This paper proposes an analytical model to describe the interaction of a bounded ultrasonic beam with an immersed plate. This model, based on the Gaussian beams decomposition, takes into account multiple reflections into the plate. It allows predicting three-dimensional spatial distributions of both transmitted and reflected fields. Thereby, it makes it easy to calculate the average pressure over the receiver’s area taking into account diffraction losses. So the acoustical parameters of the plate can be determined more accurately. A Green’s function for the interaction of an ultrasonic beam with the plate is derived. The obtained results are compared to those given by the angular spectrum approach. A good agreement is seen showing the validity of the proposed model.
We derive exact and approximate controllability conditions for the linear one-dimensional heat equation in an infinite and a semi-infinite domains. The control is carried out by means of the time-dependent intensity of a point heat source localized at an internal (finite) point of the domain. By the Green’s function approach and the method of heuristic determination of resolving controls, exact controllability analysis is reduced to an infinite system of linear algebraic equations, the regularity of which is sufficient for the existence of exactly resolvable controls. In the case of a semi-infinite domain, as the source approaches the boundary, a lack of L2-null-controllability occurs, which is observed earlier by Micu and Zuazua. On the other hand, in the case of infinite domain, sufficient conditions for the regularity of the reduced infinite system of equations are derived in terms of control time, initial and terminal temperatures. A sufficient condition on the control time, heat source concentration point and initial and terminal temperatures is derived for the existence of approximately resolving controls. In the particular case of a semi-infinite domain when the heat source approaches the boundary, a sufficient condition on the control time and initial temperature providing approximate controllability with required precision is derived.