LEP1520_00

77PHYWE Systeme GmbH & Co. KG · D-37070 Göttingen Laboratory Experiments Physics

Interference of ultrasonic waves by a Lloyd mirror 1.5.20-00

Mechanical Vibration Acoustics Mechanics

Principle:A partial packet of radiation passesdirectly from a fixed ultrasonictransmitter to a fixed ultrasonic re-ceiver. A further partial packet hitsagainst a metal screen that is posi-tioned parallel to the connecting linebetween the transmitter and receiv-er, and is reflected in the direction ofthe receiver. The two packets of radi-ation interfere with each other atthe receiver. When the reflector ismoved parallel to itself, the differ-ence in the path lengths of the twopackets changes. According to thisdifference, either constructive or de-structive interference occurs.

The received signal as a function of the reflector distance d.

Tasks:1. The sliding device is to be used to

move the reflector screen posi-tioned parallel to the connectingline between the transmitter andreceiver parallel to itself in stepsof d = (0.5-1) mm. The reflectorvoltage U is to be recorded at eachstep.

2. The d values of the various maxi-ma and minima are to be deter-mined from the U = U(d) graphand compared with the theoreti-cally expected values.

What you can learn about …

� Longitudinal waves� Superposition of waves� Reflection of longitudinal

waves� Interference

Ultrasonic unit 13900.00 1

Power supply f. ultrasonic unit, 5 VDC, 12 W 13900.99 1

Ultrasonic transmitter on stem 13901.00 1

Ultrasonic receiver on stem 13902.00 1

Digital multimeter 07134.00 1

Optical profile-bench, l = 60 cm 08283.00 1

Base f. opt. profile-bench, adjust. 08284.00 2

Slide mount f. opt. profile-bench, h = 80 mm 08286.02 2

Slide mount f. opt. profile-bench 08286.00 1

Sliding device, horizontal 08713.00 1

Swinging arm 08256.00 1

Screen metal, 30�30 cm 08062.00 1

Measuring tape, 2 m 09936.00 1

Connecting cord, l = 50 cm, red 07361.01 1

Connecting cord, l = 50 cm, blue 07361.04 1

What you need:

Complete Equipment Set, Manual on CD-ROM includedInterference of ultrasonic waves by a Lloyd mirror P2152000

LEP1.5.20

-00Interference of ultrasonic waves by a Lloyd mirror

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen 21520-00 1

Related topicsLongitudinal waves, superposition of waves, reflection of lon-gitudinal waves, interference.

PrincipleA partial packet of radiation passes directly from a fixed ultra-sonic transmitter to a fixed ultrasonic receiver. A further partialpacket hits against a metal screen that is positioned parallel tothe connecting line between the transmitter and receiver, andis reflected in the direction of the receiver. The two packets ofradiation interfere with each other at the receiver. When thereflector is moved parallel to itself, the difference in the pathlengths of the two packets changes. According to this differ-ence, either constructive or destructive interference occurs.

EquipmentUltrasonic unit 13900.00 1Power supply f. ultrasonic unit, 5 VDC, 12 W 13900.99 1Ultrasonic transmitter on stem 13901.00 1Ultrasonic receiver on stem 13902.00 1Digital multimeter 07134.00 1Optical profile-bench, l = 60 cm 08283.00 1Base f. opt. profile-bench, adjust. 08284.00 2Slide mount f. opt. profile-bench, h = 80 mm 08286.02 2Slide mount f. opt. profile-bench 08286.00 1Sliding device, horizontal 08713.00 1Swinging arm 08256.00 1Screen metal, 30x30 cm 08062.00 1

Measuring tape, 2 m 09936.00 1Connecting cord, l = 50 cm, red 07361.01 1Connecting cord, l = 50 cm, blue 07361.04 1

Tasks1. The sliding device is to be used to move the reflector

screen positioned parallel to the connecting line betweenthe transmitter and receiver parallel to itself in steps of d =(0.5-1) mm. The reflector voltage U is to be recorded ateach step.

2. The d values of the various maxima and minima are to bedetermined from the U = U(d) graph and compared withthe theoretically expected values

Set-up and procedureSet up the experiment as shown in Fig. 1, referring to the dia-gram in Fig. 2 for more clarity. Mount the ultrasonic transmitter and the ultrasonic receiver intheir slide mounts (h = 80 mm). Set them at the same height,then orient them on the optical bench so that their middle axesare concordant and in alignment with the optical bench. Usethe swinging arm to mount the reflector screen on the slidingdevice (horizontal) and ensure that the slide of this is initiallysituated at the start of the scaleAt the start of the experiment the reflector screen must bealigned parallel to the optical bench and at a distance of 2 cmfrom the middle axis of the transmitter and receiver. Carry outthis adjustment by turning the swinging arm, keeping thereflector parallel to the optical bench while doing so.

