HI : ‘ll‘l l l Ill HWIWTI \ l WI THS DEMONSTRATION OF FRESNEL ENTERFERENCE BY MEANS OF A RQPPLE TANK Thesis 1203' ”19 Degree of M. S. MECHEGAN STATE UNIVERSITY Claude M. Watson 1958 —-— IIIIWWflimflfllflfl'flfiflfl'ufllWEI ‘1 ‘‘‘‘‘ 3 1293 01743 00 DEMONSTRATION 0F FRESNEL INTERFERENCE BY MEANS OF A RIPPLE TANK By Claude M. Watson AN ABSTRACT Submitted to the College of Science and Arts of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics 1958 ‘ H. mm Approved ABSTRACT The surface wave analogy of the diffraction of light by a slit and by a grating was used to show details of these phenomena which are unobservable in optics. Details a few wave lengths from the source (corresponding to anslit in optics) were shown and compared with the pictures of the corresponding analogy of a long, thin ultrasonic transducer. Also, the alternation of phase and amplitude modulation of the wave front in the Fresnel region of a grating, the increasing complexity of the patterns with increasing DA, and the similarity between the patterns produced by amplitude gratings and phase gratings were shown. DEMONSTRATION OF FBESNEL INTERFERENCE BY MEANS OF A RIPPLE TANK By Claude M. Watson A THESIS Submitted to the College of Science and Arts of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics 1958 ACKNOWLEDGEMENT The author wishes to express his sincere appreciation to Dr. E. A. Hiedemann, who suggested the project and has given constant and helpful guidance through its development. He is also greatly indebted to Dr. M. A. Breaseale for his generous help and for his assistance with trans- lations, upon which much of this study is based. II. III. IV. TABLE OF CONTENTS Introduction. Experimental Apparatus. Beam Patterns. Periodic Change of Phase to Amplitude Modulation. Bibliography. 13 25 32 10. 11. 12. 13. 14. I5. 16. 17, 13, TABLE OF FIGURES. Velocity of Sinusoidal Waves on Surface of Liquid. Components of Optical System. Dippers. Interference Pattern from a Long Thin Source. Stroboscopic Picture of Interference Pattern from Transducer. Interference Pattern Showing Intensity Distribution Close to the Transducer. Directional Characteristics of a Rectangular Radiating Surface. Interaction of Cylindrical Waves from Ends and Plane Eaves from Face of a Transducer. Schematic Drawing of the Interference Pattern Immediately Before a Transducer Showing Intensity Distribution. Wave Length for Sinusoidal Frequencies on a Mercury Surface. Beam Pattern from 1 cm Transducer at 63nh Beam Pattern from 1 cm Transducer at 20mv. Beam Pattern from a 1.5 cm Transducer at 119Ah Beam Pattern from a 1.5 cm Transducer at 187AL Beam Pattern from 2 cm Transducer at 375AJ With Intensity of Amplifier Very Low. Beam Pattern from 2 cm Transducer at 376A/ With Intensity of Amplifier High. Interference Pattern Produced by Gratings. Interference Pattern Produced by Gratings. 11 15 15 15 17 17 17 20 22 22 23 23 24 24 27 27 19. 20. 21. 22. TABLE or FIGURES (cont.) Interference Pattern Produced by Gratings. Interference Pattern for Corrugated Dipper. Interference Pattern for Corrugated Dipper Interference Pattern Produced by a Saw-Tooth Dipper. 28 28 28 31 I. INTRODUCTION Ripple tanks are widely used for demonstrating the characteristic phenomena of wave propagation, such as reflection, refraction, diffraction and interference. For lecture demonstrations one usually uses low frequency gravity waves on a water surface. The velocity of the gravity waves depends on the depth of the water below the surface: this fact allows one to demonstrate refrac- tion. The low frequency necessitates the use of rather large troughs: 75 cm by 75 cm is the size used by R. H. Pohll. For special studies, more elaborate types of ripple tanks have been described very early. Lord Rayleigh2 studied surface tension prior to 1890 using capillary waves and a stroboscopic arrangement. J. H. Vincent394'5 worked in mercury before the turn of the century, using an instantaneous spark for illumination and photographing the results. Out of the large number of more modern papers on ripple tanks, those presented by J. Baurand6'7'8, H. Bondyo, O. Brandt and H. Freundlo, A. H. Davis11'12, w. a. Dean13, G. Kreisell4, M. s. Longuet-Higginsls'lfi, J. Larras and J. Laurent17, R. Mercierls, B. Tylerlg, J. S. van Wieringenzo, and Yamasaki21 are mentioned here as they contain interesting special techniques, experimental results or theoreticalJ considerations. -2 In using a ripple tank, since the method is an analogy, there are several factors to consider. For