Standing waves between a microwave transmitter and receiverJed Brody, Elena Villhauer, and Hugo Espiritu

Citation: American Journal of Physics 82, 1157 (2014); doi: 10.1119/1.4896355View online: https://doi.org/10.1119/1.4896355View Table of Contents: https://aapt.scitation.org/toc/ajp/82/12Published by the American Association of Physics Teachers

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Standing waves between a microwave transmitter and receiver

Jed Brody, Elena Villhauer, and Hugo EspirituDepartment of Physics, Emory University, Atlanta, Georgia 30322

(Received 17 March 2014; accepted 12 September 2014)

We investigate standing waves that develop in the space between a microwave transmitter and

receiver. Experimental results support a theoretical model of multiple reflections of spherical

waves, leading to the standing waves. This experiment, though straightforward, is richly endowed

with mathematical challenges and surprises. For example, the distance between adjacent antinodes

is only approximately constant for a standing wave. VC 2014 American Association of Physics Teachers.

[http://dx.doi.org/10.1119/1.4896355]

I. INTRODUCTION

The macroscopic wavelength of microwaves is ideal forexperiments in physical optics. Instructional experimentswith microwaves include Bragg diffraction,1–5 evanescentwaves,6 and three-dimensional standing waves.7 Perhaps, thesimplest configuration is to point a transmitter directly at areceiver, which is the basis for the work here. The PASCOmanual5 instructs students to determine wavelength by vary-ing the distance between transmitter and receiver until tenminima are observed. We have found such an experiment ismore instructive if students record intensity as a function ofposition between the two and compare the measurementswith theoretical calculations.

We begin by supposing that the transmitter emits a sectionof a spherical wave of wavelength k. The electric field alongthe line between the transmitter and the receiver is

E1 xð Þ ¼ A

xei kx�xtð Þ; (1)

where A is a constant, x is the distance from the transmitter,k¼ 2p/k is the wave number, x is the angular frequency, andt is time. The first reflected wave off the receiver is

E2 xð Þ ¼ rA

2L� xei k 2L�xð Þ�xt½ �; (2)

where r is the reflection coefficient and L is the distancebetween the transmitter and receiver (the total distance trav-eled by this reflected wave is 2L� x, as shown in Fig. 1).

We must consider whether r should be positive or nega-tive, in other words, whether the wave undergoes a phasechange upon reflection. In the case of a plane wave reflectingoff a flat metal surface, there is a phase change. However, inour experiment, a spherical wave reflects off a truncated

pyramidal surface. In our simple model, we assume that theextremely complicated superposition of reflected wavelets iseffectively a spherical wave, along the x-direction. However,the effective phase change is not obvious, and we allow r tobe positive or negative when fitting our experimental data.We find that fits obtained with positive r were significantlybetter overall than fits obtained with negative r. This empiri-cal evidence suggests, surprisingly, that there is effectivelyno phase change upon reflection.

By considering additional reflections, we can write thetotal electric field as an infinite sum of multiply reflectedwaves:

E xð Þ ¼Ae�ixtX1n¼0

r2n

2nLþ xeik 2nLþxð Þ

�

þ r2nþ1

2 nþ 1ð ÞL� xeik 2 nþ1ð ÞL�x½ �

�; (3)

where we have assumed the same reflection coefficient atboth the transmitter and receiver. This equation can be sim-plified for x¼L, the position of the receiver, to give

E Lð Þ ¼ A 1þ rð ÞL

ei kL�xtð ÞX1n¼0

reikLð Þ2n

2nþ 1; (4)

which presents an excellent opportunity for students to applytheir knowledge of infinite series. Defining z¼ reikL, theseries can be summed to give

S ¼X1n¼0

z2n

2nþ 1¼ 1

z

X1n¼0

z2nþ1

2nþ 1¼ 1

ztanh�1z: (5)

