SPH4U Transmission of Waves in One and Two Dimensions LoRusso

Transcription

1 Waves travelig travellig from oe medium to aother will exhibit differet characteristics withi each medium. Rules A wave of fixed frequecy will have a shorter wavelegth whe passig from a fast medium to a slower medium. The diagram below represets the crests of water waves movig from deep water to shallow water top view. Faster medium Slower medium f costat Wave frots are further apart Wave frots are closer Waves will reflect at the boudary of a medium i oe of two ways. Free-Ed Reflectios Fix-ed Reflectios Free-ed Reflectios: Ca be represeted by water waves reflectig agaist a break-wall. Fixed-ed Reflectios: Ca be represeted by waves travelig through a guitar strig.

2 Waves movig from a fast medium to a slow medium will experiece a reflectio as illustrated below. The pulse travelig from the faster medium sees the boudary as a fixed ed ad reflects accordigly. Waves movig from a slow medium to a fast medium will experiece a reflectio as illustrated below. The pulse travelig from the faster medium sees the boudary as a free ed ad reflects accordigly.

3 Two Dimesioal Waves Waves experiece rectiliear propagatio (trasmissio i a straight lie) provided the wave does ot ecouter a barrier or move ito aother medium. Types of frots. Circular Liear Radom Reflectio of two dimesioal waves Straight o Collisio Off-Agle Collisio Agle of icidece equals agle of reflectio

4 Refractio of Waves Whe a wave travels from a oe medium to aother, the wave will experiece refractio. Refractios is the bedig of a wave away or toward the perpedicular lie that defies the boudary betwee the two medium. For example: the distortio of the image of objects that are partially submerged i water. The apparet bedig is due to refractio. i is the agle of icidece R is the agle of refractio Faster medium Note: The agle of icidece ad refractio is agle measured betwee the wave ray (directio vector) ad the ormal. Upo closer aalysis, you will also ote that the agle of i icidece ad refractio ca be measured betwee the boudary lie ad the wave frots themselves as illustrated i the ext diagram i R slower medium R ormal i h

5 Sell s Law sii ad h si R rearragig h h sii ad h si R si i si R si i si R si i si R but v v f si i v f so Therefore v f si R v Example: A wave of wavelegth of m is travellig through a medium at.0m/s. The wave the moves ito a secod medium where the wave speed is reduced to.0m/s. The wave 0 eters a the medium at a agle of 0 to the ormal, a) What is the agle of refractio? b) What is the wavelegth of the wave i the secod medium? 0 0

6 Dispersio As discussed earlier, waves of the same frequecy travel at differet velocities i differet media. It is assumed however that the wave velocity is costat for all frequecies withi the same medium. This however is ot etirely accurate. There ca be a slight differece i wave speed withi the same medium, for differet frequecies. There are some very cocrete ad observable examples of this pheomeo. ) The Prism: The prism is a example of dispersio with light. As you kow whe light eters a prism, it is split ito its compoets, rederig the full spectrum from red to violet. This is caused by a slight discrepacy i the speed of light, withi glass. Violet light travels more slowly tha red light. As a result, violet light will refract more that red light, rederig the characteristic spectrum. ) Thuder: Durig a thuderstorm, the effect of dispersio ca be best observed for lightig strikes that are some distace away. It is ot however the light that experieces the most dramatic effects of dispersio, it is the soud. Typically whe lightig strikes, the crack of thuder is heard a few momets later, soo followed by a deep rumble. This differece i the soud is t heard for very close strikes. Those produce a sharp, ad violet explosive soud. The reaso for the differece is that speed of soud is depedet o both frequecy ad soud itesity. Thuder is a explosio, essetially causig a major disruptio i the medium. Uder these coditios, higher frequecies travel more quickly tha lower frequecies, hece the crack followed by the rumble. Total Iteral Reflectio Recallig from last years, waves travelig from a slower medium to a faster medium will experiece total iteral reflectio. Total iteral reflectio occurs whe the critical agle is exceeded. The critical agle is defied as the icidet agle that results i agle of refractio 0 of 90 i R sii v si90 v v sii v or sii Examples: ) A wave travelig at a speed of 40 m/s eters a secod medium. If the speed i the secod medium is 65m/s, fid the critical agle. ) Fid the critical agle betwee two mediums if the wavelegth of ay give wave is icreased by 40% whe travelig i the faster medium.

