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# If we solve result into equation 31.19 (0 = A' / @ = AU, p + 2kaBG ) for kaBa and plug the equation 31.18 (0 = A'ut = Aut ~k"B' _ K'Blu + n...

## Question

###### If we solve result into equation 31.19 (0 = A' / @ = AU, p + 2kaBG ) for kaBa and plug the equation 31.18 (0 = A'ut = Aut ~k"B' _ K'Blu + n"'kaBa ), we get 0 = Aut _ k"B' _ kBH _ gn"" A"o (31.20) {Note that I have renamed the index in the last term to avoid confusion with the free LL index) To see that we can teally find values of B" that solve tho equation it helps to take advantage of our freedom to orient our reference frame so

If we solve result into equation 31.19 (0 = A' / @ = AU, p + 2kaBG ) for kaBa and plug the equation 31.18 (0 = A'ut = Aut ~k"B' _ K'Blu + n"'kaBa ), we get 0 = Aut _ k"B' _ kBH _ gn"" A"o (31.20) {Note that I have renamed the index in the last term to avoid confusion with the free LL index) To see that we can teally find values of B" that solve tho equation it helps to take advantage of our freedom to orient our reference frame so that the tzaxis coincides with the direction of the wave s motion. Then ko = [~0,0,0, 0] and we can divide equation 31.20 into individual component equations as follows: 0 = A" _ kB' _ KB' _ {n"A"o = A" [email protected]' + %A? (31.2la) 0 = At _ kB _ kB' _ %ntA"o = At @B* (31.21b) 0 = Aty _ kBY _ kYBt ~ IntA"o = Aty @BY (31.21c) 0 = A" _ KB? _ kB' inA" = Atz @BZ @Bt (31.21d) The first three equations we can solve for B',B*,and BY our solution into the last directly We can then plug equation to solve for B?_ Exercise 31.2.3. Carry out this plan to express the in terms of the original wave components of Bu entirely amplitude matrix Auv

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