Thus, Eq. (4) becomes

E Lð Þ ¼ A 1þ rð ÞL

ei kL�xtð Þ 1

reikLtan h�1 reikLð Þ; (6)

and the (measureable) intensity is then

I Lð Þ � jE Lð Þj2 / 1þ r

L r

� �2

j tan h�1 reikLð Þj2: (7)

Equation (7), though exact, does not provide muchconceptual insight into the dependence of I on L. For thisreason, students benefit by considering a truncated series aswell as an infinite series. Returning to Eq. (3) and retainingterms through second order in r, we obtain

FIG. 1. A schematic diagram of the experimental apparatus. A wave that has

travelled a distance x from the transmitter will interfere with a wave that has

travelled a distance L, then reflected, and then travelled an additional dis-

tance L� x. Additional reflections are not shown but contribute to the total

electric field at position x.

1157 Am. J. Phys. 82 (12), December 2014 http://aapt.org/ajp VC 2014 American Association of Physics Teachers 1157

I xð Þ � 1

x2þ 2r

x 2L� xð Þ cos 2k x� Lð Þ½ � þ r2

2L� xð Þ2

þ 2r2

x 2Lþ xð Þ cos 2kLð Þ: (8)

The first-order term predicts that if x is varied while keeping Lconstant, local maxima in I occur whenever the differencebetween x and L is approximately an integer multiple of k/2.This is simply the condition for constructive interferencebetween the emitted wave (E1) and the first wave reflected offthe receiver (E2): to reach position x, the reflected wave hastraveled a total distance of [Lþ (L� x)]¼ (2L� x), so the pathlength difference between E2 and E1 is (2L� x)�x¼ 2(L� x).Constructive interference thus occurs when 2(L� x) is an inte-ger multiple of k, assuming no phase change upon reflection.

One of the second-order terms contains cos(2kL), which ismaximized whenever 2L is an integer multiple of k. Tounderstand this, we must consider the waves emitted by thetransmitter and the first waves reflecting back off the trans-mitter (what we would call E3). The path length difference is2L, and constructive interference occurs when 2L is an inte-ger multiple of k.

Higher-order terms account for the interference of wavesthat have undergone a greater number of reflections. Forexample, the third-order terms are 2r3 cos ½kð4L� 2xÞ�=½xð4L� xÞ� and 2r3 cos ð2kxÞ=ð4L2 � x2Þ. The formeraccounts for the interference between the emitted wave and awave that has reflected three times and the latter accounts forthe interference between a wave that has reflected once andone that has reflected twice.

Further insight is gained by examining the second-orderapproximation for the intensity at the receiver (for x¼ L),given by

I Lð Þ � 1

L

� �2

1þ 2r þ r2 þ 2r2 cos 2kLð Þ3

� �: (9)

This expression retains the second-order sinusoidal term thatis periodic in L with a period of k/2. Using the first-derivativetest, students can confirm that local maxima in intensity donot occur at precisely L¼ nk/2, where n is an integer. Simplenumerical experiments reveal that maxima occur at L slightlysmaller than nk/2. This is easily explained by the factor of1/L2, which causes I to decrease before cos(2kL) reaches itspeak. This effect, however, is less pronounced in the exactexpression for I(L) given by Eq. (7). Using k¼ 2.85 cm andr¼ 0.40 in Eq. (7), we find that local maxima occur at Lsmaller than nk/2 by up to 2.4%. This discrepancy decreasesas L increases, falling below 1% at L¼ 88 cm.

The exact expression for I(L) is compared with thesecond-order approximation in Fig. 2. Error in the approxi-mation is on the order of r3¼ 6.4%. In spite of this error, theapproximation provides insight that is obscured in the exactexpression. As we have seen, the approximation revealsthat maxima occur when L is slightly smaller than nk/2,as L¼ nk/2 is the condition for construction interferencebetween waves emitted by the transmitter and waves reflect-ing back off the transmitter.