example, one must consider the limitations on the analogy, the nature of the motion, and how the meniscus, contamination of the surface, and viscous damping affect this motion. It is obvious that analogy between surface, sound and electromagnetic waves can be expected for all phenomena characteristic for every type of wave motion. For special types of waves, certain phenomena may occur to which there do not exist analogies in other types of waves. F. Ursellz2 describes, as an example, the trapping modes which are characteristic for the theory of surface waves, but do not occur in sound. The experimental observations have con- firmed the analogy for the phenomena of reflection, refrac— tion, diffraction and interference. However, one investiga- tor, M. Bouasse23, concluded from an extended study of the diffraction of capillary waves, that the Huygens' principle had no validity for surfaceIwwves. This statement was in direct contradiction to the results of E. Grossmann and E. Biedemann24, who had found that the diffraction of capil- lary waves by a slit revealed even the finer details of the interference pattern calculated from classical diffraction theory. The interference phenomena of capillary waves studied in this thesis give a further illustration of the complete analogy with the effects observed with electro- magnetic or ultrasonic waves. -3 The velocity (v) of the surface wave is given by the relationship: v2=fz)\2=(%*‘z%)finh 11%); (1) where f is frequency, A is wave length, g is acceleration of gravity, offs surface tension, f’is density and h is the height of the liquid. For depths greater than A/2, the hyperbolic tangent approaches one, so the expression reduces to the terms in the brackets. Numerical values of the velocity can be seen from the plot of velocity versus wave length in Figure l. The curves are for water, clean mercury and dirty mercury. The value of.A.for minimum velocity is given by Amm ‘ 13%:- (2) This minimum for the three curves plotted is: >\m fm vm Water 1.73 cm. 13.4 cps 23.2 cm/sec Clean Hg 1.29 cm. 15.7 cps 19.9 cm/sec Dirty Hg .94 cm. 18.3 cps 17.2 cm/sec This minimum value is the dividing line between two classes of waves. For large wave lengths, the term 2: dominates. This is the region to the right of the minimum value in the plot. The waves here are commonly called grwvity wmves since gravity dominates. In this region one finds that the depth of the liquid is important in determining the wave length. This allows one, for w- VELOCITY (cm/Sit) VELOCITY OF SINUSOIDAL WAVES ON SURFACE OF LsQwD. ——-— WATER _____ Ccnm chuRY -— DIRTY Mucus? FIG. I. 4O /. / /' ’ 30 " / /I ’I ‘ I /‘/ / ‘ I I / ‘Ar ,’ / / /‘ / l‘ ’ /' / O I 0/ A / / \/' [I 0/ // / I /' / s I c/ I / I ——./ ’ I I I I / / I 20 ” ~ I ” IOL‘ ' O 2 4 b WAVE LENGTH (cm) example, to study refraction, since a variation in depth changes the velocity. Therefore, by putting a flat piece of solid in the shape of the lens on the bottom of the trough, one obtains an area of different refractive index in the form of a lens. For small wave lengths, the curve era' AP dominates. This is the region of capillary waves and is is to the left of the minimum value and the term well suited to the study of diffraction. This region is particularly desirable because the absorption is greater, and, therefore, small tanks can be used. Also, with small wave length, many waves can be shown in a small space. It is well known that contamination of the surface does affect the surface tension. Since surface tension does enter into the equation that is dominant for capil- lary waves, the velocity and wave length are changed for a given frequency. This change in wave length is immaterial, since the analogies still bold, but the viscous damping increases and the contamination usually shows up in photographs. In the case of mercury, the surface tension changes from 530 cgs units for clean mercury to 300 cgs units for dirty mercury. A thin film forms after a short exposure to airzs. This film may be skimmed off frequently by drawing a plastic straight edge across the surface. Scum will adhere to the straight edge and the sides of the trough, presenting a clean surface again. In the case of water, fresh water is always -6 available, so a clean surface is easier to obtain. The viscous damping not due to changes in the sur- face is a more important consideration. Davis12 has shown that the effect of viscosity on harmonic surface waves is similar to that on sound. This property is desirable in the analogy to sound, such