34
4 Where Does the Wave Equation Come From?
c
(n)
K =
∂ω n
∂k n
= N
2D
m
L
N
sin
L k n
N
2
2D
m
1 − cos
L k n
N
,
hence
c
(n)
K =
∂ω n
∂k n
= L
D
m
sin
L k n
N
2
1 − cos
L k n
N
and with (39)
c
(n)
K = c K =
sin
L k n
N
2
1 − cos
L k n
N
=
sin
2πn
N
2
1 − cos
2πn
N
.
(44)
Equations (43) and (44) describe the strict dispersion relation of a linear chain. An
approximation formula for a not too large x = L k n /N = 2πn/N can be found using
the Taylor series expansion according to sin x = x −
1
6
x
3
, cos x = 1 −
1
2
x
2
+
1
24
x
4 ,
so that
sin x
√
2 − 2 cos x
=
x −
1
6 x 3
x 2 −
1
12 x 4
=
1 −
1
6 x 2
1 −
1
12 x 2
= (1 −
1
6
x
2
)(1 +
1
24
x
2
) = 1 −
1
8
x
2
,
hence
c
(n)
K = c K
1 −
π
2
2
n
2
N 2
.
(44a)
Due to the quadratic dependency of n/N , Eq. (39) gives for n N a very good
approximation for Eq. (44a). The real dispersion in every linear chain of single
coupled masses is only detectable when the wavelength becomes small enough.
Using the aforementioned rod with N = 10
10 atoms in one metre, we receive
for an n = 10
−5 N = 10
5 for the harmonic oscillation of the nth order the value
c
10
5
K = c K (1 − 5 × 10
−10
). For a wavelength value of λ 10 5 = L/10
5
= 10
−5 m, the
deviation from the propagation velocity c K is only 5 × 10
−8 %. This deviation stays
under the value 5 × 10
−8 % for all wavelengths with λ ≥ 10
−5 m. In other words, if
the wavelength λ ≥ 10
5 a is valid for a lattice parameter a, then the dispersion can
only be measured by measuring c
(n)
K if the relative inaccuracy of the process remains
smaller than 5 × 10
−8 %.
We will now compare the oscillations of a linear chain with the longitudinal natural
oscillations of an elastic rod, as we had considered in Chap. 2. We take a homogenous rod with a constant mass density ρ and the modulus of elasticity E. For an
4 Where Does the Wave Equation Come From?
c
(n)
K =
∂ω n
∂k n
= N
2D
m
L
N
sin
L k n
N
2
2D
m
1 − cos
L k n
N
,
hence
c
(n)
K =
∂ω n
∂k n
= L
D
m
sin
L k n
N
2
1 − cos
L k n
N
and with (39)
c
(n)
K = c K =
sin
L k n
N
2
1 − cos
L k n
N
=
sin
2πn
N
2
1 − cos
2πn
N
.
(44)
Equations (43) and (44) describe the strict dispersion relation of a linear chain. An
approximation formula for a not too large x = L k n /N = 2πn/N can be found using
the Taylor series expansion according to sin x = x −
1
6
x
3
, cos x = 1 −
1
2
x
2
+
1
24
x
4 ,
so that
sin x
√
2 − 2 cos x
=
x −
1
6 x 3
x 2 −
1
12 x 4
=
1 −
1
6 x 2
1 −
1
12 x 2
= (1 −
1
6
x
2
)(1 +
1
24
x
2
) = 1 −
1
8
x
2
,
hence
c
(n)
K = c K
1 −
π
2
2
n
2
N 2
.
(44a)
Due to the quadratic dependency of n/N , Eq. (39) gives for n N a very good
approximation for Eq. (44a). The real dispersion in every linear chain of single
coupled masses is only detectable when the wavelength becomes small enough.
Using the aforementioned rod with N = 10
10 atoms in one metre, we receive
for an n = 10
−5 N = 10
5 for the harmonic oscillation of the nth order the value
c
10
5
K = c K (1 − 5 × 10
−10
). For a wavelength value of λ 10 5 = L/10
5
= 10
−5 m, the
deviation from the propagation velocity c K is only 5 × 10
−8 %. This deviation stays
under the value 5 × 10
−8 % for all wavelengths with λ ≥ 10
−5 m. In other words, if
the wavelength λ ≥ 10
5 a is valid for a lattice parameter a, then the dispersion can
only be measured by measuring c
(n)
K if the relative inaccuracy of the process remains
smaller than 5 × 10
−8 %.
We will now compare the oscillations of a linear chain with the longitudinal natural
oscillations of an elastic rod, as we had considered in Chap. 2. We take a homogenous rod with a constant mass density ρ and the modulus of elasticity E. For an