Elastic waves in a medium: An interactive graphics package

Comput. Educ. Vol. 9, No. I, pp. 1 7, 1985 Printed in Great Britain

0360-1315/85 $ 3 . 0 0 + 0 . 0 0 Pergamon Press Ltd

ELASTIC WAVES IN A MEDIUM: AN INTERACTIVE GRAPHICS PACKAGE L. BORGHIl, A. DE AMBROSIS~, G. GAZZANIGA2, L. IRONI2, P. MASCHERETTII and C. I. M A S S A R A I IDipartimento di Fisica "A Volta", Universit~ di Pavia and Gruppo Nazionale Didattica della Fisica--CNR, Via A Bassi, 6-127100 Pavia and 2Istituto di Analisi Numerica--CNR, Pavia, Italy (Received 21 February 1984; amended 19 March 1984)

Abstract--An interactive graphics package dealing with propagation of elastic waves is presented. It can be used to teach the fundamentals of mechanics at undergraduate level. The goal of the package is to provide a sound understanding of both the phenomenology of wave propagation in an elastic medium and its mathematical description.

INTRODUCTION This work deals with an application of computers in the teaching of mechanics. As regards the ways a computer can be used by a physics teacher a wide literature is available [1-7]. The package described here belongs to an area where computers can be used very effectively, i.e. dynamic computer graphics, and consists of two interactive graphics programs dealing with the propagation of transverse and longitudinal waves in an elastic medium. The first program is devoted to transverse waves in a string, the second one to the propagation of waves in a solid rod. They are meant to support a standard course of physics designed for first-year University students. Our package bridges the theoretical framework with the laboratory activity. In our work, we employed a didactic strategy that can be called "guided simulation". By such an expression we mean a tutorial dialogue based on the simulation of a phenomenon. The student shouio answer questions and take appropriate measurements in the simulated phenomenon in order to "verify" the correctness of the mathematical description. N o programming skills are required to run the programs. The didactic strategy used seems particularly effecflye in focusing the attention on the relevant aspects of elastic wave propagation. This package has been jointly developed by researchers of the Department of Physics of the University and the C A L group of the Istituto di Analisi N u m e r i c a - - C N R , Pavia. The authors started their work in 1981 with the aim of identifying the topics of elementary physics which can be most effectively learned with C A L packages. Until now, we have produced packages on mechanics [8, 9]: in this field mathematical treatments are often cumbersome, experiments tend to be too complex or delicate and mechanical quantities such as velocity and acceleration are often hard to measure with the needed accuracy [10]. A computer m a y help to overcome these difficulties. It allows the simulation of experiments too expensive or too difficult to be performed in the laboratory, to analyse the "ideal" situations which most appropriately illustrate the theory and to show details of motion which might escape observation. The programs have been written in F O R T R A N . The hardware used consists of the Honeywell DPS8 multi-purpose computer of the Computing Center of the University of Pavia and Tektronix 4012 graphics terminals combined with a Tektronix hard-copier.

A MATHEMATICAL

INTRODUCTION

TO E L A S T I C WAVES

The equation describing a wave propagating in a dispersionless medium is: ~2f v 2 0 2 f t~t2 -- ~X 2

(1)

where x is the coordinate of a point of the medium, t is t i m e , f is the displacement from equilibrium CAE 9!1 A

1

2

L. BORGHI et al.

and v the velocity (see, for instance, [11]). Equation (1) describes a great variety of physical problems such as propagation of a small disturbance in a uniform string and in an elastic solid in the absence of external forces. The string is defined to be a one-dimensional continuum in which the only interaction between different parts is a tension T. In equilibrium the string is parallel to the X-axis and small displacements perpendicular to X are considered. In the case of a solid, equation (1) describes the one-dimensional motion of the cross-sections of an elastic rod with axis parallel to X. The rod could be stretched or contracted in such a way that planes with X = constant move together in the X-direction (longitudinal vibration) or it could be disturbed in such a way that each section moves in a direction perpendicular to the X-axis (transverse vibration). The velocity of propagation of the disturbance, v, depends both on the nature of the medium and the type of vibration. For the string, t, = ~ where T is the tension and p the mass per unit len~gt_h of the string. For the rod, a longitudinal disturbance propagates with a velocity v = ~/E/,,,E/I~where # is the rod density and E Young's modulus of elasticity. A transverse disturbance travels with a velocity v = x / ~ where G is the modulus of rigidity (or shear modulus) of the material. The general solution of equation (1) has the form f (t, x ) = u~ (t - x/~,) + u2(t + x / v )

