Waves Basics. April 2001 Number 17

Transcription

1 Apl 00 Numbe 7 Waves Bascs Ths Factsheet wll ntoduce wave defntons and basc popetes. Types of wave Waves may be mechancal (.e. they eque a medum such as a o wate to popagate) o electomagnetc (whch popagate n a vacuum) Waves can be classfed as: a) Tansvese hee the dstubance s at ght angles to the decton of the wave. Examples nclude wate waves and lght (electomagnetc). wave decton Wave pulses and contnuous waves A wave pulse nvolves a shot o sngle dstubance of the medum t s tavellng n. Fo example, doppng an object n wate may poduce a wave pulse. Contnuous waves nvolve epeated dstubances of the medum. Fo example, to poduce contnuous waves n a pple tank, the dppe would have to be dpped nto the tank at egula ntevals. The est of the Factsheet wll focus on contnuous waves. The wave fomula v = f v = velocty (ms - ) = wavelength (m) f = fequency (Hz) Patcles n the wave vbate at 90 o to wave decton b) Longtudnal hee the dstubance s n the same decton, paallel to the decton of the waves. Examples nclude sound and sesmc P waves. wave decton compesson expanson (aefacton) Patcles of the wave vbate n the same decton as the wave Both longtudnal and tansvese waves can be epesented gaphcally as shown below. Exam Hnt: You wll be expected to know examples of longtudnal and tansvese waves and to descbe the dffeences between them. To see whee ths fomula comes fom, consde how fa the wave moves n one second. We know (fom the defnton of fequency) that thee ae f waves each second. Each wave s of length. So the total dstance moved n one second s f But velocty = dstance tme, so v = f Typcal Exam Queston a) What s the velocty of a wave wth wavelength 5cm and fequency Hz? [] 0.5 = = 3ms - b) A wave has a velocty of.5 ms -. Eght waves ae obseved to pass a fxed pont n seconds. Fnd ) the peod of the wave [] 8 = 0.5 ) the wavelength of the wave [] f = /T = 4 = v/f =.5/4 = 0.35 m. Glossay Ampltude: the maxmum dsplacement of a wave patcle fom ts undstubed poston. Wavelenth: the dstance between two smla ponts on a wave. Unts: metes Fequency: the numbe of waves that pass a pont n one second. Unts: Hetz (Hz) Peod: the tme taken to complete one wave cycle. Unts: second (s) The peod (T) and the fequency (f) ae elated by T = f dsplacement ampltude wavelength tough peak poston Peak(o cest): the pont of maxmum dsplacement the hghest pont Tough: the pont of mnmum dsplacement the lowest pont

2 Waves bascs Wave popetes All waves wll undego the followng pocesses accodng to the same laws: eflecton efacton ntefeence dffacton Reflecton and ts laws Thee ae two laws of eflecton. They apply to both plane (flat) and cuved mos (o othe eflectng suface fo waves othe than lght). the and eflected ay le n a sngle plane whch s pependcula to the suface at the pont of ncdence angle of ncdence () s equal to angle of eflecton () (90 o to suface) eflected ay Refactve ndex Consde a wave passng fom mateal to mateal. In mateal ts angle to the s and n mateal t s. Fom Snell s Law, we know that sn = a constant sn Ths constant s the efactve ndex of mateal wth espect to mateal, wtten n. So Snell s Law can be wtten as: sn = n sn mateal mateal NB: The angles ae always measued to the, NOT the eflectng suface Refacton Refacton s the change n decton of a wave as t cosses the bounday between two mateals (eg a and wate). Now suppose the ay s nstead tavellng fom mateal to mateal. Usng the above equaton, we would obtan: sn = n sn Ths gves: n n a bounday wate Wave speed, wavelength and efactve ndex Refactve ndex s also elated the the wave speed and wavelength n the two mateals. Ths s used to defne the efactve ndex. If the wave cossed the bounday n the opposte decton (.e. wate to a) then the wave decton would change n the opposte way: Refacton s govened by two laws: wate bounday a the and eflected ay le n a sngle plane whch s pependcula to the suface at the pont of ncdence Snell s Law : at the bounday between any two gven mateals, the ato of the sne of the angle of ncdence to the sne of the angle of efacton s constant fo ays of any patcula wavelength. v n v whee v = velocty of wave n mateal v = velocty of wave n mateal Ths defnes the efactve ndex. Also, snce fequency does not change dung efacton: n whee = wavelength of wave n mateal = wavelength of wave n mateal The efactve ndex s often gven elatve to a. So f, fo example, a queston tells you that the efactve ndex of glass s.50, t means that an glass =.50. Exam Hnt: Defnng the efactve ndex s commonly asked. Make sue you use the equaton nvolvng wave speeds, and defne all the tems.

