Trapped Modes and Active Control in Rectangular Channels with Perturbed Bottoms

Abstract Background The existence of embedded trapped modes in infinite waveguides is closely related to the geometry of the system, which may give rise to resonant phenomena. However, in finite waveguides, this behavior may be modified by the boundary conditions and the dimensions of the domain. When resonance occurs in a finite guide, it becomes necessary to implement control strategies to mitigate its effects and avoid possible compromises in the structural stability of the system. Goals To identify the trapped mode in an infinite channel with a bottom perturbation. To describe the continuous spectrum associated with the infinite guide in order to compare it with the discrete spectrum of the finite channel when its length L is much greater than its width d. To apply a velocity feedback control strategy to mitigate the oscillations of the system without compromising its dynamics. Methods Using Green’s functions, the frequency associated with the trapped mode in an infinite waveguide with a bottom perturbation is calculated. Subsequently, by means of traditional analytical methods, the behavior of the discrete spectrum is described in the limit as $$L\rightarrow\infty$$ . Numerical methods are employed to verify the existence of resonance in the finite guide when the excitation frequency coincides with the trapped-mode frequency. Finally, a velocity feedback controller is implemented to reduce the amplitude of the system oscillations. Results The frequency associated with the embedded trapped mode in the infinite channel was calculated. In addition, it was shown that, when $$L<\infty\;and\;L\gg d$$ , the discrete spectrum of the finite channel behaves similarly to the continuous spectrum of the infinite guide. Through numerical simulations, the existence of resonance in the finite guide was verified when the oscillation frequency is equal to the frequency of the embedded trapped mode. Finally, the application of the velocity feedback controller resulted in a $$23.1\%$$ reduction in amplitude growth.