimensional viewing perspectives of this srface. Choose the perspective of the srface for which yo feel the graphs in part (a) are most apparent.. A model for an infinitely long string that is initially held at the three points (, ), (, ), and (, ) and then simltaneosly released at all three points at time t is given by () with f (),, and g(). (a) Plot the initial position of the string on the interval [6, 6]. (b) Use a CAS to plot d Alembert s soltion (3) on [6, 6] for t.k, k,,,..., 5. Assme that a. (c) Use the animation featre of yor compter algebra system to make a movie of the soltion. Describe the motion of the string over time.. An infinitely long string coinciding with the -ais is strck at the origin with a hammer whose head is. inch in diameter. A model for the motion of the string is given by () with f () and g(),.,.. (a) Use a CAS to plot d Alembert s soltion (3) on [6, 6] for t.k, k,,,..., 5. Assme that a. (b) Use the animation featre of yor compter algebra system to make a movie of the soltion. Describe the motion of the string over time.. The model of the vibrating string in Problem 7 is called the plcked string. The string is tied to the -ais at and and is held at at h nits above the -ais. See Figre..4. Starting at t the string is released from rest. (a) Use a CAS to plot the partial sm S 6 (, t) that is, the first si nonzero terms of yor soltion for t.k, k,,,...,. Assme that a, h, and p. (b) Use the animation featre of yor compter algebra system to make a movie of the soltion to Problem 7..5 APACE S EQUATION REVIEW MATERIA Reread page 438 of Section. and Eample in Section.4. INTRODUCTION Sppose we wish to find the steady-state temperatre (, y) in a rectanglar plate whose vertical edges and a are inslated, as shown in Figre.5.. When no heat escapes from the lateral faces of the plate, we solve the following bondary-vale problem:, a, y b y,, y b a (, ), (, b) f (), a. (3) () ()