(2)

where functions uj and u2 depend on the boundary conditions and the way the disturbance has been produced. For a wave propagating in the +X-direction only function u~(t - x / v ) appears in (2) while a wave propagating in the - X - d i r e c t i o n is represented b y f ( t , x ) = u2 (t + x / v ) . Our package treats only the case where a disturbance propagates in the + X-direction. THE PACKAGE According to our experience as physics teachers, the most common difficulties encountered by first-year University students in studying the propagation of elastic waves occur in: (i) understanding the differences between longitudinal and transverse waves; (ii) visualizing the motion of a particle in the vibrating medium; (iii) making connections between properties of the medium and the velocity of propagation and (iv) mastering the mathematical description of the wave. The level of mathematical background of students is generally insufficient to tackle the problem in a formal way. On the other hand, it is highly appropriate that prospective physicists and engineers develop an intuitive but sound understanding of waves in the early stages of their education. The use of a computer can help them in analysing the mot'_'on of a point in the vibrating medium, in making some sense of the mathematical description and in understanding that a variety of elastic wave phenomena (with different internal dynamics) can be described mathematically by the same type of equation. In the first of the two programs of the package, we help the students to realize that the general expression for a wave traveling with velocity + v in a very long string can be written in the form f (t, x ) = u (t - x /~).

(3)

The second program stresses the differences between transverse and longitudinal waves and shows that equation (3) describes both types of wave. A detailed description of the programs is given in the following. "STRING":

transverse waves in a string

The program presents images at different times of a string in which a disturbance is propagating along +X. The velocity of propagation is chosen at random by the program in a set of values defined according to the above mentioned physical assumptions. From each set of images the student must derive the velocity of propagation. If a wrong answer is given, dotted curves will represent a wave corresponding to the answer and the student is given another try (Fig. 1). Then the student follows the displacements from equilibrium of few points of the string as a function of time. A copy of the images of the traveling waves and of a table with the displacements is automatically provided by the program (Fig. 2). The student should recognize that the motion of each point of the string is the same but has a time delay depending on the distance from a point

Elastic waves in a medium

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4

L. BORGHI et al.

chosen as reference. To this end, the program produces the plots of displacement vs time given in Fig. 2. The student can at leisure compare the new display with the copy of Fig. 2 and think about the equivalence between the representations of a wave as a function of distance and as a function of time. Then, the student considers the time corresponding to the maximum displacement from the equilibrium position of different points of the string; he or she finds that the relation between the delay At and the distance Ax = x - x0 from the reference point x0 is At = Ax/v. The student is guided to conclude that the propagation of a disturbance in the string can be represented by equation (3). " R O D " : traveling waves in a solid rod

The program deals with: (1) the propagation of a transverse perturbation in a rod; (2) the propagation of a longitudinal perturbation in a rod. Figure 3 shows how we choose to represent the rod and the transverse deformation traveling in it. In order to clarify the drawing the displacements of each cross-section are enlarged. The student is told that this type of deformation (transverse) is produced by hitting the rod at one end in a direction perpendicular to the axis of the rod; so the student can connect the type of vibration with the way the rod is perturbed. The student is required to identify both the direction of propagation of the wave and the direction of motion of the sections of the rod. Subsequently, as in the case of a string, graphs of the displacements are shown (Fig. 4) and again the student determines the velocity of propagation of the perturbation. Finally a display presents the graphs of Fig. 4 together with those seen in the program " S T R I N G " (Fig. 1). Despite great differences in the values of displacement and of velocity of propagation, this comparison shows that the displacement vs time plots are the same. Next the simulation of a disturbance, produced by hitting one end of the rod along the axial direction, is presented (Fig. 5). The student is guided to recognize that in this case two adjacent

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Fig. 3. Images at different times of a rod in which a transverse perturbation travels.

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Fig. 4. Graphs of displacements vs x of the cross-sections at different times, i

[1III11111 l[[lll[llllI[lllllllllll ]lltlll, o~s lllllll, l~S II11[1111111[111111111111111111111 11IIil1111111]IIIIII IIIlllll[lllll lllllll,-2~s lllllll, 3~s 11111111111111111111111111IIIIIIII [11111111111111IIIIIII111111I111 In w h i c h direction does the d i s t u r b a n c e propagate 2 (a) p e r p e n d i c u l a r to the a x i s o f the rod (b) p a r a l l e l to the axis of the rod (c) oblique w i t h r e s p e c t to the axis of t h e rod =b Tn w h i c h d i r e c t i o n (Answer : a, b, c) =b Good

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Fig. 5. Images at different times of a rod in which a longitudinal perturbation travels.

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Fig. 6. Undisturbed cross-sections of a rod and the same sections when they are reached by the perturbation. The table presents displacement vs position of the sections.

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Elastic waves in a medium

7

sections of the rod undergo different displacements and that displacements are in the same direction of the propagation of the wave. Again, he or she should evaluate the velocity of propagation of the wave. Finally a summary of the differences between longitudinal and transverse waves in a rod is given. In order to analyse the displacement of each section of the rod in the case of a longitudinal wave, a very small portion of the rod is shown enlarged on the display. The rod is shown with each cross-section at the equilibrium position and then with the same cross-sections in the positions they have when they are reached by the perturbation at time t (Fig. 6). The student should evaluate the displacement of each section and construct a table of displacement vs position. This procedure is repeated at different times and graphs corresponding to the values in the table are displayed (Fig. 7). The student should appreciate that these plots are similar to those seen for transverse waves in a string (Fig. 2) and in a rod (Fig. 4) but have different values of velocity and displacement. He or she should recognize that an analytical function of the t y p e f ( t , x ) = u ( t - x / v ) describes also the propagation of longitudinal waves. CONCLUSIONS The programs have been used by first-year undergraduate students of physics at the University of Pavia. Students run the programs after learning the fundamentals of wave propagation in the physics course and having carried out simple experiments with long "soft-springs". A simple manual is provided to teach the students how to use the keyboard to enter data. They work in groups of two to allow discussion of the problems which arise during program execution. The time required to run each program is about 30 min. D a t a about the students' performance are gathered by means of direct observation, questionnaires and an automatic record of information about the student-computer dialogue. Response of students to the use of programs like these has been, until now, most encouraging. The dynamic display of the images offers learning opportunities which could be obtained otherwise only by means of very sophisticated apparatus. Indeed by these programs, students can observe a wave propagating in a medium, can visualize the motion of every particle in a vibrating elastic medium and can take measurements of the velocity of propagation. The interactive nature of the programs guide students to find the appropriate solution for the traveling wave. Moreover the representation of the waves corresponding to students' answers is very effective in helping students to recognize their mistakes. In our experience the computer as a support to traditional teaching helps the understanding of mechanics and stimulates interest towards this branch of physics. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Bork A., Learning with Computers. Digital Press, Billerica, MA (1981). Bork A. and Peckam H., Comput. Educ. 3, 145 (1979). Bork A., Comput. Educ. 4, 1 (1980). Hinton T., Comput. Educ. 2, 71 (1978). McKenzie J., Elton L. and Lewis R., Interactive Computer Graphics in Science Teaching. Ellis Horwood, Chichester (1978). Staudenmaier H. M., Eur. J. Phys. 3, 144 (1982). Tawney D. A., Learning Through Computers. Macmillan Press, London (1979). Borghi L., De Ambrosis A., Gazzaniga G., Ironi L., Mascheretti P. and Massara C. I., G. Fis. 24, 57 (1983). Borghi L., De Ambrosis A., Gazzaniga G., Ironi L., Mascheretti P. and Massara C. I., Computers in physics education: an example dealing with collision phenomena. Am. J. Phys. In press. Wildenberg D. (Ed.), Computer Simulation in University Teaching, p. 107. North-Holland, Amsterdam (1981). Alonso M. and Finn E. J., Fundamental University Physics. Inter European Editions, Amsterdam (1974).