Diffraction and Interference
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Wo01:Diffraction And Interference Abdel Darwish 23 November 2015
An investigation into different cases of diffraction and interference of waves in two media, waves waves in a phy physical sical medium, shallow water inside a ripp ripple le tank and electr electromagn omagnetic etic wa wave ves, s, las laser er light light mount mounted ed on an optical optical rai rail. l. Both Both types types of wa wave vess obey the same same 1D wave wave equation sho shown wn on the right and hence are comparable. comparable. This paper prov proves es both obey the same fundamental geometry concerning young’s double slit interference pattern to an average percentage error of ±5.15% with the uncertainty being ±23.60% for the ripple tank and ±1.2848% and ±4.4273% for the laser single slit and double slit minima respectively within my uncertainty of ±5%. Drawing clear parallels between the two seemingly very different types of waves.
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In Intr trod oduc ucti tion on
Waves in a ripple tank can also be modelled like this, hence however different the two may seem they An electromagnetic wave is a transverse wave consist- follow the same fundamental physics. ing of perpendicular synchronised oscillations of electric and magnetic fields which propagate at the speed Wav aves es in a rip ripple ple tank tank are als alsoo transv transvers ersee wa wave vess of light (c = 3.00 × 108 1 0 m s 1 ) through a vacuum consisting of the the vertical oscillation of the water in a direction perpendicular to the two fields, shown molecules and the wave velocity being perpendicular visually in Figure 1. along the surface. −
In 1678 1678,, Huyg Huygen enss pr prop opos osed ed hi hiss pr prin inci cipl plee th that at ”every point which a luminous disturbance reaches becomes beco mes a source source of a spheri spherical cal wave; wave; the sum of these the se second secondary ary wa wave vess determ determine iness the for form m of the wave at any subsequent time” [2], this leads to the diffraction and interference effects to be investigated in this experiment.
:Magnetic Figure Figu re 1: [1] Electromagnetic wave, B :Electric :Velocity. component, E :Electric component and V
As the two field oscillations are in sync, the following will only consider the equivalen equivalentt a 1-dim 1-dimension ensional al wave.
Wav aves es can either either interfer interferee constructi constructively vely or destructively, due to their waves being either in phase or π out of phase respectively respectively.. Figure Figure 2 shows shows how ∂ 2 ψ 1 ∂ 2 ψ = (1) single slit diffraction can lead to an interference pat∂x 2 v 2 ∂t 2 tern. n. Huygen Huygenss pri princi nciple ple allows allows the single single slit to be (1) [4] shows the general wave equation for a 1- ter dimensional wave with ψ :wave function, v :wav :wavee ve- modelled as two sources whose waves constructively locity and x and t having their usual meaning, this interfere where 1,2 meet and destructively interfere where 3,4 meet creating this pattern. equation equat ion then implies implies (2) This This le lead adss to equa equati tion on (3 (3)) [5] [5],, with with I :Intensity, :Intensity, v = f λ (2) I 0 :max intensity, a:slit width, λ:wavelength of inci1
see, the sin2 α is the single slit envelope function with the cos2 δ being b eing the underlying underlying function. function.
(4) The sma small ll angle angle appro approxim ximati ation on (5) allows allows us to simplify simpli fy equations (3) and (4). Howeve Howeverr for | for | θ | ≈ 0.2 the predictions will incur a maximum relative error for the widest angles of 0.6% which I consider acceptable. sin θ ≈ x
Figure Figu re 2: Wave geometry behind the single slit
(5)
diffraction pattern [3]
x =
dent light and θ :angle of diffraction
nλ , a
x =
(3) Double slit interference shown in Figure 3 is slightly more complex and due to the a superposition of the single slit’s interference pattern and the interference between the two slits themselves creating the more complex interference pattern of a single slit pattern envelope function on the interference pattern of the two slit’s interference.
n ∈ Z , n =0
(n + 12 )λ d
,
(6)
n ∈ Z
(7)
Equation (6) shows the position of the dark bands for single slit diffraction pattern, and Equation (7) the position of the dark bands for the double slit. These were derived from the given formula (3) and (4) by equati equating ng eit either her the sin2 α or the cos2 δ to zero and solving for x . These will allow quantitativ quantitativee comparison with my experimental data.
v =
gλ 2π
(8)
Equation (8) shows the relation betw Equation b etween een v :wave velocity veloc ity and λ:w :wav avee length length where where is g :acceleration duewaves” to gravit gravity This Thisurface s is thetension equati equation on ”deep water asy.then is for negligible, we are on boundary however for comparison it will be useful. This experiment will attempt to: 1. Compa Compare re the interference interference patterns patterns of a laser for single and double slit to the predicted interference pattern given by the functions (3) and (4) for light. 2. Explor Exploree plane wave wave speed, reflection, reflection, refractio refraction n at change of medium, and single and double slit interfere inte rference nce in a ripple tank.
