Pulses on a Struck String

Transcription

1 8.03 at ESG Spplemental Notes Plses on a Strck String These notes investigate specific eamples of transverse motion on a stretched string in cases where the string is at some time ndisplaced, bt with a non-zero transverse velocity. Consider the Cachy Problem tt = c 2, (, 0)=f() t (, 0)=g(), (1) where the sbscripts denote partial differentiation. What follows might well annoy mathematicians. For a more rigoros treatment, inclding of corse the needed differentiability and smoothness of the fnctions, see Differential Eqations: A Modeling Approach, Robert L. Borrelli and Cortney S. Coleman (Prentice-Hall, 1987), Section 13.2 (hereinafter referred to as B&C ; a newer edition has been pblished). The soltion to Problem (1) is shown in B&C to be (, t) = 1 2 [f( ct)+f( ct)] + 1 2c +ct ct g(s) ds. (2) The remainder of these notes will consider the case f() 0, so that Eqation (2) simplifies to (, t) = 1 2c +ct ct g(s) ds. (3) Note that in the limit of small t >0, this redces to g() t. Eqation (3) is often written in the form (, t) =P ( + ct)+q( ct) (4) where P (s) = 1 2c s 0 g (s ) ds, Q(s) = 1 2c s 0 g (s ) ds. It is common to interpret Eqation (4) in terms of plses traveling in the negativeand positive- directions respectively. Part of the motivation for these notes is to show that this mathematical interpretation may lead to a physical interpretation that is somewhat conterintitive. For now, consider that in both P (s) andq(s), the lower limit of zero is arbitrary, and changing this lower limit corresponds to a constant of integration added to one plse and sbtracted from the other. 1

2 Before looking at eamples, note that the problem in (1) above is not a bondary-vale problem. Restricting, and hence the range of both f() and g() to a finite interval, wold reqire sitable periodic etensions of f() and g(); these periodic etensions will not be discssed in detail here, althogh they are clearly implied in any se of Forier Series. Althogh it may not be needed for mathematical analysis, for physical interpretation it might be helpfl to recall kinematic and dynamic properties of (, t) and its derivatives. For small displacements on a niform string of mass density µ and sbject to a tension T,wehave: Speed of Propagation (denoted c, asabove): c 2 = T µ. Potential Energy Density (denoted U): U = 1 2 T 2. Kinetic Energy Density (denoted K): K = 1 2 µ2 t. Total Energy Density U + K = 1 2 Longitdinal Power Density (denoted p): [ T 2 + µ 2 ] µ ( t = c ) 2 t. p = T t. Conservation of Energy: p (U + K)+ t =0. Transverse Momentm Density (denoted p T ): p T = û t. Five eamples will be considered in what follows. Some of the above qantities may be infinite. We won t worry. Mch. 2

3 The five eamples will be: A standing wave on a string. A nit (δ-fnction) implse on an nbonded string (no reflection). A sqare-wave implse on an nbonded string. A smoother implse on an nbonded string. The same implse on a bonded string. Standing Wave on a String As will be seen, the standing wave, being spatially periodic, may be a soltion for an nbonded string or a bonded string with linear bondary conditions. The eample is taken from the tet, Calcls: An Introdction to Applied Mathematics, H. P. Greenspan and D. J. Benney (Brekelen, 1997), Page 505. In the crrent notation, we have f() 0, g() =cos. A basic se of Eqation (3) above gives (, t) = 1 [sin( + ct) sin( ct)] 2c = 1 cos sin ct. c This standard reslt shows that a standing wave may be represented as the sm of twotravelingwaves,andviceversa. A δ-fnction Implse At the risk of offending most mathematicians and some physicists, let f() 0, g() =Aδ(). All we ll really need is that the δ-fnction is the derivative of the heaviside fnction H(), where { 1 >0 H() = 0 <0. If we made an attempt to be rigoros, we wold want H(0) = 1/2, bt that s not or goal, so the matter won t be mentioned in these notes. The reslt of sing Eqation (3) is then (, t) = A [H( + ct) H( ct)], 2 c 3