3 Waves bascs Typcal Exam Queston The efactve ndex of wate s.33. The speed of lght n a s ms -. a) Calculate the speed of lght n wate. [] 8 va = vwate vwate v wate = =.60 8 ms -.33 b) State the effect of the efacton on the fequency [] none t s unchanged Tp: The speed of lght n any othe medum should always be lowe than n vacuum (o a). Use ths to check you answe f you have got too hgh a speed, you have pobably got the v a and v mateal the wong way up n the equaton Intefeence When two sets of waves combne, we use: Pncple of Supeposton The esultant dsplacement at a pont s equal to the vecto sum of the ndvdual dsplacements at that pont. To see how ths woks, we wll look at some examples:. Constuctve Intefeence Wave + Wave Ctcal angle and total ntenal eflecton When lght tavels fom a mateal wth a hghe efactve ndex to one wth a lowe efactve ndex (eg fom glass to a), t s possble fo the angle of efacton to be 90 o. Poduces In ths case, the peaks and toughs n the two waves concde, and hence enfoce each othe. glass a. Destuctve Intefeence Wave + Wave The angle at whch ths occus s called the ctcal angle, c. sn sn c Usng the equaton = n, we fnd: glassna sn sn90 But snce we ae usually gven a n glass, not glass n a, t s moe useful to wte ths as: sn c sn90 a n glass Snce sn90 o =, we have: sn c a n glass Gven that the efactve ndex of glass s.50, we can calculate: snc = c = 4.8 o.50 (3 SF) Note: ths cannot occu fo lght tavellng fom a to glass, snce the angle to the deceases when tavellng n ths decton. What happens fo angles lage than the ctcal angle? If the angle of ncdence s geate than the ctcal angle, then the ay cannot be efacted nstead t s totally ntenally eflected eflected ay glass a In fact, a cetan amount of eflecton wll always occu at the nteface, but fo angles geate than the ctcal angle, only eflecton can occu. The ncdent and eflected ay obey the laws of eflecton. 3 Poduces In ths case, the peaks n one wave concde wth the toughs n the othe, to poduce no esultant dsplacement the two cancel. 3. Wave + Wave Poduces In ths case, peaks and toughs do not exactly concde. Phase dffeence Phase dffeence s a way of measung how fa ahead one wave s of anothe (eg half a wavelength, quate of a wavelength etc). It s gven as an angle, wth a whole wavelength coespondng to 360 o. So f one wave leads anothe by a quate of a wavelength, ths s a phase dffeence of ¼ 360 o = 90 o. If the phase dffeence s zeo (example above), they ae n phase. If the phase dffeence s 80 o (= half a wavelength), they ae completely out of phase (example ).

4 Waves bascs Souces of waves ae coheent f they mantan a constant phase dffeence and have the same fequency (eg lases). A wavefont s a lne o suface n the path of a wave moton on whch all the dstubances ae n phase. It s pependcula to the decton tavel of the wave. Dffacton of lght and the sngle slt Dffacton of lght though a slt onto a sceen leads to the poducton of lght and dak fnges. The bghtness and wdth of the fnges can be epesented gaphcally: Typcal Exam Queston a) Explan what s meant by supeposton of waves. [] Waves concdng at a pont n space Dstubances add togethe,.e. supepose b) Dstngush between constuctve and destuctve ntefeence. [4] Constuctve: waves n phase as they supepose Dstubances add to gve a lage ampltude Destuctve: waves 80 o out of phase Dstubances cancel to gve zeo ampltude c) State the condtons necessay fo souces of waves to be coheent.[] Same fequency Constant phase elatonshp Dffacton When waves pass an edge of an obstacle, o though a gap, they spead out and change shape. The wavelength, fequency and velocty emans constant. The extent of the speadng depends on the sze of the gap, as shown below: bghtness cental fnge The fnge patten has the followng popetes: It s symmetcal The cental bght fnge s much bghte than the othe fnges It s twce as wde as the othe fnges The bghtness (ntensty of lght) deceases wth dstance fom the cental fnge so the oute fnges ae the fantest. The poston of the dak fnges can be calculated usng: sn = w a whee = angle subtended n the cente (see dagam) = wavelength w = slt wdth a =,, 3. (fnge numbe) Cental lght fnge st dak fnge nd lght fnge nd dak fnge Double slt dffacton Ths poduces a dffeence n ntensty wthn each bght fnge seen n the sngle slt patten. Ths esults fom ntefeence between lght fom one slt the othe ognal one-slt fnge Exam Hnt: When dawng dagams of dffacton, make sue you keep the spacng between the wavefonts the same ths shows the fequency of the waves s unchanged. Appecable dffacton only occus f the gap s no bgge than the wavelength of the wave. Vaaton n ntensty due to two slts The naowe the slt, the geate the dffacton fo a patcula wavelength The longe the wavelength fo a constant slt wdth, the geate the dffacton Ths ntefeence esults fom the fact that lght fom the dffeent slts tavels a dffeent dstance to each a gven pont on the sceen; ths s efeed to as the path dffeence. 4

5 Waves bascs At the cente of the sceen (pont A), waves fom slts X and Y have tavelled the same dstance and theefoe ae n phase, and hence ntefee constuctvely leadng to a bght fnge. Pogessve and statonay waves The waves dscussed so fa have been pogessve.e. they move n a patcula decton, tansfeng enegy along the decton the wave s tavellng. In a statonay (o standng) wave, the wave does not move n a patcula decton and enegy s stoed by the wave. slts Y X A sceen Statonay waves ae the esult of two pogessve waves of the same fequency tavellng n opposte dectons along the same lne. The dagam below shows an example of a statonay wave. B Each pont on the we can oscllate between the two postons shown. As the dstance fom the cente changes, so does the path dffeence between the waves fom each slt. At pont B, fo example, the waves fom slt Y have tavelled substantally futhe than those fom slt X. N Oscllates between these ponts N N When the path dffeence becomes half the wavelength, then destuctve ntefeence occus, poducng a dak fnge. Futhe nceases n dstance fom the sceen nceases the path dffeence untl t becomes a whole wavelength ths makes the waves n phase agan, so constuctve ntefeence occus. Futhe nceases poduce a path dffeence of ½ wavelengths gvng destuctve ntefeence agan. So the bght fnges ae poduced fom path dffeence m, and the dak fnges fom path dffeence (m + ½), whee s the wavelength and m s any whole numbe. The fnge spacng between two adjacent bght fnges s gven by D y = d D = dstance to sceen; = wavelength; d = slt spacng In between the lght and dak fnges, the ntefeence s not pefectly constuctve o destuctve, so the ntensty of the lght changes gadually. Polasaton Nomally, the oscllatons n a tansvese wave may be n many dffeent dectons. Fo example, fo a wave tavellng out of ths page towads you, the oscllatons wll be n the plane of the page, and could be left to ght, up and down, dagonally etc. Tansvese waves may undego polasaton. A polased wave oscllates n one decton only. Longtudnal waves cannot be polased because the oscllatons ae aleady n one decton only. Exam Hnt: Ths s a key dffeence between longtudnal and tansvese waves, and s often asked. Polased lght s most easly poduced usng a pece of Polaod (as used n sunglasses). Polaod woks by only allowng though lght whch oscllates n a patcula decton A The nodes (maked N) neve move. These occu whee the two ognal waves ntefee destuctvely The antnodes (maked A) ae ponts of maxmum dsplacement. They occu whee the two ognal waves ntefee constuctvely. Table below compaes pogessve and statonay waves. Table. Pogessve and statonay waves Statonay wave Pogessve wave Stoes vbatonal enegy Tansmts vbatonal enegy Ampltude vaes Ampltude s constant All ponts between any two adjacent nodes ae n phase Phase vaes smoothly wth dstance along the path of the Nodes ae half a wavelength apat; antnodes ae mdway between nodes Typcal Exam Queston a) Explan the tems node and antnode [] Node: pont of no vbaton Antnode: pont of maxmum vbaton wave No nodes o antnodes b) Two dentcal pogessve waves ae tavellng along the same staght lne n opposte dectons. () Explan how a statonay wave patten s fomed [3] Statonay wave s fomed by the supeposton of the two waves Nodes ae ceated by destuctve ntefeence and antnodes ae ceated by constuctve ntefeence. () Compae the ampltude and phase of patcles along a statonay wave wth those of a pogessve wave. [] All ponts on a statonay wave ae n phase, ponts on a pogessve wave ae out of phase wth each othe All ponts on a pogessve wave have the same ampltude, dffeent ponts on statonay wave have dffeent ampltudes A If the lght s passed though a vetcal pece of Polaod, then the emegng ay wll be polased vetcally. It wll have half the ntensty of the ognal beam ths s why sunglasses wok. If ths ay then meets a hozontal pece of Polaod, no lght wll pass though. 5

