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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


Timoshenko system

Outline

1 IntroductionWaveguidesTimoshenko system

2 Schrödinger equation

3 Timoshenko systemGreen matrixOutgoing solutionTrapped modes



Timoshenko system

WaveguidesTimoshenko system

Embedded modes in waveguides

∆φ+ ω2φ = 0, φy|y=±d = 0, φn|r=a = 0



Timoshenko system


Waveguides

Antisymmetric modes

ω1 =π

2d, ω2 =

3π2d

a ≈ 0,352d

Linton, McIver, Wave Motion, 45(2007), 16–29



Timoshenko system


Timoshenko beam

Hagedorn, DasGupta, Vibrations and waves..., Wiley, 2007



Timoshenko 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.



Timoshenko system


Spectrum Timoshenko

ω20 = GA/I



Timoshenko 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



Timoshenko system

Shallow potential well

−ψ′′ + εV(x)ψ = Eψ, ε� 1

Landau-Lifshitz 1948, Simon 1976



Timoshenko system

Spectrum

Eigenvalue E = −β2, β ∼ ε, β > 0Eigenfunction: ψ ∼ e−β|x|, |x| � 1



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



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).



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



Timoshenko system


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

)



Timoshenko system


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

)



Timoshenko system


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)



Timoshenko system


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)



Timoshenko system


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



Timoshenko system


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).



Timoshenko system


Example 1: “square bump”

f =

{1, |x| < 10, |x| > 1

sin m + O(ε) = 0, m = nπ + O(ε), n = 1, 2, . . .



Timoshenko system


Example 2: “parabolic bump”

f =

{1− x2, |x| < 10, |x| > 1

tan m + O(ε) = m



Timoshenko system


Example 3: Gaussian

f = exp(−x2/2)∫f (x) cos mx dx =

√2π exp(−m2/2) 6= 0

NO trapped modes!


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