as in the part of the demonstration dealing with beam patterns from a transducer. In the analogy comparing light and sound in this paper, it is important to have a large area relative- ly free from damping. Lamb26 has shown that for a depth of liquid greater than half a wave length, the velocity is unaltered by viscosity (1», and the modulus of decay (T7 is given by: I T= W k: £3— Expressing T’in terms of A for water and mercury, Toaster-9‘ Azsuonds and TH; 04 /o)\2' ; or for the same amount of damping, a wave length in mercury can be one- third that of waters. Thus, for an extended field, mercury would seem to be favored slightly when using high frequencies. -7 II. EXPERIMENTAL APPARATUS The optical set—up was designed to project parallel light on the surface of a mercury pool and to project an image of the surface wave on a screen or into a camera. In Figure 2, light from the source 0 is focused by lens F1 on a circular aperture 81. Light from 31 passes through a green filter F and is focused on the barrier B by lens F2. The modulated light leaving this point is rendered parallel by the lens L3, and is reflected to the mercury surface by the mirror M1. Light reflected from the mercury surface is reflected again by mirror M2. This mirror is placed as close as possible to M1 to reduce distortion of the image by keeping the angle between incident and reflected light small and by keeping the incident angle nearly normal to the surface. An image of the surface is projected by lens L4 on the camera or screen. The camera used was an Exacta V, which has a focal plane shutter, thus permitting the removal of the lens. In order to observe traveling waves in a stationary lposition, it is necessary to provide stroboscopic illumi- Iladdon. It is possible to obtain stroboscopic illumination '53? using a mercury vapor lamp which is modulated at 120 cycles by the alternating line current and exciting the dIi-pper on the mercury surface at the same frequency. Although this procedure in this case was extremely simple .Imkw>m acuCaO do 0.328on .N 6E1 emutau «£34.30 9034 36:31.2 «5.. nzmm n23 wzuE o< fiuoadZL NZ (Jn. muluo duneuam 030.— 335 «333 2.64 mused 3oju> tam. 305.353.. smegma t menace...” 33:33 at: 225303 it was abandoned because it excluded greater variations in the experimental conditions. The use of a special stroboscope was found to be much more satisfactory. This stroboscope was made by interrupting the light periodi- cally by a barrier attached to the cone of a loud speaker. The speaker is excited by a signal from the same source as the dipper in the mercury. A continuous light source is obtained by using an incandescent lamp of such a type that the light is not modulated appreciably by the alternation of the line current. The electrical system consisted of a Hewlett Packard audio oscillator, model 2000 (so in Figure 1). The output was connected directly to the loud speaker L81 driving the dipper (D). The signal was also connected to a Knight audio amplifier, thus providing a means of controlling the amplitude of vibration of the loud speaker L82, used to strobe the light. The loud speaker L81, used to drive the dippers, was a very ruggedly constructed surplus U. 8. Coast Guard loud speaker, type COR-63—D. A.rod was bolted to the voice coil in such a manner that dippers could be plugged into the end. For loud speaker L82, a cone was removed from a five inch loud speaker and most of the frame was cut away in order to get the speaker as close to the light beam as possible. A small piece of wood, long enough to extend into the light path, was cemented to the voice coil of the speaker. -10 This loud speaker was mounted on a sliding track to permit moving the speaker into and out of the light path. Dippers were of three main types. Those used to simulate transducer sources (Figure 3a) were made of celluloid. Three of these were used, of widths .5 cm, 1.5 cm and 2 cm. These were cemented to a stiff wire which plugged into LS1, The dippers simulating amplitude grating (Figure 3b) were constructed by removing teeth from combs until the right “grating spacing” was obtained. The dippers simulating the phase grating.