Retrofocusing of Acoustic Wave Fields by Iterated Time Reversal

• Published in: SIAM journal on applied mathematics Vol. 64; no. 6; pp. 1954 - 1986
• Main Authors: B. Lars G. Jonsson, Gustafsson, Mats, Weston, Vaughan H., de Hoop, Maarten V.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2004
• Subjects: acoustic; Acoustic ducts; Acoustics; Algorithms; Alliances; Annan elektroteknik och elektronik; Applied mathematics; Boundary conditions; Control theory; Electrical Engineering, Electronic Engineering, Information Engineering; Electrical impedance; Elektroteknik och elektronik; Engineering and Technology; Half spaces; mirrors; Other Electrical Engineering, Electronic Engineering, Information Engineering; Particle velocity; Pressure distribution; Pressure pulses; retrofocusing; Teknik; time reversal; Transducers; Wave equations; Wave fronts; Wave propagation; wave splitting
• ISSN: 0036-1399; 1095-712X; 1095-712X
• DOI: 10.1137/S0036139903426964

In the present paper an iterative time-reversal algorithm that retrofocuses an acoustic wave field to its controllable part is established. For a fixed temporal support, i.e., transducer excitation time, the algorithm generates an optimal retrofocusing in the least-squares sense. Thus the iterative time-reversal algorithm reduces the temporal support of the excitation from the requirement of negligible remaining energy to the requirement of controllability. The time-reversal retrofocusing is analyzed from a boundary-control perspective where time reversal is used to steer the acoustic wave field towards a desired state. The wave field is controlled by transducers located at subsets of the boundary, i.e., the controllable part of the boundary. The time-reversal cavity and time-reversal mirror cases are analyzed. In the cavity case, the transducers generate a locally plane wave in the fundamental mode through a set of ducts. Numerical examples are given to illustrate the convergence of the iterative time-reversal algorithm. In the mirror case, a homogeneous half space is considered. For this case the analytic expression for the retrofocused wave field is given for finite temporal support. It is shown that the mirror case does not have the same degree of steering as the cavity case. It is also shown that the pressure can be perfectly retrofocused for infinite temporal support. Two examples are given that indicate that the influence of the evanescent part of the wave field is small.