170
K. S. Sreelatha
m i ¨
a = K (a i+1 − a i + α(a i+1 − a 2
i + β(a i+1 − a 3
i + · · · ) − K (a i − a i−1 + α(a i − a i−1 ) 2 + · · · )
(6)
Assuming that the inhomogeneity is very small and does not depend on time, and
making the following transformations
m i = ˜
m(1 + ρ)
(7)
ρ = ρ 1 +
2
ρ 2 + · · ·
(8)
with ˜
m is the average mass and ρ 1 , ρ 2 , . . . are functions of the lattice site i. Considering
the lattice spacings as h in the x-direction and k in the y-direction. Then, a i =
a i x, y, t We considered the following three wave motions:
(a) Slowly varying in x, y and t for quadratic nonlinearity, α = 0, β = 0
(b) Slowly varying in x, y and t for cubic nonlinearity,α = 0, β > 0
(c) Slowly varying in x, y and t for quadratic nonlinearity along with cubic nonlinearity, α > 0, β > 0.
When the wavelength is very large compared to the spacing of particles in a lattice,
one can make the Taylor expansion on a i+1 . To study the wave propagation through a
2D lattice, we consider the Taylor series expansion for two variables to expand near
a i+1 :
a i+1 = a i + ha x + ka y +
1
2
[h
2 a xx + 2kha x a y + k
2 a yy ] + · · ·
(9)
where a x and a y are corresponding derivatives of a i
Case(a): Quadratic nonlinearity α = 0, β = 0.
For β = 0, Eq. (6) becomes
m i ¨
a = K (a i+1 − a i + α(a i+1 − a i )
2
+ · · · − (a i − a i−1 )α(a i − a i−1 )
2
− · · ·
(10)
From Eqs. (6), (7), (8), (9) and (10), we get
(1 + ρ) ¨
a i =
K
˜
m
h 2 a x x + k 2 a yy + 2hka xy + 2αh 3 a x a x x + 2αhk 2 a x a yy + 2kh 2 a x x a y
+ 2αk 3 a yy a y +
h 3
12
a x x x + · · ·
(11)
Introducing change of variables x, y and t to ζ , η and τ using the following transformations:
η =
h
(x − vt), ζ =
2
k
y, τ =
3
24h
t
(12)
where v is the velocity of sound given by v = h
K
˜
m
Also