An undergraduate experiment of wave motion usinga coupled-pendulum chain

Changlin Liang,a) Weiping Ke, Minxue Fu, Changjiang Wang, and Xi ChenTeaching Center for Experimental Physics, Department of Physics, Tsinghua University, Beijing 100084,China

(Received 24 February 2014; accepted 27 December 2014)

We report on the design and construction of a series of coupled pendula to demonstrate various

aspects of wave motion, such as normal modes, standing waves, a dispersion relation, and

evanescence waves. The experiment can help undergraduate students obtain a better understanding

of the properties of wave motion. VC 2015 American Association of Physics Teachers.

[http://dx.doi.org/10.1119/1.4905842]

I. INTRODUCTION

Numerous devices have been used to teach undergraduatestudents about vibrations and waves.1–15 Nonetheless, thereis still a lack of well-designed experiments to demonstratethe core concepts in wave motion, such as normal modes,standing waves, dispersion, and evanescence waves. In thispaper, we describe the design and construction of a series ofcoupled pendula—a coupled-pendulum chain—as a labora-tory teaching instrument to exhibit the rich physics and fasci-nating experimental phenomena in wave motion. Such anapparatus is a representation of a model system with wideapplications in scientific research and engineering,16–18

including, for example, the phonon model in a solid.19 Acoupled oscillator or coupled pendulum chain is commonlyintroduced into a physics laboratory course as a simulationexperiment,20 but a working experimental apparatus used inthe laboratory for teaching has not been reported.

II. THEORETICAL BACKGROUND

The coupled pendulum chain as a one-dimensional vibra-tion system is schematically shown in Fig. 1 and consists ofa series of identical simple pendula of mass M and length L.In equilibrium, all pendula are equally spaced and separatedby a distance of a. The pendulum bobs are connected hori-zontally with ideal springs of spring constant j andunstretched length a. The motion of each bob is in the samevertical plane, and un denotes the horizontal displacement ofthe nth mass away from its equilibrium position.

A. Dynamic equation and dispersion relation

As shown in Fig. 1, without considering the boundary con-ditions, the dynamic equation for un is written as21

€un þ ð2x2s þ x2

pÞun ¼ x2s ðunþ1 þ un�1Þ; (1)

where x2p ¼ g=L and x2

s ¼ j=M, corresponding to the squaresof the angular frequencies of a single pendulum and a springoscillator, respectively. Here, g is the gravitational fieldstrength.

The equation of motion for all masses is given in Eq. (1).When the system vibrates freely, there must be some simplemotions where all pendula oscillate harmonically with thesame angular frequency x and the same phase constant d. Anormal-mode solution to Eq. (1) can be written as

~un ¼ Ae�iðkna�dÞeixt; (2)

where A and d are the vibrational amplitude and initial phase,both of which are independent of n and determined by theinitial conditions. Additionally, k ¼ 2p=k is the wave num-ber, with k representing the wavelength. This solutionimplies that a traveling wave with wave number k and ampli-tude A is generated on the system. By substituting Eq. (2)into Eq. (1), we obtain the dispersion relation of the coupled-pendulum chain as

x ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

p þ 4x2s sin2 ka

2

� �s: (3)

B. Boundary conditions and normal modes

Assuming that the chain has N pendula, its total lengthshould be (N–1)a. If n¼ 0 for the leftmost bob, thenn¼ (N–1) for the rightmost one. Because there is only onenearest neighbor for each of the pendula on the ends, free-end boundary conditions are applied, leading to

€u0 þ ðx2s þ x2

pÞu0 ¼ x2s u1 (4)

and

Fig. 1. Schematic diagram of a coupled-pendulum chain. The pendula of

mass M and length L are coupled with linear springs of spring constant j.

The separation between adjacent pendula in equilibrium is equal to the

unstretched spring length a. The horizontal displacement of the nth pendu-

lum away from its equilibrium position is denoted by un.

389 Am. J. Phys. 83 (5), May 2015 http://aapt.org/ajp VC 2015 American Association of Physics Teachers 389

€uN�1 þ ðx2s þ x2

pÞuN�1 ¼ x2s uN�2: (5)

Under these conditions, the steady-state solution for thesystem is a standing wave. According to Eq. (2), this will bea superposition of a wave ~uþn ¼ Ae�iðkna�d�xtÞ travelingin the positive direction, and its reflected wave ~u�n¼ Ae�i½kað2l�nÞ�d�xt� traveling in the negative direction. Thereflection arises at the right boundary of the chain withreflection coefficient þ1 for its free end. Here la denotes thereflecting point location and should satisfy ðN � 3=2Þa � la� ðN � 1=2Þa. The solution is therefore

~un ¼ ~uþn þ ~u�n ¼ 2A cos½kaðn� lÞ�eiðxt�klaþdÞ: (6)

Substituting this solution into Eqs. (4) and (5) and consider-ing the dispersion relation Eq. (3) and the physical require-ment that k> 2a, we obtain

l ¼ N � 1

2(7)

and

kq ¼q

N

pa; (8)

where q is an integer satisfying 0 � q < N. The systemtherefore has a total of N normal modes.