(Figures 3c and 3d) were constructed by running thin strips of metal through the gear cogs used for corrugating in a sheet metal shop. Various shapes, from a sine wave to a saw tooth, could be obtained by varying the types of gears and the pressure. Metal dippers used in mercury must be coated with lacquer to prevent amalgamation. Final adjustments were made with the dipper vibrating in the mercury. The best adjustment is obtained with a symmetrical dipper, perfectly level, and just touching the surface of the mercury. Since this adjustment was so critical, the loud speaker was mounted on a stand with adjustable leveling screws in the three feet. The stroboscope was easily adjusted by moving it into the light beam. Best contrast was obtained by moving it to the point where the light was almost extinguished. Tm can 3; at a. CELLULOID TRANSDUCER _mm-—~J' \. I ‘ l ' I I [v ‘ v 1 v W 1r”? _D__ I). AMPLiTu DE GRRTINC: ___.D____. /\\\J/W ToP View or Mme; Sum C. Sim: Wave PHASE Gamma W A. SRW'TOOTH PHASE GRHTWG Fm. 3. DIPPERS -12 In order to keep the mercury surface clean, occasional scumming is desirable. After each use the mercury was filtered. -13 III. BEAM PATTERNS Fresnel diffraction may be observed near a slit illuminated by parallel light. The Fresnel region in a sound field near a vibrating quartz is analogous to the interference near a slit. Using the analogy dis- cussed in the introduction, this Fresnel region can be produced by vibrating a thin, rigid dipper in water or in mercury. The width of this dipper is determined by the appropriate D/A ratio for the analogous acoustical or optical case. Since A changes with the frequency, a range of ratios D/A is available for any particular width of the dipper. Although the nature of Fresnel diffrac- tion has been thoroughly investigated, the ripple study .is very useful because this ease in changing the ratio D/A makes it possible to observe a wide range of D/A readily, and even to demonstrate the change in pattern Vi th D/A to a large audience. Fresnel diffraction for optics is well known and the intensity distribution in the Fresnel field can be calculated. However, only those structures in the optical infiterference pattern can be observed which are large cOlnpared with the wave length. The wave length of ultrasonic or of capillary waves is very large compared wi‘th the wave length of light. Therefore, even the finest details of an interference pattern produced by ultrasonic or capillary waves should be observable by -14 optical methods. Early investigators attempted to determine experi- mentally the ultrasonic intensity pattern in front of a slit or transducer. Placing a measuring device in the field disturbs the field, however. Boyle, Lehmann and Reid27 observed how small particles were driven by the radiation pressure in the sound beam and collected at points of minimum energy. This method allows one to find the direction of the main lobes of a beam pattern, but gives no finer details. The development of optical techniques for the study of sound waves permitted the observation of waves with- out disturbing them. Hiedemann and Osterhammelzs’zg, with these techniques, used long thin quartzes with light passing along the longest dimension. The resulting cylindrical symmetry enabled them to photograph sound intensity distributions which could be compared directly with light intensity distributions behind slits having the same ratios of D/A. The symmetry inherent in sur4 face waves makes it possible for one to make the same observations and analogies as did Hiedemann and Osterhammel. Some of their experimental results are illustrated in Figures 4, 5 and 6. Details of the waves themselves are shown in the stroboscopic pictures, Figures 4a and 5. The complete pattern of the beam can best be seen in Figure 4b, which has the same D/A used geflr Q. \ ‘ “ \- 5—. a. Stroboscopic picture b. Patterns of lobes. of wave fronts. D = 4A. D 2 4.2%" Figure 1. Interference Pattern from a Long Thin Source. ” Figure é. Stroboscopic Elyure Q. Interference Picture of Interference Pattern Showing Intensity Pattern from a Transducer. Distribution Close to the D : 3.4%. Transducer. D : 12.8A- -16 for the stroboscopic picture, Figure 4a. This picture shows the features far from the source, as well as the near field. Figure 6, obtained by Osterhammelg0 shows the details close to the source of maximum intensity. From these pictures, one can see the intensity distri- bution both in the Fresnel and the Fraulnhofer regions and how the features change with a change of the ratios Dfl\. The theoretical prediction for an extended beam pattern is illustrated in Figure 7. Much of the beam pattern can be observed at low intensities for certain ratios BAA in the ripple tank. At higher intensity, the fine structure of the pattern close to the source is made apparent. Osterhammel analyzed this pattern for Fresnel