Figure Figu re 3: Wave geometry behind the double slit
diffraction pattern [3]
3. Draw paralle parallels ls and differences differences betw b etween een the two two types of wa waves ves resulting resulting interfere interference nce patterns patterns and compare to functions (3) and (4).
This leads to equation (4) [5] with the same meanings as before and with d :slit separation. As you can 2
2 2. 2.1 1
Ex Exper perim imen enta tall me meth thod od
the the ba bacckgro kgroun und d re read adin ingg whic which h ca can n then then la late terr be subt subtra ract cted ed from all our our da data ta.. This This will will ei eith ther er be perfo per form rmed ed just just befor beforee ever every y sl slit it,, ho howe weve verr if it is found that the data is the same we will no longer continue to repeat the runs.
Lase Laser r Exper Experim imen entt
The laser is mounted at the front in figure Figure 4, it will impinge on one of the slits on the rotating slit
whe wheel in se the middle, middle whi sev severa l ht slit iswidth widths Uncertainties arise from small angle approximation and and elsl slit it sepa para rati tion ons, s,, which this thischdiffr dihas ffrac acte ted deral light lig then thens (5), from the linear translator, the slit’s and the wavedetected by the light sensor mounted on the linear length len gth of the laser. laser. Howe Howeve verr all of which which should should be translator trans lator,, which which will measure the x-coordinat x-coordinate, e, at reasonable small. the end of the optical rail. Regarding safety the laser should remain mounted so not to move freely, turned off when not in use and The light sensor also has adjustable slits in front of eyes should be kept away from the laser beam itself. it to define its resolution, resolution, differ different ent resolut resolutions ions will be tested for the best data. The The se sens nsor or and and tran transl slat ator or outp output utss are are fed fed into into the interface box at the back left of Figure 4 and then the n to the PC which which will simult simultane aneous ously ly log the x-coordinat x-coor dinatee and the light light intensit intensity y. This will allo allow w the the us usee of a hi high gh samp samplin lingg rate rate (200 (20000H z ) in this ca case se,, to crea create te a ne near ar cont contin inuo uoss da data ta plot plot of the the interference pattern. The rota The rotati ting ng slit slit whee wheell pro provide videss a varie arietty of single and double slits all labeled for use with the experiment experim ent.. Informati Information on about the later lateral al position of the sensor and the intensity of light are sent to a Pasco interface box which sends the information onto the computer to be viewed by the Pasco Capstone software for export into a .dat file allowing proper data analysis through the software package Octave. A plot of the theoretical function imposed onto the experim expe rimen entt data data will be cre create ated, d, also also the minim minimaa will be recorded from the plot and taken note to also be compared to the theoretical minima in (6) and (7) .
Figure 4: Fig 3: experimental setup: Laser (front)
impinges imping es on one of the slits on the rotat rotating ing slit wheel (middle). (middle). Diffra Diffracted cted light is detected by the light sensor mounted on the linear translator (back). The light sensor xoutputs are fed into the interface box (back left) and hence to the PC.
Measuremen Measu rements ts of the following following slits will be tak taken: en: 1. 0.02mm 0.02mm Single slit slit 2. 0.04mm 0.04mm Single slit slit 3. 0.08mm 0.08mm Single slit slit 4. 0.16mm 0.16mm Single slit slit
2.2
5. a=0.04mm a=0.04mm d=0.50mm d=0.50mm double slit
Rippl Ripple e Tank Tank Experi Experimen mentt
Figure 5 shows the ripple tank to be used, this allows visualisation of the waves. 7. a=0.08mm a=0.08mm d=0.50mm d=0.50mm double slit The ripple tank consists of a shallow tray that is filled fille d with with clean clean tap water water and has a transp transpare arent nt 8. a=0.08mm a=0.08mm d=0.25mm d=0.25mm double slit ba base se that that is ill illum umin inat ated ed fr from om un unde dern rnea eath th by a Before Bef ore record recording ing an any y of the diffrac diffractio tion n patter patterns, ns, strobos stroboscop copic ic LED, LED, thi thiss has tw twoo functi functions ons,, firstly firstly,, a norma normalisa lisatio tion n run will will be perf perform ormed ed where where the as it’s light travels through the shallow tray filled linear line ar transl translato atorr will be mo move ved d acr across oss rec record ording ing wit with h wate water, r, the the lig light ht becom becomes es ei eith ther er focus focusse sed d of 6. a=0.04mm a=0.04mm d=0.25mm d=0.25mm double slit
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defocussed as it hits a trough or a peak, projecting visible dark and light bands on the drawing table and secondly, secondly, the frequency frequency is synchroni synchronised sed to the frequency frequ ency of the wave wave source, source, the dipper, allo allowing wing the ”freezing” of the waves making higher frequencies visible to the naked eye.
