4 Where Does the Wave Equation Come From?
31
m · ¨
s 1 = D N s N − 2D N s 1 + D N s 2 ,
m · ¨
s 2 = D N s 1 − 2D N s 2 + D N s 3 ,
. . .
m · ¨
s i = D N s i−1 − 2D N s i + D N s i+1 ,
. . .
m · ¨
s N = D N s N −1 − 2D N s N + D N s N +1 .
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(31)
(Here, s N +1 ≡ s 1 as explained above).
Equation (31) is a coupled system of N differential equations for the motions of
the N masses. The general solution of these equations is searched for as (N − 1)
harmonic oscillations of the single masses (a trivial solution is always the rigid
motion of the whole chain). The nth oscillation is written for the ith mass as s
(n)
i (t).
To determine the eigen frequencies ω n of a linear chain, we will try the following
ansatz,
s
(n)
i (t) = cos (ω n t) cos 2(
i
N
n 2π) , n = 1, 2, · · · , N − 1.
(32)
This ansatz fulfils the conditions of periodicity: The (i + N )th mass oscillates according to (32) just as the ith mass oscillates. The position of the ith mass is according
to (30) at x i =
L
N
i. Equation (32) shows us the largest ‘wavelength’ λ 1 with its
corresponding smallest wave number k 1 = 2π/λ 1 according to,
λ 1 = L ,
k 1 =
2π
L
(33)
and for the nth harmonic vibration the following is valid,
λ n =
L
n
,
k n =
2πn
L
,
n = 1, 2, · · · , N − 1.
(34)
We insert Eq. (32) into Eq. (31) and discover, whilst shortening the common factor
cos(ω n t),
m ω
2
n cos(i
n
N
2π) = D N [cos(i
n
N
2π −
n
N
2π) + cos(i
n
N
2π +
n
N
2π)
− 2 cos(i
n
N
2π)]
= D N [cos(i
n
N
2π) cos(
n
N
2π) + sin(i
n
N
2π) sin(
n
N
2π)
+ cos(i
n
N
2π) cos(
n
N
2π) − sin(
n
N
2π) sin(
n
N
2π)
− 2 cos(i
n
N
2π)] ,