12 Superposition Notes

- ¡Æ n o t in H 1 syllabus light. (T he structure and use of the spectrometer are not required . ) (mr describe the use of a diffraction grating to determ ine the w avelength of principa©¥ m axim a produced by a diffraction grating . (I) ' re ca ll a n d u se the eq u atio n d sin e = n to lo c a te th e p o s itio n s o f th e single aperture. (k) ' re c a ll a nd use the R ayleigh criterion e . À / b fo r th e re so ©¥ving p ow e r o f a m inim a for single slit diffraction. ü) " re c a ©¥©¥ a n d u s e th e e q u a tio n s in ©¬ ¡¤ j b to lo c a te th e p o s itio n o f th e firs t interference using light. (i) recal©¥ and solve problem s using the equation À = ax /D fo r d o u b ©¥e - s ©¥it interference fń nges to be observed. (h) show an understanding of the conditions required for tw o - s o u r c e new in H 2 syllabus interference using w ater, ©¥ight and m icrow aves. (g) show an understanding of experim ents w hich dem onstrate twsource and a narrow gap . including the diffraction of w ater w aves in a ripple tank w ith both a w ide gap show an understanding of experim ents w hich dem onstrate diffraction (e) exp©¥ain the m eaning of the term diffraction. identify nodes and antinodes. (d) exp©¥ain the form ation of a stationary w ave using a graphical m ethod, a n d waves using m icrow aves, s tre tched strings and alr co©¥um ns. (c) show an understanding of experim ents w hich dem onstrate stationary di¨¤erence and path difference. (b) show an understanding of the term s Interference, c o herence, phase (a) exp©¥ain and use the princip©¥e of superposition ©¥n sim ple app©¥ications, e Learnïng O utcom es S i ng l e- s ©¥it and multip©¥s©¥it di¨¤raction T w o - s o u r c e Interference D i ffrac t ion s t at ion a r y wa v es P r inc lp©¥e of sup e r po s ition S yllabus C ontent 12 S U P E R P O S IT IO N . . H ìplca 9 74 9 C hapter Y E A R 5 - 6 P H Y S IC S D E P A R T M E N T J R A F FL E S IN S T IT U T IO N : \ . ' (

0 displacem ent is the vector sum of the displacem ents due to each w ave. F ©¥g . 1 . W hen tw o w aves arrive at a point at the sam e tim e, the resultant 11 the principle of superposition. sources) m eet at a point in space? T he answer to this question is provided by W hat happens when tw o w aves of the sam e kind (e. g . s o u n d waves from two m agnetic field. electrom agnetic w aves, the disturbance refers to the varying electric and displacem ent of the particles of the m edia from their equilibrium position. F or S uperposition m echanica©¥ w aves, like sound and w ater w aves, the disturbance refers to the P rinciple of A w ave is a disturbance that travels through a m edium or vacuum . F or Y E A R 5 - 6 P H Y S IC S D E P A R T M E N T R A F F L E S ©¥N S T ©¥T U T ©¥O N

treatm ent can be found in A ppendix A . F ig . 3 . S uperposition of two w aves displaced by % Ä apart. A m athem atica©¥ 4 \©¥©© . 1 » \ ©¥l ©¥1n oa A t a particular instant of tim e R esu©¥tant e of tim e w e m ay get a resu©¥tant w aveform as show n in F ig . 3 T he princip©¥e of superposition applies at each point and at a particular instant T he superposition of the tw o w aves is observed along the line of propagation. lirf W ©©udc Ąirr f¡Ærdukn 12 . 2 S tationau w aves Y E A R S - 6 P H Y S IC S D E P A R T M E N T R A F F L E S ÏN S T ©¥T U T IO N

4 N ote A n envelope of a rapidly varying signal is a curve outlining its am pïitudes. draw n on displacem ent - distance axes) w ave (w here the m axim um disp©¥acem ent is