7. Double source standing wave Two separate vibrators will be used to obtain a standing wave that can be seen when the stroboscope is switched off. Uncertainties will arise from the measurement of
wavelength as it will be rather ambiguous defining the wavefron wavefronts, ts, this will be ov overco ercome me by countin countingg a nu num mber of wa wave vele leng ngth thss and and divid dividing ing do down wn fo forr one wavele wavelengt ngth, h, thi thiss wil willl als alsoo divide divide the error to bring bring it do down wn to a manage manageabl ablee magnit magnitude ude.. Errors Errors will also arise whilst obtaining the scaling factor as measurements will be done with a ruler who’s error be ±00.005m however this is negligible compared One is also able to change the shape of the will be ± dippe di pper, r, or add obstac obstacle les. s. This This will allow allow for for the the to the uncertainty in defining wavelengths. following experiments: Us Usee of a ca came mera ra moun mounte ted d dire direct ctly ly abov above allo allows ws quantitat quan titative ive measureme measurement nt of the wa wavel velength ength,, done by placing an object of known length into the tray, taking a picture and thus allowing the calculation of the equivalent equivalent length of a pixel.
Thiss is a rel Thi relati ative vely ly saf safee experim experimen ent, t, ho howe weve verr a 1. Plane wave and the speed of water waves small risk is present when working with water around Using the long straight dipper and ensuring it is electronic ronics. s. Reasonable Reasonable precaution precaution should be taken taken in the water evenly, the frequency of the vibra- elect tor and hence the stroboscope will be selected selected at throughout. a reasonable starting value and increased by a set value after each photo is taken for a reasonable range. range. The exact range range and intervals intervals will be chosen for best results. 2. Reflection A barrier will be placed inside the tray at an angle to the dipper and a photo taken. Analysis of the photo will allow for the confirmation that the angle of incidence is equal to the angle of reflection. Figure 5: Fig 2: PHYWE model ripple tank with
3. Refraction at a change in medium This will be achieved by placing an acrylic slab fully submerged in the water tray, as in water wave velocity depends on depth. Slabs of different planar shapes will be tried.
labelled components
3
4. Single slit interference A barrier with a gap will be created by use of two smaller barriers, and the effects will be photographed for different gap widths but also different fere nt frequencies frequencies for the same gap width.
3.1
Dis iscu cuss ssio ion n Laser Laser experi experimen mentt
Regarding the single slit laser diffraction patterns the resultss are very result very pleasing. As seen in figures figures Figure 6, Figure Fig ure 7, Figure Figure 8 and Figure Figure 9, qualita qualitativ tively ely the obtained data for single slit laser diffraction fit very well to their theoretical plots given by Equation (3). 5. Double slit interference Barriers will be placed int the water tray to pro- There are only 2 issues, as seen in figures Figure 6 and Fig Figur uree 7, the the se sens nsor or becom becomes es sa satu tura rate ted d and and duce a double slit and the resulting diffraction and does not reach the peak intensity, this is purely an pattern patte rn will be photograph photographed ed aesthetic issue and has no negative effect on the data obtain obt ained ed for the minima, minima, second secondly ly,, the minima minima in 6. Double source interference A do doub uble le di dippe pperr will will be used used to produ produce ce the the Figure 8 and Figure 9 don’t reach zero intensity as the theory predicts even after normalisation, this is equivalent double source diffraction pattern. 4
due to the resolution of the sensor, to reach zero the sensor resolution would have to be much less of what it was but again this is purely an aesthetic issue and has no negative effect on the data obtained for the minima.
in Table Table 12. Then Then usi using ng octave octave I plotte plotted d the data data with wit h it it’s ’s line line of best best fit fit,, de deri rivi ving ng an equa equati tion on fo forr the the cha hang ngee of wa wave ve speed speed wi with th fr freq eque uenc ncy y sh show own n on Fig Figure ure ??. This This does does not not at al alll co comp mpar aree to the the de deep ep wa wate terr equa equati tion on gi give ven n in (8 (8)) ha havi ving ng an opposit oppo sitee gradie gradient nt to the predic predicted ted,, clearl clearly y thi thiss is more complex and further study would be required over a larger range of frequencies .
