(all k/m substitutes with k here)
Xi'' = k Xi-1 -2k Xi + k Xi+1
X0, X n+1 were controlled vibration of cos & sin wave
X0 = Gj cos Wj t
Xn+1= Hj cos Wj t
let Xi be linear, Rij cos Wj t + Qij sin Wj t, composed of different j
Xi'' + 2k Xi = (2k-Wj^2)(Rij cos Wj t + Qij sin Wj t) = k Xi-1 + k Xi+1
k R(i+1)j - (2k-Wj^2) Rij + k R(i-1)j =0
consider characteristic kx^2 - (2k-Wj^2) x + k =0 , with x solution a, b
When Wj^2 > 4k => a,b belongs to Real number
Rij = Gj (a^(n+1-i) - b^(n+1-i)) / (a^(n+1) - b^(n+1)) + Hj (a^i - b^i) / (a^(n+1) - b^(n+1))
When Wj^2 =0 => a=b=1
Rij = Gj (n+1-i) / (n+1) + Hj i / (n+1)
When 0< Wj^2 < 4k => a,b belongs to Imaginery number
(Wj^2 - 2k) / 2k +- Wj x (Wj^2-4k)^0.5 /2k
-1 <(Wj^2 - 2k) / 2k <1
=> let cos Y = (Wj^2 - 2k) / 2k, sin Y = Wj (4k - Wj^2)^0.5 /2k
Rij = Gj sin ((n+1-i)Y) /sin ((n+1)Y) + Hj sin( iY) /sin( (n+1)Y)
共振頻率為 W=2(sin (hπ/[2(n+1)])) k^0.5,
當n夠大時,最高頻趨近 2k^0.5
最低頻 2(sin (π/[2(n+1)])) k^0.5,剛好從0點到(n+1)點形成半波長駐波