The most general motion of the system can be expressedas a superposition of all normal modes, as in

~un¼XN�1

q¼0

Cqcos kqa n�Nþ1

2

� �� �ei xqt�kqa N� 1=2ð Þð Þþdq½ �;

(9)

where Cq is a constant determined by the initial conditions.The frequencies of the normal modes can be acquired con-veniently through a Fourier analysis on the free oscillations.In this work, we employ the Fast Fourier Transform (FFT).

C. Allowed and forbidden frequency bands

The dispersion relation of Eq. (3) shows that the system

has a maximum angular frequency xmax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

p þ 4x2s

qand

a minimum angular frequency xmin ¼ xp; the bandwidth is

Dx ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

p þ 4x2s

q� xp. Thus, the pendulum chain acts as a

band-pass filter, as only vibrations with frequencies insidethe band are allowed to propagate without decay. Vibrationswith frequencies outside the band will decay exponentiallywhile propagating forward along the chain, which forms anevanescent wave. Specifically, if the angular frequency islower than xmin, then

~un ¼Ax2

min�x2

2x2s

þ 1

!�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

min�x2

2x2s

þ 1

!2

� 1

vuut264

375

n

� ei xtþdð Þ; ð10Þ

and the oscillations are in phase. If the angular frequency ishigher than xmax, then

~un¼Ax2�x2

max

2x2s

þ1

!�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2�x2

max

2x2s

þ1

!2

�1

vuut264

375

n

� ei xt6npþdð Þ; ð11Þ

and the adjacent objects oscillate out of phase. In both cases,the amplitudes decay exponentially.

III. EXPERIMENTAL SETUP

Figure 2(a) shows our experimental apparatus. It consistsof three parts installed on a rectangular frame, the coupled-pendulum chain, the electromagnetic driver, and the motiondetector.

To ensure that all pendula vibrate in the same verticalplane, we use thin strips of aluminum alloy instead of stringsto suspend the masses as shown in Fig. 2(b), leading to aneffective pendulum length L. Bearings are used to support thestrips on the top beam of the frame to minimize mechanicaldamping. The lower end of each strip is sandwiched betweentwo stainless steel blocks, forming a pendulum bob with massM. All pendula are carefully aligned along the horizontal linewith equal spacing. The coupling springs are all identical withspring constant j, and their ends are spiraled to form smallrings so that they can be fastened to the bobs using screws.This design effectively constrains all pendula to vibrate in thesame vertical plane with negligible damping. Moreover, thestructure is easy to adjust to form a coupled system with dif-ferent numbers of vibration units. For our coupled chain, thereare fifteen vibration units and its ends are free. The detailedstructural parameters are shown in Table I.

Taking into account the primary factors that influence theproperties of the coupled system, such as the block radius Rand height H, the suspension strip mass m, and the springmass m0 [see Fig. 2(b) and Table I], the single pendulumangular frequency xp and the spring oscillator angular fre-quency xs should be corrected as1,22–25

xp ¼ffiffiffig

L

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ m

2M1� R

L

� �

1þ 1

4

R2

L2þ 1

12

H2

L2þ 1

3

m

M1� R

L

� �2

vuuuuuut (12)

and

xs ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

jM þ m=2þ m0=3

r: (13)

An electromagnetic interaction is utilized to excite thechain by driving the leftmost pendulum (n¼ 0), as shown inFig. 2(c). The two coils shown are connected in series andmounted on the bottom beam of the frame. A pair of smallbutton-shaped permanent magnets are attached to the sidesof a thin plastic strip below the 0th stainless steel block bytheir mutual attraction and located in the center of the coils.Powered by an alternating current, the coils generate a mag-netic field and create an alternating driving force on the 0thpendulum through the permanent magnets. The wholechain then begins to vibrate under the influence of this force.This noncontact electromagnetic driving scheme has theadvantages of lower noise and simpler structure comparedwith the universal electromotor driving.4,6

390 Am. J. Phys., Vol. 83, No. 5, May 2015 Liang et al. 390

In order to measure the motion of each pendulum, a com-mercial device called position sensitive detector (PSD) isused, which is an optoelectronic position sensor utilizingphotodiode surface resistance.26 This device provides contin-uous position data and features high resolution and high-speed response. Usually, a PSD detects the position of anincident spot of light, which, in our case, would require thateach pendulum carry a light source, and thus would greatlycomplicate the design. Therefore, we use a PSD in thecomplementary mode where it detects the position of adark spot rather than a bright one. As shown in Fig. 2(d),a parallel light source, such as an LED line array, and aone-dimensional PSD (PSD-2534) are placed face-to-face ona stand to form the motion detector; the suspension strip

vibrates inside the gap between the two devices. A guide railis horizontally installed on the frame, along which the detec-tor can be moved and locked at arbitrary position, as seen inFig. 2(a). When a parallel light beam emitted from the LEDarray illuminates PSD, a shadow caused by the pendulumstrip is formed on the PSD’s light sensitive surface. The PSDthen outputs a current signal, whose intensity is proportionalto the displacement of the shadow relative to the center ofthe light sensitive surface.

A homemade controller provides a sinusoidal ac currentfor the electromagnetic coils with an adjustable frequencyranging from 0.3 to 6.0 Hz, which is wider than the bandwidthof our coupled system. The maximum peak current is 1.0 A,which is determined by the driving power requirement. In

Fig. 2. (a) Photograph of the entire experimental apparatus, which includes a coupled pendulum chain, an electromagnetic driver, a motion detector, a control-

ler, and a digital oscilloscope. Close-up view of (b) an elementary oscillation unit, (c) the electromagnetic driver, and (d) the motion detector.

Table I. The structural parameters of the coupled-pendulum chain. The tested values before installation and the design values are labeled with the superscripts

T and D, respectively.

Pendulum

Spring SeparationBob Suspension strip

Mass

M (g)

Radius

R (mm)

Height

H (mm)

Effective length

L (mm)

Mass

m (g)

Spring constant

j (g/mm)

Mass

m0(g)

Relaxed

length (mm) a (mm)

148.33 T 15.00D 30.00D 500.0D 29.63T 1.717T 1.911T 50.0D 80.0D

391 Am. J. Phys., Vol. 83, No. 5, May 2015 Liang et al. 391

addition, the controller supplies a constant dc current to thelight source of the motion detector and linearly converts thePSD output current into voltage. A dual-trace digital oscillo-scope with an FFT function is used to monitor the driver andto display and measure the vibration of the pendulum.

IV. EXPERIMENTS

Based on the instrument described above, students areable to carry out various experiments related to vibration andwave motion. Several are illustrated in the following discus-sions. For convenience, we use the frequency f ¼ x=2p,instead of x.

A. Frequency spectrum analysis of a one-dimensionalcoupled system

Free vibration can be excited by slightly swinging anypendulum of the chain by hand. The motion of a certain pen-dulum is detected by the detector and is displayed on theoscilloscope (upper panel of Fig. 3). The peaks in the FFTspectrum (lower panel of Fig. 3) correspond to the frequen-cies of the normal modes.

In order to observe all normal modes of the system, theabove process should be repeated several times, as an arbi-trary trigger cannot ensure exciting all normal modes. And inthe case of the detected bob located at a standing-wave nodeof a particular normal mode, its corresponding amplitudeshould be zero and thus no information on such a normalmode can be obtained. Table II shows the frequencies of allfifteen modes determined by FFT.

Substituting the structural parameters (see Table I) into Eqs.(12) and (13) resulted in fp¼ 0.7157 Hz and fs¼ 1.613 Hz,respectively. Thus, the normal mode frequencies and the

corresponding wavelengths can be calculated using Eqs. (8)and (3) and are included in Table II. The experimental datafFFT are quite consistent with the theoretical results fCal.

B. Standing waves

Standing waves are excited on the chain by using the elec-tromagnetic driver with a driving frequency within its bandand is especially apparent when the frequency is carefullyadjusted to one of the experimental values (in Hz) 0.80, 1.23,1.77, 2.27, 2.70, 3.04, or 3.24, which correspond to wave-lengths (in units of a) 30, 10, 6, 30/7, 10/3, 30/11, or 30/13,respectively (see Table II). Some pendula are at the nodalpositions and remain (nearly) motionless, which makes thevisualization of a standing wave more pronounced. Forexample, Fig. 4 shows a photo of the chain oscillating with afrequency of 1.23 Hz. It is evident that the 2nd, 7th, and 12thpendula are almost at rest, even though the other pendula areoscillating briskly. These pendula correspond to the nodalpositions of the standing wave with a wavelength of 10 a.

C. Dispersion relation

The dispersion relation of Eq. (3) connects frequency withwave number. The frequency is measured directly by the os-cilloscope, while the motion detector records the oscillationof each pendulum and determines its amplitude. We can thenobtain both the wavelength k and the parameter l (recallingthe restriction on l), both in units of a, by fitting the relativeamplitude Un (in unit of voltage) as a function of the bobposition n (in units of a) with the function

Un ¼ A cos 2pn� l

k

� �; (14)

[see Fig. 5(a)]. Consequently, we get the wavenumber k (inunits of p=a) for this particular frequency.

We then obtain the dispersion relation by repeating aboveprocess while driving the system at different frequencies,as shown in Fig. 5(b). Fitting the experimental data usingEq. (3) yields fp¼ 0.725 Hz and fs¼ 1.611 Hz, which areclose to the theoretical estimations of 0.716 Hz and 1.613 Hzobtained from Eqs. (12) and (13). Accordingly, the lowerand upper cut-off frequencies are fmin¼ 0.725 Hz andfmax¼ 3.303 Hz.

In addition, the mean value of l is 14.485 in our experi-ments, which agrees quite well with the theoretical result of14.5 obtained using Eq. (7) with N¼ 15. Physically, thisresult tells us that the reflection point for a positive travelingwave of any allowed wavenumber k happens at the locationof (N–1/2)a. If an elementary oscillation unit of the systemconsists of a pendulum and two springs, each having a lengthof a/2, then this reflection location seems to be the rightboundary of the (N–1)st oscillation unit.

Fig. 3. Frequency analysis of the oscillator chain using the oscilloscope

(screen capture). The upper panel shows the displacement of a pendulum as

a function of time, and the lower panel shows the FFT of the displacement.

Table II. The normal-mode frequencies, wavelengths, and node positions. The frequencies obtained from the FFT spectral analysis and the theoretical calcula-

tions are identified with fFFT and fCal, respectively. Index q is the serial number of normal mode.

Index (q) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

k (a) 1 30 15 10 15/2 6 5 30/7 15/4 10/3 3 30/11 5/2 30/13 15/7

fFFT (Hz) 0.73 0.80 0.99 1.23 1.50 1.77 2.02 2.27 2.50 2.70 2.89 3.04 3.16 3.24 3.29

fCal (Hz) 0.7157 0.7912 0.9810 1.227 1.495 1.765 2.027 2.275 2.502 2.707 2.885 3.033 3.151 3.236 3.288

Node n 7 2, 7, 12 1, 4, 7, 10, 13 7 2, 7, 12 7 7

392 Am. J. Phys., Vol. 83, No. 5, May 2015 Liang et al. 392

D. The phenomenon of cut-off in evanescent waves

We can also observe evanescent waves in the coupled pen-dulum system. If the driving frequency is lower than thelower cut-off frequency fmin, all pendula vibrate in phasewhile the amplitudes decay exponentially along the chain, asshown in Fig. 6(a). If the frequency is higher than the uppercut-off frequency fmax, the amplitudes alternate in sign and

attenuate exponentially, as shown in Fig. 6(b). In all cases,the further the driving frequency is away from the cut-off,the shorter the amplitude attenuation length is. As seenin Fig. 6, the experimental data are well fitted by Eqs. (10)and (11).

It is worth noting that if the driving frequency is equal (orvery close) to half or double any normal mode frequency,then sub-harmonic or harmonic resonance is generated,which makes the motion of the chain very complicated.

V. CONCLUSION

We have designed and built an experimental apparatusthat contains a series (or chain) of coupled pendula for a labthat teaches about wave motion. The normal modes, standingwaves, dispersion relation, and evanescent waves are all welldemonstrated using this apparatus. The experiment has beenused to help students at Tsinghua University understandwave motion.

ACKNOWLEDGMENTS

The authors thank Professor J. L. Lu for his advice. Thiswork was financially supported by the lab construction fundof the Ministry of Education of China and TsinghuaUniversity.

Fig. 4. Photograph of the oscillator chain vibrating with a frequency of

1.23 Hz. The 2nd, 7th, and 12th pendula are at rest and represent the nodes of

a standing wave with a wavelength of 10 a. (Continuous exposure for 10 s

using Cannon EOS600D camera.).

Fig. 5. (a) An example for the test procedure of wavenumber. The relative

amplitude Un (dots) is plotted as a function of the bob position n, and the

solid curve is the fitting function Un ¼ A cos 2p n� lð Þ=k� �

. (b) The meas-

ured dispersion relation (dots) of the system and its fitting function

f ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2p þ 4f 2

s sin2 ka=2ð Þq

, from Eq. (3).

Fig. 6. Evanescent waves for the driving frequencies (a) lower than fmin and

(b) higher than fmax. The dots are the measured amplitudes and the curves

are fits using Eqs. (10) and (11).

393 Am. J. Phys., Vol. 83, No. 5, May 2015 Liang et al. 393

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