Understanding Boundary Conditions in Physics
Explore the concept of boundary conditions in physics through a series of equations and descriptions. Delve into symmetric boundary conditions and complete sets of functions, illustrated with helpful images. Uncover the complexities and nuances of these fundamental principles in this detailed analysis.
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Presentation Transcript
y V =0 a V =V0 x b b V =0 z ?2? = 0
?2? = 0 ?2? ??2 + ?2? ??2 = 0 ? ?,? = ? ? ?(?) ?2(??) ??2 + ?2(??) ??2(?) ??2 + ??2(?) ??2 = 0 ??2(?) ??2 + ??2(?) ??2 = 0 ??2 = 0 ??2(?) ????2 + ??2(?) ????2 = 0 ?2(?) ???2 + ?2(?) ???2 = 0
?2(?) ???2 + ?2(?) ? ? = ????+ ?? ?? ???2 = 0 Y ? = ????(??) + ????(??) ?2(?) ???2 = ?1and ?2(?) ???2 = ?2 ?1= ?2and?2= ?2 Completely general up to this point ?2(?) ???2 = ?2and ?2(?) ???2 = ?2
? ? = ????+ ???? Y ? = ????(??) + ????(??) Boundary conditions Symmetric in x: ? = ? ? ? = ?(???+? ??) ? ? = 2?(???+? ??) = 2?cosh(??) 2 ? ?,? = 2?cosh(??)(???? ?? + ???? ?? ) ? ?,? = cosh(??)(???? ?? + ???? ?? )
? ?,? = cosh(??)(???? ?? + ???? ?? ) ? ?,0 = 0 = cosh(??)(???? 0 + ???? 0 ) ? ?,0 = 0 =???? 0 ) ? ?,? = ?cosh(??)??? ?? ? ?,? = 0 = ?cosh(??)??? ?? ?? = ?? ? ?,? = ?cosh(???/?)??? ???/?
? ?,? = ?cosh(???/?)??? ???/? ? = 0,1,2,3 15 10 5 0 -5 150 150 100 100 50 50 0 0 One boundary condition to go
? ?,? = ?cosh(???/?)??? ???/? 15 10 5 ? = 0,1,2,3 0 -5 150 150 100 100 ? ?,? = ?0= ?cosh(???/?)??? ???/? Never going to happen 50 50 0 0 ? ?,? = ??cosh(???/?)??? ???/? ? ?,? = ??cosh(???/?)??? ???/? ?=1 ? ?,? = ?0= ??cosh(???/?)??? ???/? ?=1
?0= ??cosh(???/?)??? ???/? ?=1 15 10 5 0 -5 150 150 100 100 50 50 0 0
?0= ??cosh(???/?)??? ???/? ?=1 ?0??? ? ??/? = ??cosh(???/?)??? ???/? ??? ? ??/? ?=1 ? ? ?0??? ? ??/? ?? = ??cosh(???/?)??? ???/? ??? ? ??/? ?? ?=1 0 0 ? ? ?0??? ? ??/? ?? = ??? ???/? ??? ? ??/? ?? ??cosh(???/?) ?=1 0 0
? ? ?0??? ? ??/? ?? = ??? ???/? ??? ? ??/? ?? ??cosh(???/?) ?=1 0 0 ? ??? ???/? ??? ? ??/? ?? = 0 ??? ? ? 0 =? 2??? ? = ? Orthogonal, complete set of functions
? ? ?0??? ? ??/? ?? = ??? ???/? ??? ? ??/? ?? ??cosh(???/?) ?=1 0 0 ? ??? ???/? ??? ? ??/? ?? = 0 ??? ? ? 0 =? 2??? ? = ? ? ?0??? ? ??/? ?? = ?? cosh(? ??/?)? 2 ? 0 ?0??? ? ??/? ?? = ??cosh(? ??/?)? 2 0 ? ? =2??0 ?? ?0??? ? ??/? ?? = ?0 ??? ? ??/? ?? = 0 ?? ? ?? ???? ?? ? ?? ??? 0 0
? ?0??? ? ??/? ?? = ??cosh(? ??/?)? 2 0 2??0 ??? ? ? 2 ??? ??? ? = ??cosh ?? 4?0/?? cosh(???/?) ??= ? ?,? = ??cosh(???/?)??? ???/? ?=1,3,5 4?0 ?? cosh(???/?) cosh(???/?)sin(???/?) ? ?,? = ?=1,3,5
4?0 ?? cosh(???/?) cosh(???/?)sin(???/?) ? ?,? = ?=1,3,5 15 10 5 4?0 ? cosh(??/?) cosh(??/?)sin(??/?) 0 -5 150 150 100 100 50 50 0 0 15 10 4?0 ? cosh(??/?) cosh(??/?)sin(??/?) +4?0 cosh(3??/?) cosh(3??/?)sin(3??/?) 5 0 3? -5 150 150 100 100 50 50 0 0