II. EXPERIMENT

For the experiment, we use a PASCO microwave transmit-ter and receiver from the Basic Microwave Optics System

(WA-9314B). The receiver contains an analog meter and ananalog output. In use, we observe the arrow on the meter tooscillate about an average position, making measurementsdifficult. We therefore connect the analog output to aNational Instruments DAQ device (USB-6009) and collectthe data in LabVIEW. To obtain the average intensity at aparticular position, we record intensity measurements 100times per second for 10 s.

To measure intensity at the position of the receiver, wesimply record the output of the receiver. However, to mea-sure intensity at another position, we place a PASCO micro-wave probe at the desired position. By design, the probe is tobe wired to another receiver, which is kept out of range ofthe microwave beam so as not to introduce any additionalreflections. This second receiver’s output indicates themicrowave intensity at the position of the probe.

To test Eq. (7), we simply measure intensity at thereceiver as a function of the distance between the transmitterand receiver. To test Eq. (8), we measure intensity at a probeplaced between the transmitter and receiver. Initially, wemoved the probe while maintaining a fixed distance betweenthe transmitter and receiver. But in addition, we kept theprobe at a fixed distance from the transmitter while varyingthe distance between the transmitter and receiver.

The output of the receiver is nonzero even when the trans-mitter is off. This is true regardless of whether a probe iswired to the receiver. The nonzero offset becomes negligiblewhen an emitted beam travels directly into the receiver.However, when measuring intensity with a probe, the offsetbecomes significant. We account for this offset by includingan additive constant in the fits to data obtained with a probe.Moreover, the offset drifts with time. When the receiver isturned on, the offset is initially high; it then decreasesroughly 30% within about half an hour. Thereafter, the offsetis more stable, within about 5–10%. To avoid the initial driftin the offset, we turned the receiver on about an hour beforerecording data.

The transmitter, receiver, and probe are mounted oncomponent holders positioned on the arms of a goniometer(whose angle is held at 180�). Distances are measured byuse of a ruler attached to the goniometer arms. The effec-tive locations of the transmitter and receiver are indicatedby the manufacturer’s marks on the base of the componentholders. Uncertainty in our distance measurements is0.1 cm.

FIG. 2. Comparison of the exact solution [Eq. (7), solid] and second-order

approximation [Eq. (9), dashed] for I(L), with k¼ 2.85 cm and r¼ 0.4.

1158 Am. J. Phys., Vol. 82, No. 12, December 2014 Brody, Villhauer, and Espiritu 1158

III. RESULTS AND DISCUSSION

Figure 3 shows the intensity at the receiver as a functionof the distance between the transmitter and receiver. Themeasured data were fit with Eq. (7), the exact expression forinfinite reflections. There are three fitting parameters, thewave number k, reflection coefficient r, and a multiplicativeconstant. We find k¼ 2.2225 6 0.0005 cm�1, from whichk¼ 2.8239 6 0.0006 cm. The wavelength specified in thePASCO manual is slightly higher, 2.85 cm. We suppose thatthe 0.9% difference is within the manufacturing tolerance forthe instructional transmitter.

We may compare our model with the simpler approach ofcalculating the average distance between antinodes. The av-erage distance obtained by a least-squares fit to the measuredpositions of antinodes is 1.44 6 0.02 cm. Under the naiveassumption that the distance between antinodes is k/2, wefind k¼ 2.88 6 0.04 cm. Note that the precision is muchlower than that of our model. More significantly, the naiveassumption of plane-wave interference predicts a perfect si-nusoid, which is clearly not observed in Fig. 3. The simplis-tic model simply does not fit the data at all.

In addition to the wavelength, the reflection coefficient rcan be determined from our data. The fit shown in Fig. 3indicates a reflection coefficient r¼ 0.398 6 0.013. Thus, ifwe wished to use the second-order approximately given byEq. (8), the error would be r3 � 6%.