Rega Regard rdin ingg the the do doub uble le sli slitt lase laserr diffr diffrac acti tion on the the results are also very pleasing as seen in Figure 10, Figure Fig ure 11, Figure Figure 12 and Figur Figuree 13, ho howe weve verr not so much much as for the single single slit. Qualita Qualitativ tively ely,, the they y follow the theoretical plot given by Equation (4)but as with the single slits, the resolution of the sensor is a limiting factor as minima are not as well defined as the theory predicts, for all except Figure 11, this was not an issue however for Figure 11 the minima were too close for the sensor’s resolution and some ov overlappe erlapped, d, making making it very very difficult difficult to differe differentia ntiate te between them and to know what nth order minima I was looking at. Ambiguit Ambiguity y in the defining the position of the minima lead to uncer uncertain tainties ties sometime sometimess larger than the value itself so I considered it useless attempting to record them as you can see in Table 7. Recording of the major minima (this is the single slit envelope) was also very difficult to define where it was due to their shallow shape, hence why I rarely record rec orded ed them. This This could could be ov overc ercome ome by using a linear translator and sensor of both higher resolution and precision.
For pa part rt tw two, o, re refle flect ctio ion, n, I was was su succ cces essf sful ully ly able able to confirm that angle of incidence = angle of reflection to to an average error of 6 .6%, however as you can see see in ta tabl blee Table able 13, 13, it is only only the the se seco cond nd re resu sult lt that contributes to this error, which was due to the the fact that the boundary was nearly parallel to the dipper meaning the wa wave ve fronts began to interfer interferee with each other, distorting the measurements. For part three, photographs are in my lab book, the effects effe cts can cle clearl arly y be observ observed ed of both refrac refractio tion, n, where the wavelength increases and the effects of the lens shaped acrylic. acrylic. Pa Part rt fo four ur,, si sing ngle le sl slit it,, ac achi hiev eved ed no vi visib sible le re resu sult lt,, with only with only a diffra diffract ctio ion n pa patt tter ern n bu butt no vi visi sibl blee in in-ter terfer ferenc encee patter pattern. n. Theore Theoretic ticall ally y thi thiss is because because they are all compressed to very small angles (≈ ( ≈ 1 ), attained from the use of equation (6). ◦
Part art five five, dou oubl blee sl slit it in intter erffer eren encce, as ca can n be seen seen in my la lab b book, book, th thee phot photos os sh shoow no cl clea earr As seen in table Table 5 for single slit laser diffrac- interference pattern, it can be seen they do in fact tion, the percentage error has an average of ± of ±1.2848% interfere but none obvious enough to take any form measur sureme ement nt,, thi thiss is due to a combin combinati ations ons of and a maximum error of ±1.8245%, this is largely due of mea to the resolution of the sensor itself which was 0 .001m reasons, one being that the waves loose allot of their allowing me toof take in the±measurement of the position theuncertainty minima as being 0.0005m, this accounts for all of the errors. As seen in table Table 10 for double slit laser diffraction, the percentage error has an average of ± of ±4.4273% and a maximum error of ±6.4554%, the percentage error is larger simply due to the considerably smaller values of the minor minima, however as before these errors still lie within the uncertainty of our measurements.
energy having impinged the make barriers thus are much less visible, alsoonto you can outand another two wave sources on the sides as the barrier does not stop all the wave, leading to a even less visible diffraction pattern.