7 Iterferece Patter From Two Poit Sources Represets the crests Represets the troughs The diagram represets two poit sources geeratig waves i phase with each other. The distace betwee two cosecutive crests or troughs is oe wavelegth. The two sources are set oe wavelegth apart. Where the crests ad troughs meet, destructive iterferece occurs, resultig i odal lies that take the shape of a hyperbola. I this example, oly a few odal lies will be produced. The umber of odal lies will icrease whe: a) The frequecy icrease separatio of poit sources remais costat b) The frequecy remais costat the separatio of the poit sources icreases This pheomeo ca be predicted easily by makig the followig observatio. Pick ay poit alog the odal lies ad cout the umber of wavelegths that separate each poit source from that poit. Take for istace the poit o the odal lie to the right. This poit is 4 away but the poit o the right is 4.5 away. Therefore the path differece is A path differece of results i destructive iterferece. This will occur ay time the waves are out of phase (phase agleπ ) I geeral, Path differece ( ) Usig the diagram o the ext page we are goig to prove this relatioship Procedure:. Draw all the odal lies o the diagram ad label them,, etc. Sice the patter will be symmetrical, there will be two ad two.. Determie the path differece i wavelegths ad attempt to prove the formula.

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9 This iterferece patter geerates 8 odal lies, or 4 3 symmetrical pairs. The path differece betwee the poits located o the odal lies ad the two sources are 4 0.5,.5,.5 ad, 3.5 for odal lies,,3 ad 4 respectively. S A The above demostratio of this physical pheomeo does actually lead to a importat set of formulae that are quite importat i optics. As metioed the earlier, ay poit o ay odal lie will be exactly ( ) apart. The problem with the above aalysis is that it is very difficult to measure these path differeces accurately. Therefore aother procedure is ecessary. Cosider the followig situatio. C d θ B θ L S x P To complete the aalysis of this questio requires a few assumptios that will oly work whe P is very far away from the poit sources compared to their separatio. The reaso for this is the assumptio that must be made about S AS. We are goig to use two relatioships. AS si θ ad d si θ There is oe small problem with our assumptio about si θ, is ot a right agle d ΔS AS triagle techically. So the sie relatioship really does t apply uless we assume P is very far away. The sides of the isosceles triagle, PAB, approximately become parallel whe P is very large compared 0 to d. If we make this assumptio the S ~ AS 90, allowig us to treat the triagle like a right agle triagle. x L AS AS let s first cosider si θ d d siθ AS but AS is the path differece betwee P S ad P S. Recallig earlier path differece ( ) dsiθ ( ) or ( ) d si θ

10 ΔS EC Δ CEB A θ α θ + α S θ E α C B θ S siθ Now cosider the formula ( ) d siθ Therefore ( ) d x L ( ) d x L d ( ) θ θ Examples:. Two poit sources are separted by distace of.0cm. You pick a poit alog the first odal lie. The poit is.0 m away from the midpoit betwee the two poit sources. The poit is also 0cm away from the perpedicular bisector that separted the two poit sources. Fid the wavelegth of the wave.. Fid the wavelegth of the iterferig waves usig the above formulae ad aforemetioed techiques.

11 Diffractio We have previously discussed wave behaviour i terms of reflectio ad refractio. Waves also experiece a pheomeo called diffractio. Diffractio is the ability for waves to bed aroud barriers or to expad out from small opeigs. It s diffractio that eables us to hear soud aroud corers for example. Diffractio ad Path Restrictios Waves passig through small opeigs i barriers produce a iterestig yet uexpected effect. Straight waves eterig a small opeig will become curved whe they emerge o the other side. Diffractio is depedet o both wavelegth ad aperture size. Rules:. Wavelegth icrease, diffractio icreases. Aperture icreases, diffractio decreases For ow, it suffices to simply describe the diffractive behaviour of waves. I the ext sectio we will discuss the theories that attempt to explai diffractios.