In the remaining experiments, we made measurementswith a probe and found the noisy, nonzero offset to be verysignificant. Therefore, we wanted to use the strongest trans-mitter possible, so we used a different transmitter than wehad used for the previous experiment. Although we hadobserved that the transmitters had different intensities, wehad not expected the wavelengths to also be different. Infact, as shown below, the wavelength emitted by the secondtransmitter was 2.863 6 0.002 cm. This wavelength is 0.5%above specification, whereas the first transmitter’s wave-length was 0.9% below specification (providing some sup-port for our supposition that the manufacturing tolerance isresponsible for the differences).

In Fig. 4, we see the standing wave itself—the intensitybetween a fixed transmitter and receiver. The standard devia-tion of each intensity measurement is so high that error barsare impractical, so we represent the standard deviation with

dashed lines instead. We fit the data using the second-orderapproximation given by Eq. (8), because the exact expres-sion is obtained only when x¼ L. (We tried adding the third-order terms, but the fit did not improve.) Although the fit isnot terribly impressive, the periodicity is clear and we obtaina wavelength of 2.860 6 0.003 cm. The reflection coefficient,however, is poorly resolved as 0.47 6 0.18.

In Fig. 5, we show the intensity at fixed probe positions asthe receiver position varies. The fits are disappointing, butthe wavelength is still determined, with the upper curveyielding k¼ 2.864 6 0.010 cm, and the lower curvek¼ 2.869 6 0.004 cm. Within error, these two values agreewith each other and are very close to the wavelengthobtained from Fig. 4. The weighted average of the threeresults is 2.863 6 0.002 cm.

Perhaps more interesting is the observation that the twocurves in Fig. 5 are approximately 180� out of phase. This isnot a coincidence and in fact is predicted by Eq. (8). Thedominant sinusoidal term in Eq. (8) is cos[2k(x�L)], and the

FIG. 3. Measured intensity as a function of the distance between the trans-

mitter and the receiver; the fit is obtained using Eq. (7).

FIG. 4. Dependence of intensity on probe position x, with L fixed at 95 cm.

The dashed lines represent standard deviation above and below the data

points. The fit is obtained using Eq. (8); an additive constant was included as

a fitting parameter.

FIG. 5. Intensity at probe position x, as L varies. The dashed lines represent

standard deviation above and below the data points. The fit is obtained using

Eq. (8); an additive constant was included as a fitting parameter. The two

curves are approximately 180� out of phase because the difference between

probe positions (0.7 cm) is approximately k/4.

1159 Am. J. Phys., Vol. 82, No. 12, December 2014 Brody, Villhauer, and Espiritu 1159

argument changes by p when (x� L) changes by k/4 �0.7 cm. To demonstrate this fact, we chose two probe posi-tions separated by 0.7 cm.

IV. CONCLUSIONS

Although the data is not superb, it supports a theoreticalmodel of multiple reflections of spherical waves between atransmitter and receiver, allowing one to determine themicrowave wavelength. Important experimental and theoret-ical skills are learned in this exploration of wave optics.Experimentally, students observe the importance of com-puter interfacing, due to fluctuations in an analog meter.Students therefore learn how to use a virtual instrument (VI)in LabVIEW. In fact, an early task in a student’s LabVIEWtraining could be the creation of the simple VI required forthis experiment. Subsequently, the data analysis gives stu-dents practice with curve fitting.

This experiment also highlights the importance of identi-fying undesired influences on measurements. Specifically,we recognize a nonzero offset that varies significantly whenthe receiver is first turned on. If time allows, studentsmay be allowed to discover this undesired influence ontheir own.

Theoretical calculations allow students to solve a practicalproblem by applying their skills in infinite series. Just as im-portant, they see how an approximation provides usefulinsights that are obscured by an exact solution. The approxi-mation demonstrates, for example, that the peak intensitiesat the receiver are due to constructive interference betweenthe emitted wave and the first wave reflected back off thetransmitter.

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1160 Am. J. Phys., Vol. 82, No. 12, December 2014 Brody, Villhauer, and Espiritu 1160