Fig.1: Experimental set-up

LEP1.5.20

-00Interference of ultrasonic waves by a Lloyd mirror

21520-00 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen2

Finally move the transmitter and receiver so that their frontedges are symmetrical to the sides of the screen and at a dis-tance of 29.4 cm apart (see Fig. 2). As the active parts of theultrasonic elements are approx. 3 cm behind their protectivegrids, their effective distance apart is now 30 cm.Connect the transmitter to the TR1 diode socket of the ultra-sonic unit and operate it in continuous mode “Con“. Connectthe receiver to the left BNC socket (prior to the amplifier).Connect the signal received to the analog output of the digitalmultimeter to have it displayed subsequent to amplificationand rectification. To ensure proportionality between the inputsignal and the analog output signal, avoid operating the ampli-

fier in the saturation range. Should such a case occur and the“OVL“ diode light up, reduce either the transmitter amplitudeor the input amplification. To start with, for control and avoid-ance of overloading, use the sliding device to bring the screento the area of the 1st maximum of the measurement curve.To now record the measurement curve, use the sliding deviceto move the screen away from the middle axis of the systemin steps of ∆d = (0.5 – 1) mm, measuring the appropriatereceiver voltage U at each step. The results of such a mea-surement series are shown in Fig. 3.

Theory and evaluationFig. 4 shows the paths of the partial packets from the soundwave emitted by the transmitter which interfere with eachother at the receiver. Part of the wave reaches the receiverdirectly, whereas a second part is first reflected by a metalscreen. According to the difference in the path lengths of thetwo packets either constructive or destructive interferenceoccurs. With a constant distance between transmitter andreceiver, the difference in the paths lengths ∆ (and with this theinterference conditions) is a function of the distance d of thereflector from the middle axis (see Fig. 4).

The following is valid:

(1)

Constructive interferences (maxima) are given, when ∆ corre-sponds to a whole-numbered wavelength of the ultrasonicwave:

(2)

Destructive interferences (minima) are given, when ∆ corre-sponds to an odd-numbered half-wavelength of the ultrason-ic wave:

(3)

S d � B a 2n � 14

b 2

l2 �2n � 1

2 lx ; n � 0, 1, 2, 3, ...

¢ �2n � 1

2l � 2 a21d2 � x2 2 � x b

S d � Bn2l2

4� nlx ; n � 0, 1, 2, 3, ...

¢ � nl � 2 a21d2 � x2 2 � x b

¢ � 2 1y � x 2 � 2 a21d2 � x2 2 � x b

Fig.3: The signal received U as a function of the reflector dis-tance d.

Fig.4: Diagram for the calculation of the path length differenceof the two partial packets (t = transmitter, r = receiver, sc = screen)

Fig. 2: Diagram of the experimental set-up(t = transmitter, r = receiver, sd = sliding device, sa =swinging arm, sc = screen)

LEP1.5.20

-00Interference of ultrasonic waves by a Lloyd mirror

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen 21520-00 3

Table 1 lists the d values for the maxima and minima in Fig. 4,and also, for comparison purposes, the d values calculatedusing equations (2) and (3)..

Table 1: d values at the maxima and minima

The following values were used to calculate d: x = 15.0 cmand l = 0.86 cm.As the transmitter emits at a frequency of f = 40 Hz, it followsfrom c = l · f (c = 343.4 ms-1 at T = 20°C) that the ultrasoundwavelength is: l = 0.858 cm 0.86 cm.As only the d values of the extremes are determined in thisexperiment and not their absolute intensity, factors such asabsorption by air and the type of wave (plane or spherical)need not be considered here.

�

Maxima Minima

n dexp./cm dtheor./cm dexp./cm dtheor./cm

0 0 2.50 2.55

1 3.55 3.62 4.35 4.44

2 5.00 5.15 5.60 5.78

3 6.15 6.35 6.65 6.88

4 7.15 7.38

LEP1.5.20

-00Interference of ultrasonic waves by a Lloyd mirror

21520-00 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen4