[Paper Review] Semi-infinite herringbone waveguides in elastic plates

Stewart G. Haslinger, I. S. Jones+2 more|arXiv (Cornell University)|Dec 4, 2017

Ultrasonics and Acoustic Wave Propagation37 references4 citations

TL;DR

This paper proposes a novel semi-infinite herringbone waveguide in elastic Kirchhoff plates to enhance flexural wave localisation and guiding. Using a dipole approximation for closely spaced rigid pins and a wave scattering method with Wiener-Hopf analysis, it demonstrates that tuning the dipole orientation and spacing can convert a reflective grating pair into a waveguide mode, significantly improving localisation and waveguiding efficiency, especially with convex herringbone entrances.

ABSTRACT

The paper includes novel results for the scattering and localisation of a time-harmonic flexural wave by a semi-infinite herringbone waveguide of rigid pins embedded within an elastic Kirchhoff plate. The analytical model takes into account the orientation and spacing of the constituent parts of the herringbone system, and incorporates dipole approximations for the case of closely spaced pins. Illustrative examples are provided, together with the predictive theoretical analysis of the localised waveforms.

Motivation & Objective

• To design and analyze a new semi-infinite herringbone waveguide structure in elastic Kirchhoff plates for enhanced flexural wave control.
• To investigate how geometric parameters—specifically dipole orientation and spacing—affect wave localisation and guiding in structured plates.
• To develop and validate a dipole approximation method for closely spaced rigid pins in waveguide systems.
• To demonstrate the transition from reflection-dominated to waveguide-dominated behavior in grating systems via structural tuning.
• To provide a theoretical framework using wave scattering and Wiener-Hopf techniques for semi-infinite periodic waveguide arrays.

Proposed method

• A wave scattering approach is used to model flexural wave interaction with a semi-infinite herringbone array of rigid pins in an elastic plate.
• The dipole approximation is applied to model closely spaced pin pairs, treating each pair as a dipole with orientation and strength determined by spacing and position.
• A system of linear algebraic equations is derived for flexural displacement coefficients using the boundary conditions at the pin locations.
• The discrete Wiener-Hopf method is employed to solve the scattering problem, with kernel matrices constructed from quasi-periodic Green’s functions.
• Bloch modes for infinite systems are used to inform the solution of the semi-infinite problem, with zeros of the kernel determinant identifying guided modes.
• Numerical solutions are validated by comparing source and dipole coefficients and analyzing displacement fields for waveguiding and localisation effects.

Experimental results

Research questions

• RQ1 How does the orientation of dipole pairs in a herringbone waveguide affect flexural wave localisation and guiding efficiency?
• RQ2 Can a herringbone configuration transform a reflective two-grating system into a waveguide mode under the same incident wave?
• RQ3 To what extent does the dipole approximation accurately model wave scattering in closely spaced rigid pin arrays in elastic plates?
• RQ4 How does the convexity or concavity of the herringbone entrance influence the amplitude and spatial distribution of localised flexural waves?
• RQ5 What role do the spacing and relative positioning of pin pairs play in tuning the waveguiding and localisation characteristics?

Key findings

• The herringbone waveguide design significantly enhances wave localisation compared to a simple two-grating system, with peak amplitudes up to 30% higher in optimal configurations.
• For a dipole orientation of θ = 0.805 and spacing vector s = (0.01, 0.03), the waveguide achieves strong localisation and efficient energy confinement.
• A convex herringbone entrance (θ = π/4) leads to stronger localisation at the front of the system than a concave entrance (θ = 3π/4), with higher displacement peaks observed in the former.
• The dipole approximation is validated numerically: the envelope of dipole coefficients closely matches the displacement field profile, confirming its accuracy for small |s|.
• The transition from reflection to waveguiding is achieved by replacing source terms with dipoles, demonstrating a design strategy to convert blockage into leaky waveguide behavior.
• The kernel matrix zeros from the Wiener-Hopf formulation correspond to Bloch modes, enabling identification of guided modes and improving solution convergence.