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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

252 For notation, see

252 For notation, see Fig. A.2. The reflected wavefront is the envelope of the family of circles F(x,y,9) -- (x-cos 9) 2 + (y-sin 9) 2 - (t-l+sin 9) 2, 9 ~ [t3, ~/2], parametrized by (suitably scaled) time t. The singularity of the wavefront for all reflection points 9 ~ 0 is a Morse critical point. For tp = 0 we have 8 4 2 1 4 . 2 20 2 j4F = ~4t4+(ax - y2)V2+(8y3 -Txy- 8y)V+~-xy --~-y +4x +-~-- y 1 where ~ -- 9 + y • The wavefront (envelope) is the intersection of a swallowtail with a paraboloid, Fig. A.3. Fig. A.4 shows a schematic of the reflected and diffracted wavefronts, and an accurate computer-generated picture is given in Fig. A.5.. Fig. A.3 Intersection of a swallowtail and a paraboloid. bY diffracted Fig. A.4 Caustics by reflection and diffraction. The reflected wavefront at time t is given by I x

253 -- tsin @ )+(t-'+~in~>t_~os2q> j ' The diffracted wavefront at time t > 1 is given by I tp ~ [0, rW2l. 0 ~ [0, ;x/21. L /!/ ,,,'/XYlJWI Fig. A.5 Computer- generated picture of caustics by reflection and diffraction. We investigate the smoothness of the reflected and diffracted wavefronts at the point where they join. For the reflected wavefront, i = atp'~-----Iq a=0= 2(t-l), XI~= o = 3, ~" I~a=0 = -8(t-l) dy 5, = ~ 1,~oo = o, For the diffracted wavefront, Now we calculate ~ltp=o = 4(t-l), y" Itp=o = 12. Rle=o= -(t-l), xl0__o = 1, "x" le=0 = t-l, p Io= o = o, dy ~, d2y ~-j,~ d3y dx x dx 2 .3 x dx 3 y 10= 0 = t-l, ~," 10= 0 = -2. • ~. k2 _ 21 "x" i -3(p ~ - ~ ~) .5 x

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