Part art si six, x, do doub uble le so sour urce ce in inte terf rfer eren ence ce,, whic which h is analogous to the double slit geometry due to Huygens?Fresnel principle and hence as you can see in the picture and in table Table 11, this showed clear interfer inte rference ence patterns patterns which which followed followed equation equation (7) to a maximum percentage error of 8% which when 3.2 Rippl Ripple e tank tank experim experimen entt compar com pared ed to the percen percentag tagee uncert uncertain ainty ty ha havin vingg a maxim max imum um of 44.64% 64% is ac acce cept ptab able le as you you can can se seee Part Part one of the experimen experiment, t, pla plane ne wa wave vess and the in Table able 11. 11. The The aver averag agee perce percent ntag agee er erro rorr wa wass speed of water waves, the speed of the water waves was was ca calc lcula ulate ted d fo forr 5 diffe differe rent nt wave wavele leng ngth thss to a 5.15% and with the uncertainty being 23.60% this is acceptable. maximum percentage uncertainty of of ±4% seen here 5
4
Co Conc nclu lusi sion onss
References
https://commons.wikimedia.o ns.wikimedia.org/wiki/File: rg/wiki/File: In conclusion of my ripple tank results, I can say I [1] https://commo Onde_electromagnetique.svg (23 (23 Nove Novembe mberr was able to derive an expression relating wavelength 2015). and wave speed in a ripple tank to an uncertainty of ≈ 4%, I was successfully able to confirm that angle http://www.atoptics.co.uk/d optics.co.uk/droplets/ roplets/ [2] http://www.at of incidence = angle of reflection to to an average error huygens.htm of 6.6% which is within it’s uncertainty, successfully explored explo red refraction refraction of waves waves and diffraction. diffraction. I was [3] http://hyperp http://hyperphysics.phy-astr.gsu.ed hysics.phy-astr.gsu.edu/ u/ also able to confirm the equation (7) to an average hbase/ percentage error of 5.15% with the uncertainty being [4] 1. Hecht E: Optics. Addison-W Addison-Wesley esley (Second Edi23.60%. tion, 1987). My la larg rges estt sour sourcces of erro rror were sim simply ply the [5] Pedrotti FL and Pedrotti LS: Introduction Introduction to Opambig am biguit uity y behind behind the measur measuring ing me method thod,, a mo more re tics. Prentice Hall (Second Edition, 1996). systematic approach should be devised next time in tackli tac kling ng this. For the wave wave speed, speed, an error could could [6] Young HD: Universit University y Physics. Physics. Addison Addison Wesley have been calculated by turning the stroboscope off (Eighth Edition, 1992). and recording the waves, to later calculate their real velocity. Results In conclu conclusio sion n of my laser laser res result ults, s, con confirm firmed ed the 5 equations (7) and (6) for the position of minima to Laser Experi Experimen mentt an error of just ±1.2848% and ±4.4273% respectively 5.1 Laser again within my average percentage uncertainty of 5%. Confirmed Confirmed the theoretical theoretical functions functions (3) and (4) qualitatively against my experimental plots.
I do not feel much improvement is necessary for this experiment however, some slit widths could have used a sensor with a much finer aperture and therefore a linear translator of also a higher precision. The need for this can be clearly seen as the data points tend to bunch up into vertical lines of varying intensities however the same position, clearly the linear translator is not registering the movement. However this was only a limitations for very densely packed minima. Finally, I was able to draw clear parallels between wa waves ves in a ripple tank and electromagnet electromagnetic ic wa waves ves,, clarifying that they both obey the same basic geometry and theory. This section should contain a brief summary of the experiment and a report of your MAIN results and conclusions. In practice, the conclusion is a reworded version of the abstract. However, it may also contain a brief discussion of key limitations of the current experiment and possible improvements which could be made. LARGE ARGER R RANG RANGE E OF FREQ FREQ FOR WAVE SP SPEE EED D AND AND RECO RECORD RD TH THE E WAVES VES WI WITH TH STROBE STRO BE OFF 6
