Fields in the Neighborhood of a Caustic
Fields in the Neighborhood of a Caustic
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1?) 7(s) -I cos J dt/p(t), sin I dt/p(t) (, and (3. 16) ^ = N(3)/p(3) =^ -sin I dt/p(t), cos j dt/p(t)^ /p(s) From (3. IU) and (3. 15), using the power series expansions for the sin and cos functions about zero we obtain (3. 17) f - Is - p/2(a+l)^(2a-. 3)V°''^ - ... , r\/(a-H)(a+2)'j s*"*^ * rn/(p+l)(p+2)ls^^^ + ... I (3. 18) f = h -ir|x/(a+l)j s''*^ Jn/(P-Hl)j sH^ + ... , [x/(a+l)1 s'"'^ + L/(P+1)1 s^*^ ^ ... 1 .
- 12 - Writing P = (x, y) we then have for the phase (3. 19) [P -r]. T +s = X + f...yX/(a+l)]s''''^ + [yvL/(P+l)]s^*^ . [xX^x/(c. L)(p. L)] s^-^P-^2^[xV(a^2)(2a. 3)] s^""'^ \Ihen P is at the cusp x = y = 0, and the phase becomes 2a+3 + (3. 20) rxV(a+2)(a+3)] Along a ray y = 0, and the phase becomes near the cusp (3. 21) X - lxXV2(a+l)^ s^""^ + ...
- rxxV2(a-M)2J We can also consider various other cases in which we approach the cusp along different paths, e. G. , along x = 0, and in each case we take from (3. 19) only the lowest order term in 3. For these other approaches a more precise relationship between a and p must be given to decide which term is actually of lowest order.
What to read after Fields in the Neighborhood of a Caustic?
You can find similar books in the "Read Also" column, or choose other free books by Irvin W Kay to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
1?) 7(s) -I cos J dt/p(t), sin I dt/p(t) (, and (3. 16) ^ = N(3)/p(3) =^ -sin I dt/p(t), cos j dt/p(t)^ /p(s) From (3. IU) and (3. 15), using the power series expansions for the sin and cos functions about zero we obtain (3. 17) f - Is - p/2(a+l)^(2a-. 3)V°''^ - ... , r\/(a-H)(a+2)'j s*"*^ * rn/(p+l)(p+2)ls^^^ + ... I (3. 18) f = h -ir|x/(a+l)j s''*^ Jn/(P-Hl)j sH^ + ... , [x/(a+l)1 s'"'^ + L/(P+1)1 s^*^ ^ ... 1 .
- 12 - Writing P = (x, y) we then have for the phase (3. 19) [P -r]. T +s = X + f...yX/(a+l)]s''''^ + [yvL/(P+l)]s^*^ . [xX^x/(c. L)(p. L)] s^-^P-^2^[xV(a^2)(2a. 3)] s^""'^ \Ihen P is at the cusp x = y = 0, and the phase becomes 2a+3 + (3. 20) rxV(a+2)(a+3)] Along a ray y = 0, and the phase becomes near the cusp (3. 21) X - lxXV2(a+l)^ s^""^ + ...
- rxxV2(a-M)2J We can also consider various other cases in which we approach the cusp along different paths, e. G. , along x = 0, and in each case we take from (3. 19) only the lowest order term in 3. For these other approaches a more precise relationship between a and p must be given to decide which term is actually of lowest order.
What to read after Fields in the Neighborhood of a Caustic?
You can find similar books in the "Read Also" column, or choose other free books by Irvin W Kay to read onlineMoreLess
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