F ig . 4 b A graphica©¥ representation of a stationary ìé争 X distance axes) . tim e interval of T /16 (on disp©¥acem ent - T he w aveform of a stationary w ave at equa©¥- ig . 4 a ßµ «À X w ith each other. 7 . P artic©¥es in neighbouring segm ents vibrate 1 8 0 ¡¤ (or n rad) out of phase 6 . D istance betw een tw o adjacent nodes (or antinodes) ls yaÄ . = wam plitude. the sam e instant. N ote that these particles do not have the sam e «¤国ûö they reach their respective m axim a, m inim a and equilibrium positions at W ithin tw o consecutive nodes, e v e ry partic©¥e osclllates ©¥n phase, I. e. ,5 ¡¤ are labe©¥©¥ed ' N ' o n Flg . 3 . 4 . A ©¥iaaa is a point ©¥n a standing w ave w here the» nipm irdii Ë m d T hey at the antinodes ls double that of the com ponent w aves, am p©¥itude . T hese are ©¥abe©¥led ' A ' o n F ig . 3 . T he am plitude of oscillation ' : fhă dm un particles at the antinodes vibrate w ith the greatest A n inHnod ls a point In a standing w ave w here the Ëiim©­li ߣ画画 s the3 ¡¤ com ponent w aves. *©¥fferent amp©¥©¥tud T he frequency i8 the sam e as that of the tw o their respective ę quilibrium posillons w ith the * ©¥m e rrequoncĺ . B ut E very partic©¥e of the w ave m ere©¥y osci©¥lales (except at the nodes) about2 ¡¤ know n as a stationary w ave or stand©¥ng w ave. 1 . T he w ave profile does not propagate. A s such, the resultant w ave is statio m arrv w v @ P roperties of a T he resultant w aveform has the follow ing features Y E A R 5 4 P H Y S ©¥C S D E P A R T M E N T R A F F L E S I N S T ïT U T IO N

5 10 Đ W . ' are 1 . 5 m apart. W hat is the speed of the progressive w aves? produce a system of stationary w aves in w hich adjacent nodes P rogressive w aves of freq uency 3 0 0 H z are superposed to (J90/P 1113) betw een a pair of adjacent nodes : g = ' ' " tinodes. ©¥s the distance between ual to tw ice the distanceW avelength to it. progressive w ave that gives rise frequency of the w ave. w ith the sam e * equency as the s. h. m . w ith the rest, a ©¥©¥ points vibrate In 8 . h. m . A ©¥©¥ points vibrate in E xcept for the nodes w hich are atF requency difference of 1¡¼ rad , segm ents have a phase different phases phase. P articles in adjacent w avelength have adjacent nodes have the sam e A I©¥ partic©¥es w ithin one A ©¥©¥ particles betw een tw oP hase am p©¥itude. A ntinodes. w ith the sam e nodes to the m axim um at the E very point osclllates A m plitude varies from O at theA m plitude E nergy T ransports energy D oes not transport energy ve©¥ocity of the w ave W aveform P ropagates w ith the D oes not propagate property l = ss©¥ve w ave stationary w ave stationary w ave . T he table below com pares the properties of a progressive w ave w ith those of a = 4 Ľ " w aves and P rogressive stationary w aves com paring Y E A R 5 - 6 P H Y S ©¥C S D E P A R T M E N T R A F F L E S ©¥N S T ©¥T U T IO N

instrum ent. * In re a lity , m u ltip©¥e harm onies give the tim bre or characteristics of an A t the fixe d en d s , there mu st be nodes (since the string cannot vibrate) . N ote F ig . 5 overtone ' " Ĺ)(n - 1) ¡£ 画画ÞÌ 国国国 f¡£ = . (ðÞ) ¡£ ¡£ ¡£, . 2 nd o ve rto n e L = . Ĺ) 1. = , . . " " . = L 1 st o ve rto n e nx% , ' " ĺţ) I, = ¡¤ , " . " " vibration R epresentation W avelength F requency A '' X ' " ^ stationary w aves on a string m edium , o btain the frequency f of the stationary w ave in term s of v and L . S tep 4 U sing f- v íÀ w here v is the speed of the progressive waves in the string or S tep 3 D erive the w ave©¥ength Ä of the stationary w ave in term s of ©¥ength L of the string S tep 2 D raw the graphical representat©¥on of the stationary wave S tep 1 S elect the m ode of v©¥bratïon of the stationary w ave m odes of vibration of a stationary w ave in general. T he fo©¥lowing steps can be used to determ ine the * equency of the different overtones. fundam enta©¥ frequency . H igher m odes of vibration are known as the T he stationary w ave with the lowest frequency ©¥s said to be vibratlng with M e transverse wave set up In the string, o r a c o m bination of various harm onlcs. T he frequency of sound produced ls equal to the frequency of the stationary string are possible. S tretched S trings p©¥ucking ©¥t at different points along the w ire , S evera©¥ m odes of vibration of the S tationary W