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Introduction
Schrödinger equation
Timoshenko system
Trapped modes for an inﬁnite nonhomogeneous
Timoshenko beam
Hugo Aya1 Ricardo Cano2 Peter Zhevandrov2
1Universidad Distrital
2Universidad de La Sabana
H. Aya, R. Cano and P. Zhevandrov Timoshenko beam

Introduction
Timoshenko system
Outline
1 Introduction
Waveguides
Timoshenko system
2 Schrödinger equation
3 Timoshenko system
Green matrix
Outgoing solution
Trapped modes

Introduction
Timoshenko system
Waveguides
Timoshenko system
Embedded modes in waveguides
∆φ + ω2
φ = 0, φy|y=±d = 0, φn|r=a = 0

Introduction
Timoshenko system
Waveguides
Timoshenko system
Waveguides
Antisymmetric modes
ω1 =
π
2d
, ω2 =
3π
2d
a ≈ 0,352d
Linton, McIver, Wave Motion, 45(2007), 16–29

Introduction
Timoshenko system
Waveguides
Timoshenko system
Timoshenko beam
Hagedorn, DasGupta, Vibrations and waves..., Wiley, 2007

Introduction
Timoshenko system
Waveguides
Timoshenko system
Timoshenko
ψ + kGA(y − ψ) + ω2
ρIψ = 0,
kGA(y − ψ ) + ω2
ρAy = 0.
(1)
Here ω is the frequency, A is the area of the cross-section, I is its
second moment, G is the shear modulus, k is the Timoshenko shear
coefﬁcient. 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.

Introduction
Timoshenko system
Waveguides
Timoshenko system
Spectrum Timoshenko
ω2
0 = GA/I

Introduction
Timoshenko system
Waveguides
Timoshenko system
Spectrum Timoshenko
Second branch: ω > ω0 :
ψ
y
∝ e±ik1,2x
k2
1,2 =
1
2
I +
1
G
ω2
±
1
4
ω4 I −
1
G
2
+ ω2A
First branch: 0 < ω < ω0 :
ψ
y
∝ e±imx
, e±lx
ω2
= ω2
0 − β2
, 0 < β 1
m = γ
√
A + O(β2
), l = β/γ + O(β3
), γ = G +
1
I

Introduction
Timoshenko system
Shallow potential well
−ψ + V(x)ψ = Eψ, 1
Landau-Lifshitz 1948, Simon 1976

Introduction
Timoshenko system
Spectrum
Eigenvalue E = −β2, β ∼ , β > 0
Eigenfunction: ψ ∼ e−β|x|, |x| 1

Introduction
Timoshenko system
The Green function
G(x, ξ) =
1
2β
e−β|x−ξ|
Look for the solution in the form:
ψ = G(x, ξ)A(ξ) dξ
For A we obtain:
A(x) = − V(x) G(x, ξ)A(ξ) dξ
= − V(x) Gr(x, ξ)A(ξ) dξ −
2β
A0V(x)
A0 = A(ξ) dξ, Gr = G −
1
2β

Introduction
Timoshenko system
Eigenfunction
Integral equation: (1 + ˆT)A = −
2β
A0V(x),
ˆTA = V Gr(x, ξ)A(ξ) dξ, ˆT C[−1,1] ≤ const
Neumann series: A = (1 + ˆT)−1
−
2β
A0V(x)
Integrating and multiplying by β, we have
β = −
2
V(x) dx + O( 2
).

Introduction
Timoshenko system
Green matrix
Outgoing solution
Trapped 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
I
γA
+ O(β3
),
b =
Iβ2 + m2
GAm
=
γ
G
√
A
+ O(β2
),
γ = G +
1
I

Introduction
Timoshenko system
Green matrix
Outgoing solution
Trapped modes
Green matrix
ˆLG = δ(x − ξ)E
ˆL =
∂2
x − GA + Iω2 GA∂x
−GA∂x GA∂2
x + Aω2 , E =
1 0
0 1
Rewrite system (1) as
ˆLΨ = − ω2
f(x)JΨ, J =
I 0
0 A
, Ψ =
ψ
y

Introduction
Timoshenko system
Green matrix
Outgoing solution
Trapped modes
Solution
Look for the solution in the form
Ψ = G(x, ξ)A(ξ) dξ, A =
B
D
(2)
We obtain for A the equation
A = − ω2
fJ Gr(x, ξ)A(ξ) dξ +
2β
ω2
fγ−1
B0
1
0
,
where B0 = B(x) dx, Gr = G +
1
2β
1
γI 0
0 0

Introduction
Timoshenko system
Green matrix
Outgoing solution
Trapped modes
Solution
This can be rewritten as
(1 + ˆT)A =
2β
ω2
fγ−1
B0
1
0
, ˆT C[−1,1] ≤ const
The solution: Neumann series
A = (1 + ˆT)−1
2β
ω2
fγ−1
B0
1
0
≡
β
B0
U
V
(3)
Integrating and multiplying by β, we have
β = U(x) dx =
2
ω2
0γ−1
f(x) dx + O( 2
) (4)

Introduction
Timoshenko system
Green matrix
Outgoing solution
Trapped modes
Trapped modes
Assume f(x) even and take G := ReG.
We have G =
G11 G12
G21 G22
where G11 and G22 are even and G12 and G21 are odd. Repeating the
same procedure we obtain
A =
B
D
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

Introduction
Timoshenko system
Green matrix
Outgoing solution
Trapped modes
Trapped modes
We put
W(m) = 0 (5)
and this guarantees that Ψ ∈ L2(−∞, ∞). In the leading term this
means
f(ξ) cos mξ dξ + O( ) = 0

Introduction
Timoshenko system
Green matrix
Outgoing solution
Trapped modes
Main result
Theorem
Let f(x) be even, f(x) dx > 0. Let
A =
B
D
be given by (3) and
Ψ =
ψ
y
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).

Introduction
Timoshenko system
Green matrix
Outgoing solution
Trapped modes
Example 1: “square bump”
f =
1, |x| < 1
0, |x| > 1
sin m + O( ) = 0, m = nπ + O( ), n = 1, 2, . . .

Introduction
Timoshenko system
Green matrix
Outgoing solution
Trapped modes
Example 2: “parabolic bump”
f =
1 − x2, |x| < 1
0, |x| > 1
tan m + O( ) = m

Introduction
Timoshenko system
Green matrix
Outgoing solution
Trapped modes
Example 3: Gaussian
f = exp(−x2
/2)
f(x) cos mx dx =
√
2π exp(−m2
/2) = 0
NO trapped modes!

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