( ) 2 2. Multi-Layer Refraction Problem Rafael Espericueta, Bakersfield College, November, 2006

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1 Mult-Layer Refracto Problem Rafael Espercueta, Bakersfeld College, November, 006 Lght travels at dfferet speeds through dfferet meda, but refracts at layer boudares order to traverse the least-tme path. I ths paper we ll play lght s game, fdg the least-tme path through a umber of layers of meda, where each meda duces a dfferet speed. Oe eample of such a problem s the lfeguard problem sometmes see calculus courses. The lfeguard ca travel at a gve speed rug through sad ad a dfferet speed swmmg, ad the problem s to fd the path that mmzes the tme t takes to reach a drowg chld. -Layer Problem We start by solvg the geeral two-layer case. Gve values y, y, ad d, as show the fgure, ad also gve the fed speeds v ad v through each layer, we seek the uque value of resultg the least tme path from A to B. As show the fgure, let d = + y ad d = d + y, ad let + y d + y d d = =, ad t = = v v v v t (These are the tmes t takes to traverse the two layers) We wsh to mmze the total travel tme fucto f defed by d d + y d + y t t f v v v v + = + = + =,

2 .e., we must fd the mmum of the fucto f ( ) = ( + y ) + ( d ) + y v v Takg the dervatve ad settg t equal to zero, we obta: f ' = ( + y ) + ( d ) + y d = 0 v v d = 0 v + y v d + y v d + y d v + y = 0 v d + y = d v + y ( + ) = ( ) ( + ) v d y d v y v v + d v v + vd + vy vd vy + vdy vd y = Beg a fourth degree polyomal, t ca be solved eactly, wth the help of a computer algebra system. The soluto s rather complcated, but umercal methods eed ot be voked. As a partcular case, let y = 4, y = 6, d = 0, v =, ad v =. The Maple yelds 4 eact solutos, of whch two are real ad two are comple. Oe of the real solutos for s egatve, ad the other s the oe we seek, whch equals eactly appromately equal to , Notce how much loger the path s through the secod layer, where the speed s doubled, order to mmze the total tme of the trp.

3 -Layer Problem Net we tur to the geeral, -layer case. I the geeral case we must optmze a fucto of several varables. Closed form solutos ca t be foud; we must tur to umercal methods. For fed postve y, v, d, where v s the speed through the th layer, whch has thckess as show the fgure. y, Notce that the dstace from A to B s d + y = + y. = = = Let ad let d = + y, for =,...,. (The legths of the least-tme path through each layer), d t =, for =,...,. v (The tmes t takes to traverse the layers) We wsh to mmze d t = = = f = = v = v + y (,, ) subject to the costrats G (,, ) = = d. ( Amazgly, each lttle photo solves ths problem stataeously! ) We have: =

4 + y + y f = = = + v v v = = = f = = v e ( + y ) v( + y ) = = G = = = e = = = =,,, ( y ) Ivokg the method of Lagrage multplers, we have f = λ G, hece: e ( + ) v y ( λ ) = λ, for =,, = λ + v y = λ v + y v = λ v y. Ψ λ v y λv y y = = = λ v λ v λ v y = ( λ v ) Ths mples that λ <,. λ must be less tha the least of the recprocals of the v (else v the dstaces become comple umbers). Pluggg these values to the costrat equato, we get: Let ψ = = = λ = y v = d. The the Newto trasform of ψ wll have terates that = y v d ted to coverge to our desred λ. Sce the Newto trasform s gve by : 3 ψ ' = v v = 3,

5 T = = = ( ) y v d 3 3 ( ) y v v NOTE: The followg eample umercal solutos to partcular cases were calculated ad plotted wth the assstace of the computer algebra system, Maple. Numercal Eample of a 3-Layer Refracto Problem Net we eame a partcular three-layer umercal eample, where we ca observe the terates of T covergg to λ, whch allows us to solve for each of the usg equatos Ψ above. Let v = 4, v = 6, v3 = 3, y = y = y3 = 0, d = 30. Recall that λ must be less tha the smallest recprocal of the v, so ths case must be less tha =. We select λ 0 = 6 00 (rouded to 50 dgts) for our tal guess. The Newto trasform T the yelds the followg successve terates: λ0 = λ λ λ λ λ λ λ = λ 7 So we kow λ to 50 decmal places, ad we ca use ths value of lambda to calculate the = y 3 ( λ v ) = = = Ths path of least tme s llustrated the fgure: :

6 The clever photo kows to make use of ts faster pace the mddle layer. Numercal Eample of a 4-Layer Refracto Problem Net we eame a partcular three-layer umercal eample, where we ca observe the terates of T covergg to λ, whch allows us to solve for each of the usg equatos Ψ above. Let: v = 5, v = 8, v = 3, v = 0, 3 4 y = 0, y = 4, y = 5, y = 8, ad d = Recall that λ must be less tha the smallest recprocal of the v, so ths case must be less tha 0. 0 =. We select λ 0 = = 0.09 (rouded to 50 dgts) for our tal guess. The Newto trasform T the yelds the followg successve terates: λ0 = λ λ λ λ λ λ λ = λ 7 So we kow λ to 50 decmal places, ad we ca use ths value of lambda to calculate the :

7 = y 3 4 ( λ v ) = = = λ = Ths path of least tme s llustrated the fgure: Implemetg Numercal Solutos of the -Layer Problem Usg Maple The followg shows the Maple worksheet that computed the above 4-layer soluto, complete wth both the Maple commads ad commets. Defe the umber of layers '', the speeds through each layer 'v', the depths of each layer 'y', ad the total horzotal dsplacemet 'd': Calculate a good tal guess 'K' to be used Newto's method later. The tal guess has to be less tha the smallest of the recprocals of the speeds through the layers. Emprcally, a lttle less seems to work well: Defe the Newto trasform whose terates coverge to the that solves the Lagrage multpler mmzato problem:

8 Iterate the above Newto trasform utl subsequet teratos dffer by less tha. The tal guess s K (defed above): Ths ca be used to solve for the -coordates of the mmum tme path's tersectos wth the layers:

9 Graph the optmal path through the layers: