Lecture 8 Interference superposition of waves in same direction graphical and mathematical phase and path length difference application to thin films in 2 3 D Beats interference of slightly different frequencies Summary of traveling vs standing wave Wave particles undergoing collective correlated SHM Traveling i same A for all x and ii at given t different x at different point of cycle a sin kx t in general function of x vt Standing i A varies with x and ii at given t all x at same point in cycle e g maximum at cos t 1 A x 2a sin kx cos t in general function of x function of t Interference basic set up standing waves superposition of waves traveling in opposite direction not a traveling wave is one type of interference 2 waves traveling in same direction distance of observer from source 1 D1 x1 t D2 x2 t a sin kx1 t 10 a sin kx2 t 20 a sin 1 a sin 2 Phase constants tell us what the source is doing at t 0 Interference graphical aligned crest to crest and trough to trough in phase D1 x D2 x 1 2 2 m Dnet 2D1 or D2 A 2a crest of wave 1 aligned with trough of wave 2 out of phase D1 x D2 x Dnet x 0 Phase and path length difference t drops out net phase difference determines interference 2 contributions i path length difference x x2 x1 distance between sources extra distance traveled by wave 2 independent of observer ii inherent phase difference 0 10 20 Example Two in phase loudspeakers separated by distance d emit 170 Hz sound waves along the x axis As you walk along the axis away from the speakers you don t hear anything even thought both speakers are on What are three possible values for d Assume a sound speed of 340 m s independent of observer in 1D cf 2 3D Identical sources 0 0 maximum constructive interference x m perfect destructive interference x m 1 2 Interference mathematical D D1 D2 A sin kxavg t 0 avg xavg x1 x2 2 0 avg 10 20 2 A 2a cos 2 independent of x of observer traveling wave xavg x of observer A varies from 0 for 2m 1 perfect destructive interference to 2a for 2m maximum constructive interference confirms graphical In general neither exactly out of nor in phase Application to thin films reflection from boundary at which index of refraction increases speed of light decreases phase shift of 2 reflected waves interfere from air film and film glass boundaries nair 1 nf ilm nglass x 2 0 f 2dn 2 wave 2 travels twice thru film in air Use 0 0 common origin f n constructive for C 2nd m m 1 2 2nd destructive for D m 1 2 m 1 2 anti reflection coating for lens Interference in 2 3 D Similar to 1D with x r D r t a sin kr t 0 constant a approx after T 2 crests troughs points of constructive destructive interference same r and A depends on observer cf in 1 D measure r by counting rings Picturing interference in 2 3 D Antinodal and nodal lines line of points with same r A traveling wave along it Strategy for interference problems assume circular waves of same amplitude note 0 draw picture with sources and observer at distances r1 2 r x for 1D Example Two out of phase radio antennas at x 300 m on the x axis are emitting 3 0 MHz radio waves Is the point x y 300 m 800 m a point of maximum constructive interference perfect destructive interference or something in between Beats superposition of waves of slightly different f so far same f e g 2 tones with f 1 Hz single tone with intensity modulated loud soft loud Simplify a 1 D1 a1 sin k1 x 1 t 10 D2 a2 sin k2 x 2 t 20 a2 a x 0 10 20 D D1 D2 2a cos mod t sin avg t 1 avg 2 1 2 average frequency 1 mod 2 1 2 modulation frequency Conditions for beats 1 2 avg 1 or 2 mod avg 2a cos mod t is slowly changing amplitude for rapid oscillations sin avg t Note heard is favg avg 2 f1 2 but loud soft loud modulation periodic variation of variation of amplitude waves alternately in and out of phase initially crests overlap A 2 a but after a while out of step A 0 due to unequal f s Beats frequency I A2 cos2 mod t Each loud soft loud is 1 beat 1 2 1 cos 2 mod t fbeat 2fmod f1 f2
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