0.02mm single single slit slit las laser er diff diffrac ractio tion n patter pattern, n, Red points :Experiment mental al data point points, s, Blue Figure 6: 0.02mm Figure points:Experi line:Theoretical function plot
0.04mm single single slit slit las laser er diff diffrac ractio tion n patter pattern, n, Red points :Experiment mental al data point points, s, Blue Figure Fig ure 7: 0.04mm points:Experi line:Theoretical function plot
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0.08mm single single slit slit las laser er diff diffrac ractio tion n patter pattern, n, Red points :Experiment mental al data point points, s, Blue Figure 8: 0.08mm Figure points:Experi line:Theoretical function plot
Figure Figure 9: 0.16mm points:Experi 0.16mm single sinplot gle slit slit las laser er diff diffrac ractio tion n patter pattern, n, Red points :Experiment mental al data point points, s, Blue function line:Theoretical
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Figure Figu re 10: Double slit laser diffraction pattern (a=0.04mm, d=0.25mm), Red poin points ts:Experimental data points, Blue line:Theoretical function plot
Figure Figu re 11: Double slit laser diffraction pattern (a=0.04mm, d=0.25mm), Red poin points ts:Experimental data points, Blue line:Theoretical function plot
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Figure Figu re 12: Double slit laser diffraction pattern (a=0.08mm, d=0.25mm), Red poin points ts:Experimental data points, Blue line:Theoretical function plot
Figure Figu re 13: Double slit laser diffraction pattern (a=0.08mm, d=0.50mm), Red poin points ts:Experimental data points, Blue line:Theoretical function plot
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Figure 14: Normalization data set for laser
11
6
s d n a b k r a d r e s a l t i l s e l g n i s m m 2 0 . 0 : 1 e l b a T
0 5 9 1 . 0
6
5 7 9 0 . 0
6
8 8 4 0 . 0
4 4 4 4 2 2 0 0 . . 6 0 0
5 2 6 1 . 0 5 7 0 9 0 2 3 1 . 1 . 4 0 0
5
3 1 8 0 . 0 0 5 6 0 . 0
2 0 0 6 4 4 0 . . 0 0 0 5 3 5 2 2 3 3 0 . . 0 4 0 0
3 0 3 0 2 2 0 . 0 . 0 0 5 4 3 6 6 1 1 0 . 0 . 4 0 0
5 7 5 7 9 9 0 0 . . 3 0 0
9 8 8 8 4 4 0 0 . . 3 0 0
2 4 4 4 2 2 0 0 . . 3 0 0
4 2 2 2 1 1 0 0 . . 3 0 0
9 0 4 5 6 6 0 0 . . 2 0 0
7 5 2 2 3 3 0 0 . . 2 0 0
1 3 6 6 1 1 0 0 . . 2 0 0
4 1 8 8 0 0 0 0 . . 2 0 0
5 5 2 2 3 3 0 0 . . 1 0 0 1 2 2 5 3 3 0 0 . . 1 - 0 - 0 3 4 0 5 6 6 0 0 . . 2 0 - - 0 6 5 5 7 9 9 0 0 . . 3 - 0 - 0 9 7 0 2 0 3 1 1 . . 4 - 0 - 0 -
5 -
2 5 6 1 . 0 0 5 9 1 . 0 -
6 ) m ) ( m x ( l x a l t n a e i c t m i r e r e o p e x T n E h
4
s d n a b k r a d r e s a l t i l s e l g n i s m m 4 0 . 0 : 2 e l b a T
2 3 6 6 1 1 0 0 . . 1 0 0 5 8 6 3 1 1 0 0 . . 1 - 0 - 0 3 2 5 2 3 3 0 0 . . 2 0 - - 0 1 8 8 8 4 4 0 0 . . 3 - 0 - 0 3 4 0 6 5 6 0 0 . . 4 - 0 - 0 -
5 -
s d n a b k r a d r e s a l t i l s e l g n i s m m 8 0 . 0 : 3 e l b a T
2 1 8 8 0 0 0 0 . . 1 0 0 8 1 1 8 0 0 0 0 . . 1 - 0 - 0 7 6 5 3 1 1 0 0 . . 2 0 - - 0 2 4 4 4 2 2 0 0 . . 3 - 0 - 0 9 2 1 5 3 3 0 . . 0 4 - 0 - 0
s d n a b k r a d r e s a l t i l s e l g n i s m m 6 1 . 0 : 4 e l b a T
2 1 4 4 0 0 0 0 . . 1 0 0 4 4 1 0 0 0 0 . . 1 - 0 - 0 3 8 1 8 0 0 0 0 . . 2 0 - - 0 1 2 2 2 1 1 0 0 . . 3 - 0 - 0 2 6 3 6 1 1 0 . 0 . 4 - 0 - 0 -
1 3 8 0 . 0 -
6 1 0 4 4 0 0 0 . . 5 - 0 - 0
0 3 0 3 2 2 0 0 . . 5 - 0 - 0 -
5 7 9 0 . 0 -
8 8 4 0 . 0 -
4 4 4 4 2 2 0 0 . . 6 - 0 - 0 -
6 ) m ) ( m x ( l x a l t n a c e i t m i e r e r e p o h x T n E
6 ) m ) ( m x ( l x a l t n a c e i t m i e r e r e p o h x T n E
12
) m ) ( m x ( l x a l t n a e i c t m i e r e r e p o x T n E h
r e s a l t i l s e l g n i S , r o r r E e g a t n e c r e P
: 5 e l b a T
) % ( r o r r 5 4 8 7 4 5 8 1 6 4 E 2 5 1 4 e 8 2 1 0 1 . . . . . g 1 1 1 1 1 a t n e c r e P ) m m e ( g h a 6 8 4 t 2 r 1 0 0 0 . . . . d i 0 0 0 0 e v A W t i l S
: 6 e l b
a T
6 9 5 1 9 1 0 . 0 . 8 0 0
0 5 6 0 . 0
3 1 8 0 . 0
8 1 7 1 1 1 0 0 . . 5 0 0
6 0 4 0 . 0
5
6 0 4 0 . 0
5
7 1 1 0 . 0
9 5 0 0 . 0
1 9 0 0 .
