Bom! Agora vamos testar valores...
Para
n = 1
:
τ n = 9 h 1 2 π 2 sen π 3 = 9 h 3 2 π 2
Para
n = 2
:
τ n = 9 h 2 2 π 2 sen 2 π 3 = 9 h 3 8 π 2
Para
n = 3
:
τ n = 9 h 3 2 π 2 sen 3 π 3 = 0
Para
n = 4
:
τ n = 9 h 4 2 π 2 sen 4 π 3 = 9 h 3 32 π 2
Para
n = 5
:
τ n = 9 h 5 2 π 2 sen 5 π 3 = 9 h 3 50 π 2
Para
n = 6
:
τ n = 9 h 6 2 π 2 sen 6 π 3 = 0
⋮
⋮
Dessa forma, podemos escrever
W ( x , t )
como...
W x , t = 9 3 2 π 2 h sen π x l cos c π t l + 9 3 8 π 2 h sen 2 π x l cos 2 c π t l - 9 h 3 32 π 2 sen 4 π x l cos 4 c π t l - ⋯
Passo 4
Aí sim, rapaziada!
Vamos então, achar as condições de contorno!
Para
t = 0
, temos que
W ( x , 0 )
vai valer...
W x , 0 = 9 3 2 π 2 h sen π x l cos c π 0 l + 9 3 8 π 2 h sen 2 π x l cos 2 c π 0 l - 9 h 3 32 π 2 sen 4 π x l cos 4 c π 0 l - ⋯
W x , 0 = 9 3 2 π 2 h sen π x l + 9 3 8 π 2 h sen 2 π x l - 9 3 32 π 2 h sen 4 π x l - 9 3 50 π 2 h sen 5 π x l ⋯
Dessa forma, a nossa curva fica assim...
P a r a t = l 4 c
:
W x , l 4 c = 9 3 2 π 2 h sen π x l cos c π l 4 c l + 9 3 8 π 2 h sen 2 π x l cos 2 c π l 4 c l - 9 3 32 π 2 h sen 4 π x l cos 4 c π l 4 c l ⋯
W x , l 4 c = 9 3 2 π 2 h 2 sen π x l - 9 3 32 π 2 h sen 4 π x l + 9 3 50 π 2 h 2 sen 5 π x l
P a r a t = l 3 c
:
W x , l 3 c = 9 3 2 π 2 h sen π x l cos c π l 3 c l + 9 3 8 π 2 h sen 2 π x l cos 2 c π l 3 c l - 9 3 32 π 2 h sen 4 π x l cos 4 c π l 3 c l ⋯
W x , l 3 c = 9 3 h 4 π 2 sen π x l - 1 4 sen 2 π x l + 1 16 sen 4 π x l - 1 25 sen 5 π x l
P a r a t = l 2 c
:
W x , l 3 c = 9 3 2 π 2 h sen π x l cos c π l 2 c l + 9 3 8 π 2 h sen 2 π x l cos 2 c π l 2 c l - 9 3 32 π 2 h sen 4 π x l cos 4 c π l 2 c l ⋯
W x , l 3 c = - 9 3 h 8 π 2 sen 2 π x l + 1 4 sen 4 π x l
P a r a t = l c
:
W x , l 3 c = 9 3 2 π 2 h sen π x l cos c π l c l + 9 3 8 π 2 h sen 2 π x l cos 2 c π l c l - 9 3 32 π 2 h sen 4 π x l cos 4 c π l c l ⋯
W x , l 3 c = 9 3 h 2 π 2 - sen π x l + 1 4 sen 2 π x l - 1 16 sen 4 π x l + 1 25 sen 5 π x l
Resposta
P a r a t = l 4 c
W x , l 4 c = 9 3 2 π 2 h 2 sen π x l - 9 3 32 π 2 h sen 4 π x l + 9 3 50 π 2 h 2 sen 5 π x l
P a r a t = l 3 c
W x , l 3 c = 9 3 h 4 π 2 sen π x l - 1 4 sen 2 π x l + 1 16 sen 4 π x l - 1 25 sen 5 π x l
P a r a t = l 2 c
W x , l 3 c = - 9 3 h 8 π 2 sen 2 π x l + 1 4 sen 4 π x l
P a r a t = l c