6 Waves bascs Exam Wokshop Ths s a typcal poo student s answe to an exam queston. The comments explan what s wong wth the answes and how they can be mpoved. The examne s answe s gven below. a) () Explan the tem 'wave font' [] t s at ght-angles to the wave decton 0/ () Ths statement s tue, but does not explan wavefont. When the student ead the next pat of the queston, s/he should have ealsed that ths was not an adequate answe. State the elatonshp between the oentaton of a wave font and the decton n whch the wave s tavellng [] at ght angles / b) A longtudnal wave of fequency 30 khz has a speed of 340ms - when tavellng n a. Its wavelength when tavellng n wate s 0.05m. () Calculate the mnmum dstance between two ponts on the wave that dffe n phase by 60 o when t s tavellng though a.[3] 60 o = one sxth of wavelength = 0.008m /3 The fst pat of the method s coect, but the student has used the wavelength fo wate waves. Read the queston! Ths shows the advantage of showng wokng wthout t, no maks would have been awaded. () Calculate the speed of the wave n wate [].5ms - 0/ The student has pobably faled to convet khz to Hz. If s/he had shown wokng, one mak mght have been awaded, snce s/he could have demonstated the knowledge that fequency s unchanged. The answe should have woed hm/he! () A pulse of the wave lasts fo 0ms. Calculate the numbe of complete waves that t contans. [] = 300 / The student has evdently now ealsed that the fequency s 30 khz, not 30 Hz, but has neglected to change the eale answe! Always make tme to check. Examne s Answes a) () A suface n whch all oscllatons ae n phase () They ae pependcula b) () = v/f =340/30000 =0.03m Phase dffeence of coesponds to /6 =.89mm () speed n wate = f wavelength n wate = =500ms - () Numbe of waves = tme of pulse fequency = =300 Questons. a) Explan the dffeence between tansvese and longtudnal waves. b) Gve two examples of tansvese waves and two examples of longtudnal waves.. Explan the dffeence between a wave pulse and a contnuous wave. 3. Explan what s meant by the followng tems: Ampltude Wavelength Peak Tough Fequency 4. Defne the efactve ndex fo a wave tavellng fom mateal to mateal. 5. Explan what s meant by the Pncple of Supeposton. 6. Explan what s meant by dffacton. 7. Explan why sound waves cannot be polased. 8. Gve two dffeences between a statonay wave and a pogessve wave. 9. A wave has speed 50ms - and wavelength m. Calculate ts peod. 0. The efactve ndex of glass s.50. a) A ay of lght passes fom a to glass. It makes an angle of 0 o to the just befoe enteng the glass. Calculate the angle the makes wth the. b) The speed of lght n a s ms -. Calculate the speed of lght n glass. c) Calculate the ctcal angle fo glass. Answes. See page. See page 3. See page 4. See page 5. See page 6. See page 4 7. See page 5 8. See page 5 9. f = 50/ = 5Hz T = /f = 0.04 s 0. a) sn0/sn =.5 sn = sn0/.5 = 0.8 = 3 o b) c a / c glass =.50 c glass = /.50 = 0 8 ms - c) snc = /.5 = c = 4 o ( SF) Acknowledgements: Ths Factsheet was eseached and wtten by Nnde Hunjan Cuculum Pess, Unt 305B The Bg Peg, 0 Vyse Steet, Bmngham B8 6NF. Physcs Factsheets may be coped fee of chage by teachng staff o students, povded that the school s a egsteed subscbe. They may be netwoked fo use wthn the school. No pat of these Factsheets may be epoduced, stoed n a eteval system o tansmtted n any othe fom o by any othe means wthout the po pemsson of the publshe. ISSN