0 5 6 0 .
1 9 0 0 .
5 2 3 0 .
4 6 4 4 0 0 0 0 . . 4 0 0
5 2 3 0 . 0
8 8 4 0 . 0
4 0 0 4 4 2 0 . 0
4 2 0 . 0
3
5 6 0 0 . 0
3 3 2 0 0 0 0 . . 3 0 0
3
5 6 0 0 . 0
0 9 4 0 3 0 0 0 . . 2 0 0
3 6 1 0 . 0
3 6 1 0 . 0
2
9 4 5 3 0 2 3 2 3 0 0 . 0 . . 0 0 0
9 2 1 0 0 0 0 0 . . 2 0 0
3 1 1 2 3 6 0 6 1 0 0 s . 1 . . d 0 0 0 n
6 1 3 1 0 0 0 0 . . 1 0 0
1 8 0 0 . s 0 d n
7 7 0 0 0 0 0 . . 0 1 0 0
1 8 0 0 . 0
1 -
7 5 3 0 0 2 6 6 1 0 0 . 1 . . 0 - 0 - 0 -
0 3 1 0 1 0 0 0 . . 1 - 0 - 0 -
1 8 0 0 . 0 -
6 0 0 7 0 0 0 . . 0 1 - 0 - 0
1 8 0 0 . 0 -
2 -
0 7 5 2 0 2 3 2 3 0 . 0 . 0 . 0 0 - - 0 -
8 9 3 0 3 0 0 . 0 . 2 0 - - 0 -
3 6 1 0 . 0 -
0 2 2 0 0 0 . 0 . 0 2 - 0 - 0
3 6 1 0 . 0 -
3 -
3 3 0 0 . 0 -
8 8 4 0 . 0 -
3 -
5 6 0 0 . 0 -
4 4 2 0 . 0 -
1 3 3 3 0 0 0 0 . . 3 - 0 - 0
4 4 2 0 . 0 -
4 -
6 4 0 0 . 0 -
0 5 6 0 . 0 -
4 -
1 9 0 0 . 0 -
5 2 3 0 . 0 -
6 4 4 0 0 4 0 . . 0 4 - 0 - 0
5 2 3 0 . 0 -
5 -
9 5 0 0 . 0 -
3 1 8 0 . 0 -
8 7 1 1 1 1 0 0 . . 5 - 0 - 0
6 0 4 0 . 0 -
9 5 0 0 . 0 -
6 0 4 0 . 0 -
6 -
2 7 0 0 . 0 -
5 7 9 0 . 0 -
5 5 9 1 1 9 0 . . 0 8 - 0 - 0
0 5 6 0 . 0 -
2 7 0 0 . -
8 8 4 0 . -
7 6 9 6 1 1 0 0 . . 7 0 0
8 3 1 1 . 0
3 3 4 4 1 1 0 0 . . 6 0 0
5 7 9 0 . 0
6 1 7 1 1 0 1 0 . 0 . 5 0 9 1 8 9 0 0 0 0 . . 4 0 0
3 1 8 0 0 . 0 5 6 0 . 0
4 0 0
4 5 6 6 0 0 0 0 . . 3 0 0
8 8 4 0 . 0
7 8 5 9 2 3 2 3 3 0 3 s 0 s 0 0 0 0 d . . . d . n 2 0 0 0 0 n a a
b k r a d r e s a l t i l s e l b u o d m m 5 2 . 0 = d m m 4 0 . 0 = a
5 7 9 0 . 0
6
6
3 4 1 0 . 0
2 7 0 0 . 0
1
b
4 3 3 3 k r 6 1 6 1 a 1 0 1 0 d 0 0 0 0 . . . . 1 0 0 0 0 r e 9 3 3 5 0 1 6 0 1 1 0 6 0 . 0 . 0 . . 0 1 0 0 0 - - - - 0 5 2 4 3 9 3 2 5 0 0 3 3 0 0 0 0 . . . . 2 0 0 0 - - - - 0 0 5 6 6 0 0 0 0 . . 3 - 0 - 0 -
8 8 4 0 . 0 -
4 1 8 9 0 0 0 0 . . 4 - 0 - 0 -
0 5 6 0 . 0 -
0 1 1 1 7 1 0 . 0 . 5 - 0 - 0 -
3 1 8 0 . 0 -
e p o l t e i l v i s t n l s e e t l i e t i b l s l l u g s e i o l n e D b s l g u l l n a o a i t t s d n d e n d e e e m t m t i i c c r r i e i d e d p x P e r e r E p x P n E
s a l t i l s e l b u o d m m 0 5 . 0 = d m m 4 0 . 0 = a : 7 e l b
a T
e p o l t e i l v i s t l e s n e t l i t e b l i l s g l u e i o l n s e D b s l g u l l n a o a i t t d s n e d n e d e m t e m t i i c c r r i e i d e d x p P r e E x p P r e n E
a b k r a d r e s a l t i l s e l b u o d m m 5 2 . 0 = d m m 8 0 . 0 = a : 8 e l b
a T
e p o l t e i l v i s t l e s n e t l i t e b l i l s g l u e i o l n s e D b s l g u l l n o a a i t t d s n n e d e d e m t e m t i i c c r r i e i d e d r e E x p P r e x p P n E 13
a b k r a d r e s a l t i l s e l b u o d m m 0 5 . 0 = d m m 8 0 . 0 = a : 9 e l b
a T
5 -
8 8 4 0 . 0
6 ) ) m m ) ) ( ( m m a ( a ( m a m a i i n i n m m i i i n n m i m i r r m o m o r j n o a r i o n m j m i a l m l m a a t t l l n a a e i e i c n c t t m m i e i e r e r e r o r o e E x p h e x p h n E T T
r e s a l t i l s e l b u o D , r o r r E e g a t n e c r e P
: 0 1 e l b a T
) % ( r o r r E e g a t n 3 2 6 4 0 e 7 3 0 5 0 c r 2 2 3 0 5 4 1 1 0 e 4 . . . . . P 6 0 6 5 4 ) m m 5 0 5 0 ( 2 . . 5 . 2 . 5 d 0 0 0 0 e ) g a m r 8 8 4 4 e m ( 0 0 0 0 v . . . . a 0 0 0 0 A
5.2
Rippl Ripple e Tank Tank Experim Experimen entt
14
% y t n i a t r e c n U e g a t n e 6 4 9 4 c 7 0 4 r . 4 . 3 . 6 . 3 e 0 . 4 4 1 . 5 P 8 1 5 4 1 7 k n a t e l p p i r e c r u o s e l b u o d m 6 6 4 0 . 0 = a : 1 1 e l b a T
% r o r r E g e a t n e c 2 7 8 0 0 5 r e 6 . . 7 . 6 . 0 . 1 . 7 P 2 2 6 8 6 4 ) ◦ ( e l g n A l a t n e m i r 6 3 0 2 3 e 0 . 7 . 7 p 5 . . 2 . 8 . 2 2 x 6 1 7 4 1 9 3 E - - - 1 3 6 ) ◦ ( e l g n A l a c i t e r 7 7 6 7 7 6 o . 3 . 3 . 6 e 1 . . 1 . 6 0 2 h 4 4 0 2 1 3 6 T - - - 1 3 6 - 1 2 3 - 1 - 2 n 3
% y t n i a t r e c n U e g a t 9 n 3 e 8 c 2 r . e 0 P 3 3 3 3 4 + λ 5 ) 4 1 3 2 . 8
−
=
v
i n o t a u q e g n i v i g k n a t e l p p i r r o f d e e p s e v a W : 2 1 e l b a T
−
s m
±
( y t i n a t 4 0 6 r 2 8 7 e c 6 0 6 0 0 8 0 8 0 0 0 0 n 0 . . . 0 . . U 0 0 0 0 0 )
1
−
s m (
d e e p s 3 9 1 9 e 5 2 1 v 0 1 1 2 1 3 5 3 a 2 2 2 2 . . . 2 . . W 0 0 0 0 0 ) m ( h t g 2 3 3 3 3 n e 0 - 0 - 0 - 0 - 0 l e E E E E E v 1 5 0 0 0 a 0 . 4 . 4 . 6 . 9 . W 1 8 7 6 5 )
z H (
y c n e u q e r 0 5 5 0 F 2 2 0 3 3 4
15
n o i t c e fl e r f o e l g n a d n a e c n e d i c n i f o e l g n A
: 3 1 e l b a T
) ◦ ( n o i t c e
fl e R f o e 7 4 0 l g 0 . . 0 . 2 n 6 8 4 A 5 7 5 ) ◦ ( e c n e d i c n i f o e 0 3 0 l g 0 . . 0 . 1 n 6 5 3 A 5 7 4
Figure 15: Wavelength plotted against Wavespeed for ripple tank, Red poin Figure points ts:Experimental data points, Blue line:Line of best fit with function v = −8.2345λ + 0 .2839
16
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