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Factorized dispersion relations for two coupled systems

Alexander Figotin

Abstract

We establish that the dispersion relations of any physical system composed of two coupled subsystems, governed by a space-time homogeneous Lagrangian, admit a factorized form G_{1}G_{2}=γG_{\mathrm{c}}, where G_{1} and G_{2} are the subsystem dispersion functions, G_{\mathrm{c}} is the coupling function, and γis the coupling parameter. The result follows from a determinant expansion theorem applied to the block structure of the coupled system matrix, and is illustrated through three examples: the traveling wave tube, vibrations of an airplane wing, and the Mindlin-Reissner plate theory. For the Mindlin-Reissner example we carry out a complete asymptotic analysis of the coupled dispersion branches, establishing that the factorized form provides a precise quantitative measure of mode hybridization: all four branches carry the imprint of both subsystem factors for any nonzero coupling, while asymptotically recovering the identity of pure uncoupled modes at large frequencies and wavenumbers. We further analyze the universal local geometry of the coupled dispersion branches near their intersection - the cross-point model - showing it is generically hyperbolic, and present a mechanical analog in which the wavenumber is replaced by a scalar parameter, exhibiting the same factorized structure and avoided crossin

Factorized dispersion relations for two coupled systems

Abstract

We establish that the dispersion relations of any physical system composed of two coupled subsystems, governed by a space-time homogeneous Lagrangian, admit a factorized form G_{1}G_{2}=γG_{\mathrm{c}}, where G_{1} and G_{2} are the subsystem dispersion functions, G_{\mathrm{c}} is the coupling function, and γis the coupling parameter. The result follows from a determinant expansion theorem applied to the block structure of the coupled system matrix, and is illustrated through three examples: the traveling wave tube, vibrations of an airplane wing, and the Mindlin-Reissner plate theory. For the Mindlin-Reissner example we carry out a complete asymptotic analysis of the coupled dispersion branches, establishing that the factorized form provides a precise quantitative measure of mode hybridization: all four branches carry the imprint of both subsystem factors for any nonzero coupling, while asymptotically recovering the identity of pure uncoupled modes at large frequencies and wavenumbers. We further analyze the universal local geometry of the coupled dispersion branches near their intersection - the cross-point model - showing it is generically hyperbolic, and present a mechanical analog in which the wavenumber is replaced by a scalar parameter, exhibiting the same factorized structure and avoided crossin
Paper Structure (25 sections, 3 theorems, 179 equations, 4 figures)

This paper contains 25 sections, 3 theorems, 179 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Review of Lagrangians with higher derivatives and the dispersion relations
  3. The Lagrangian and the Euler equations
  4. The dispersion relations
  5. Coupled systems and the factorized form of the dispersion relations
  6. Setting up the coupled system
  7. Factorized form of the dispersion relation
  8. Physically appealing examples
  9. Traveling wave tube
  10. Vibration of an airplane wing
  11. Mindlin-Reissner theory for plates
  12. Hybridization of modes
  13. Dispersion relation $f(k,\omega)=0$.
  14. Dispersion relation $A(k,\omega)=0$.
  15. Asymptotic analysis: hybridization of modes.
  16. ...and 10 more sections

Key Result

Theorem 1

Let $A$ and $B\left(b\right)$ be two $n\times n$ matrices with $n\geq2$. Assume also that $b$ is a complex number and matrix $B\left(b\right)$ depends on $b$ polynomially, that is where $m\geq0$ is an integer and $B^{\left(s\right)}$ are $n\times n$ matrices. Then $\det\left\{ A+bB\left(b\right)\right\}$ is a polynomial function of $b$ satisfying the following representation where Coefficient $

Figures (4)

  • Figure 4.1: Dispersion relation $f(k,\omega)=0$ for data sets (\ref{['eq:pLdisdat1a']}). Dashed blue: $b=0$; solid crimson: $b=0.1$; solid dark green: $b=0.2$. The coupling lifts the solid branches off the origin.
  • Figure 4.2: Dispersion relation $A(k,\omega)=0$ for data set (\ref{['eq:pLdisdat1a']}). Dashed blue: $b=0$; solid crimson: $b=0.1$; solid dark green: $b=0.2$. (a) Full view, $k\in(-0.3,0.3)$; (b) zoomed view of lower branches near origin, $k\in(-0.05,0.05)$, with dotted curves showing the parabolic approximation $\omega=k^{2}/b$. The two upper branches are lifted off the origin by coupling, while the two lower branches are pinned at the origin with parabolic tangency.
  • Figure 5.1: Cross-point principle model dispersion relations (\ref{['eq:GGgamG2e']}) for $g_{1}=1$, $g_{2}=10$, $\gamma=0.4,2,4$: (a) $g_{\gamma}=+1>0$; (b) $g_{\gamma}=-1<0$. Dashed blue: uncoupled reference lines ($\gamma=0$); solid royal blue, crimson, dark green: coupled curves for $\gamma=0.4,2,4$ respectively. Curves farther from the dashed reference lines correspond to larger values of $\gamma$.
  • Figure 5.2: Eigenfrequencies $\omega_{\pm}\left(p\right)$ for the parameter set (\ref{['eq:dataMech']}) and $b=0$ (black), $0.2$ (blue), $0.4$ (green), $0.6$ (orange). Solid lines: upper branch $\omega_{+}$; dashed lines: lower branch $\omega_{-}$. The open circle marks the bare crossing point at $p^{*}=1/11$, $\omega=\Omega^{*}\approx1.044$. The avoided crossing for $b>0$ is the mechanical analog of the cross-point dispersion relation (\ref{['eq:GGgamG2e']}).

Theorems & Definitions (6)

  • Theorem 1: determinant of the coupled systems matrix
  • proof
  • Remark 2: factorized dispersion relation
  • Remark 3: centroid
  • Theorem 4: Laplace expansion theorem
  • Lemma 5