IntroductionSchrödinger equation
Timoshenko system
Trapped modes for an infinite nonhomogeneousTimoshenko beam
Hugo Aya1 Ricardo Cano2 Peter Zhevandrov2
1Universidad Distrital
2Universidad de La Sabana
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
Outline
1 IntroductionWaveguidesTimoshenko system
2 Schrödinger equation
3 Timoshenko systemGreen matrixOutgoing solutionTrapped modes
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
WaveguidesTimoshenko system
Embedded modes in waveguides
∆φ+ ω2φ = 0, φy|y=±d = 0, φn|r=a = 0
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
WaveguidesTimoshenko system
Waveguides
Antisymmetric modes
ω1 =π
2d, ω2 =
3π2d
a ≈ 0,352d
Linton, McIver, Wave Motion, 45(2007), 16–29
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
WaveguidesTimoshenko system
Timoshenko beam
Hagedorn, DasGupta, Vibrations and waves..., Wiley, 2007
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
WaveguidesTimoshenko system
Timoshenko
{ψ′′ + kGA(y′ − ψ) + ω2ρIψ = 0,
kGA(y′′ − ψ′) + ω2ρAy = 0.(1)
Here ω is the frequency, A is the area of the cross-section, I is itssecond moment, G is the shear modulus, k is the Timoshenko shearcoefficient. We assume that the density ρ has the form
ρ = ρ0(1 + εf (x)
),−∞ < x <∞, ε� 1,
and f (x) (the perturbation) belongs to C[−1, 1].We can assume kG ≡ G, ρ0 ≡ 1.
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
WaveguidesTimoshenko system
Spectrum Timoshenko
ω20 = GA/I
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
WaveguidesTimoshenko system
Spectrum Timoshenko
Second branch: ω > ω0 :(ψy
)∝ e±ik1,2x
k21,2 =
12
(I +
1G
)ω2 ±
√14ω4
(I − 1
G
)2
+ ω2A
First branch: 0 < ω < ω0 :(ψy
)∝ e±imx, e±lx
ω2 = ω20 − β2, 0 < β � 1
m = γ√
A + O(β2), l = β/γ + O(β3), γ =
√G +
1I
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
Shallow potential well
−ψ′′ + εV(x)ψ = Eψ, ε� 1
Landau-Lifshitz 1948, Simon 1976
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
Spectrum
Eigenvalue E = −β2, β ∼ ε, β > 0Eigenfunction: ψ ∼ e−β|x|, |x| � 1
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
The Green function
G(x, ξ) =1
2βe−β|x−ξ|
Look for the solution in the form:
ψ =∫
G(x, ξ)A(ξ) dξ
For A we obtain:
A(x) = −εV(x)∫
G(x, ξ)A(ξ) dξ
= −εV(x)∫
Gr(x, ξ)A(ξ) dξ − ε
2βA0V(x)
A0 =∫
A(ξ) dξ, Gr = G− 12β
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
Eigenfunction
Integral equation: (1 + εT̂)A = − ε
2βA0V(x),
T̂A = V∫
Gr(x, ξ)A(ξ) dξ, ‖T̂‖C[−1,1] ≤ const
Neumann series: A = (1 + εT̂)−1[− ε
2βA0V(x)
]Integrating and multiplying by β, we have
β = − ε2
∫V(x) dx + O(ε2).
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
Green matrixOutgoing solutionTrapped modes
Outgoing Green matrix
G(x, ξ) =1
2(l2 + m2)×(
−a−1e−l|x−ξ| − ib−1eim|x−ξ| sgn(x− ξ)(−e−l|x−ξ| + eim|x−ξ|)
sgn(x− ξ)(e−l|x−ξ| − eim|x−ξ|) ae−l|x−ξ| − ibeim|x−ξ|
)
a =Iβ2 − l2
GAl= β
IγA
+ O(β3),
b =Iβ2 + m2
GAm=
γ
G√
A+ O(β2),
γ =
√G +
1I
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
Green matrixOutgoing solutionTrapped modes
Green matrix
L̂G = δ(x− ξ)E
L̂ =(∂2
x − GA + Iω2 GA∂x
−GA∂x GA∂2x + Aω2
), E =
(1 00 1
)Rewrite system (1) as
L̂Ψ = −εω2f (x)JΨ, J =(
I 00 A
), Ψ =
(ψy
)
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
Green matrixOutgoing solutionTrapped modes
Solution
Look for the solution in the form
Ψ =∫
G(x, ξ)A(ξ) dξ, A =(
BD
)(2)
We obtain for A the equation
A = −εω2fJ∫
Gr(x, ξ)A(ξ) dξ +ε
2βω2fγ−1B0
(10
),
where B0 =∫
B(x) dx, Gr = G +1
2β
( 1γI 00 0
)
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
Green matrixOutgoing solutionTrapped modes
Solution
This can be rewritten as
(1 + εT̂)A =ε
2βω2fγ−1B0
(10
), ‖T̂‖C[−1,1] ≤ const
The solution: Neumann series
A = (1 + εT̂)−1[ε
2βω2fγ−1B0
(10
)]≡ ε
βB0
(UV
)(3)
Integrating and multiplying by β, we have
β = ε
∫U(x) dx =
ε
2ω2
0γ−1∫
f (x) dx + O(ε2) (4)
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
Green matrixOutgoing solutionTrapped modes
Trapped modes
Assume f (x) even and take G := ReG.
We have G =(
G11 G12G21 G22
)where G11 and G22 are even and G12 and G21 are odd. Repeating thesame procedure we obtain
A =(
BD
)where B is even and D is odd. The solution Ψ is defined as above. For|x| → ∞, its components are proportional to
sin mxW(m), W(m) =(
b−1∫
B(ξ) cos mξ dξ −∫
D(ξ) sin mξ dξ)
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
Green matrixOutgoing solutionTrapped modes
Trapped modes
We putW(m) = 0 (5)
and this guarantees that Ψ ∈ L2(−∞,∞). In the leading term thismeans ∫
f (ξ) cos mξ dξ + O(ε) = 0
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
Green matrixOutgoing solutionTrapped modes
Main result
Theorem
Let f (x) be even,∫
f (x) dx > 0. Let
A =(
BD
)be given by (3) and
Ψ =(ψy
)be given by (2). Let β > 0 be a solution of (4) and m be a solution of(5). Then Ψ is a finite energy solution os system (1).
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
Green matrixOutgoing solutionTrapped modes
Example 1: “square bump”
f =
{1, |x| < 10, |x| > 1
sin m + O(ε) = 0, m = nπ + O(ε), n = 1, 2, . . .
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam
IntroductionSchrödinger equation
Timoshenko system
Green matrixOutgoing solutionTrapped modes
Example 2: “parabolic bump”
f =
{1− x2, |x| < 10, |x| > 1
tan m + O(ε) = m
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam