Dynamic imaging of a capillary-gravity wave in shallow water using amplitude variations of eigenbeams

Abstract

Dynamic acoustic imaging of a surface wave propagating at an air-water interface is a complex task that is investigated here at the laboratory scale through an ultrasonic experiment in a shallow water waveguide. Using a double beamforming algorithm between two source-receiver arrays, the authors isolate and identify each multi-reverberated eigenbeam that interacts with the air-water and bottom interfaces. The waveguide transfer matrix is recorded 100 times per second while a low-amplitude gravity wave is generated by laser-induced breakdown at the middle of the waveguide, just above the water surface. The controlled, and therefore repeatable, breakdown results in a blast wave that interacts with the air-water interface, which creates ripples at the surface that propagate in both directions. The amplitude perturbations of each ultrasonic eigenbeam are measured during the propagation of the gravity-capillary wave. Inversion of the surface deformation is performed from the amplitude variations of the eigenbeams using a diffraction-based sensitivity kernel approach. The accurate ultrasonic imaging of the displacement of the air-water interface is compared to simultaneous measurements with an optical camera, which provides independent validation.

Surface perturbation inverted from angle variations of eigenbeams in an ultrasonic waveguide

“…The waveguide is 55 mm deep and 0.6 m long for the FM experiment, and 55 mm deep and 1 m long for the IP experiment. The surface perturbation in the IP experiment is independently measured by optical means for validation purposes (not shown here; see van Baarsel et al, 2019). The optical measurements are carried out using two cameras: one that films the water surface from the top and records the frequency-wavenumber information of the perturbation; the second that views the perturbation from the side and records the height of the perturbation.…”

“…For each eigenbeam, the variation of the doublebeamformed pressure dp DBF ðh e ; h e ; tÞ is the difference between the pressure p DBF ðh e ; h r ; tÞ in the perturbed waveguide and the rest state p DBF 0 ðh e ; h r ; tÞ in the unperturbed waveguide. In the small perturbation assumption (van Baarsel et al, 2019), this difference can be linked to the surface perturbation Dhðr 0 Þ for each point at the surface r 0 using the SK formulation, i.e.,…”

“…(3). The study of the travel-time and amplitude variations dp DBF due to a surface perturbation from the SK approach has been discussed previously (Roux and Nicolas, 2014;Sarkar et al, 2012;van Baarsel et al, 2019).…”

“…As the purpose of this paper is to invert surface fluctuations from angle variations only, we do not consider from here on the travel-time perturbations in the recovery of the surface perturbation of the waveguide. For discussions about surface fluctuations inverted with travel-time or amplitude perturbations, please refer to previous studies (Roux and Nicolas, 2014;van Baarsel et al, 2019).…”

“…In this paper, we propose to use only the two new angle observables, the emitting and receiving angles of each eigenbeam, to localize and estimate the perturbation of the air-water interface at the surface of an ultrasonic waveguide. This is achieved using the SKs for the two-dimensional (2D) surface approach, which has been used previously for traveltimes and amplitudes (Roux and Nicolas, 2014;Sarkar et al, 2012;van Baarsel et al, 2019), and is here developed for the eigenbeam angles.…”