4 clearly a plse of height A/2c epanding in both the positive- and negative- directions with speed c. Unfortnately, the energy density, while localized, is infinite (as is its integral; δ-fnctions are like that). Momentm is finite and conserved. Consider the epression in Eqation (4); as defined, both P (s) andq(s) are zero for s<0and P (s) > 0for s>0, Q(s) < 0for s>0(for A>0). This gives, as epected, (, t) =0for >ct, bt for <ct, this zero net wave is the sm of two nonzero waves in the region beyond where the signal cold have propagated. This is (sort of) mathematically correct, bt the physical interpretation is perhaps conterintitive. The plses are represented in the figre below, the plse corresponding to P ( + ct) 0in ble, moving to the left in the figre, and Q( ct) 0in green moving to the right, with the sm in cyan (the ble and green plses are displaced for clarity). A certain symmetry might be restored by adding 1/2toQ(s) and sbtracting 1/2 fromp (s), bt this still has the traveling waves P ( + ct)andq( ct) nonzero in regions beyond the implse at t = 0. Any bondary conditions wold not affect this reslt. It shold be noted that the factor A introdced mst have dimensions of length 2 /time; A may be thoght of as the -component of the imparted momentm divided by µ. For frther physics interpretations, whenever any displacement of awavetravelingonastretchedstringisshownwithasharpedge,thetransverse velocity profile will inclde a δ-fnction part; if sch fnctions are not desirable, then sch waves shold not be sed. Sqare-Wave Implse Represent the implse in terms of Heaviside fnctions, so that f() 0, g() =B [H ( + 0 ) H ( 0 )]. In words, g =0for 0 > 0, g = B for 0 < 0. 4

5 The soltion to Problem (1) may then be fond by varios means. In the animation which accompanies these notes, the same program that generated the animation calclated the integral in Eqation (3) withot complaint. Calclation by hand is often left as an eercise (as in B&C, Page 520, Problem 1). Presented below is a graphical interpretation. As can be seen, as the time inreases, the plse will spread, bt the energy and momentm will remain constant. In this eample, it s clearer that the plse is smoother than the implse. Mathematically, this is represented by the integral in D Alembert s soltion (Eqation (2)). Physically, this if often interpreted by recognizing that at some point with 0 > 0, the implse imparted to different parts of the string take different times to be seen. (A more detailed discssion, with acknowledgement of physicists sensibilities, is in B&C, Page 519.) The temptation to contine with other polynomials for g() restricted to a finite interval will be only partially resisted. Analytic calclations will, for now, remain as eercises. However, the case of an implse that is a parabolic fnction of will be considered via compter-generated animations. Parabolic Implse The initial condition sed to generate the implse was f() 0, g() =[H ( +1) H ( 1)] ( 1 2) (for the prposes of generating the animiation, all parameters and dimensions are set to nity). The animation shows both the smoothing and spreading of the plse. Parabolic Implse - Bonded String The previos for eamples did not consider bondary conditions, basically assming an infinite string, or a finite string bt for times sfficiently short that any reflections from bondaries are not considered. To see the effect of reflection, 5

6 this eample assmes that plses are inverted pon reflection, so that a Forier sine series may be sed. The Cachy Problem is now a bondary-vale problem. For animation prposes, the initial data are f() 0 g() =[H ( 1/4) H ( 3/4)] (s 1/4)(3/4 s) (0, t)=(1, t) 0. The Forier coefficients C n of g() were fond by compter, and as long as a compter was being sed, the first fifty terms of the series were smmed, with the reslt shown here. Note that at this resoltion, the Forier sm is barely distingishable from the parabola The animation depicts 50 n=1 C n nπ sin(nπ)sin(nπt) and is looped at the fndamental period of T = 2. The animation shows both the spreading of the plse and the larger slopes ( nsmoothing ) at the bondaries de to inversion pon reflection. 6