A perturbative approach to acoustic scattering from a vibrating bounded obstacle
Francesca Mariani
Dipartimento di Matematica ``Ulisse Dini''
Università di Firenze
Viale Morgagni 67/a, 50134 Firenze, Italy
e-mail : fdmariani@libero.it
Maria Cristina Recchioni
Istituto di Teoria delle Decisioni e Finanza Innovativa
Università di Ancona
Piazza Martelli 8, 60121 Ancona, Italy
e-mail : recchioni@posta.econ.unian.it
Francesco Zirilli
Dipartimento di Matematica "G. Castelnuovo"
Università di Roma "La Sapienza"
00185 Roma, Italy
e-mail : f.zirilli@caspur.it
In this paper we study a mathematical model to describe a three dimensional acoustic scattering problem associated to a "vibrating" obstacle that is a bounded simply connected domain contained in the three dimensional real Euclidean space whose shape changes in time. In particular we propose a highly parallelizable numerical method based on the operator expansion method to solve the mathematical model considered. A "vibrating" obstacle is an obstacle whose shape depends continuously on time, and is a periodic function of time. We distinguish a time independent shape of the obstacle, called obstacle at rest, that is the "mean value" of the vibrating obstacle. The shape of the vibrating obstacle and its acoustic boundary impedance are known for all times. The acoustic boundary impedance is assumed to be a non-negative constant. The limit cases of acoustically soft and of acoustically hard vibrating obstacle are considered. We assume that for each value of the time variable the boundary of the obstacle is a locally Lipschitz surface. The vibrating obstacle is contained in the three dimensional real Euclidean space filled with a homogeneous isotropic medium. We consider the following scattering problem: find the acoustic wave scattered by the vibrating obstacle when hit by a known incoming time harmonic acoustic wave. From the knowledge of the scattered wave we compute the acoustic Doppler spectrum associated to the far field pattern. Under some assumptions, the mathematical model proposed to describe the scattering problem considered reduces the solution of the scattering problem to the solution of an exterior boundary value problem depending on a parameter, the time variable, for the Helmholtz equation with a boundary condition prescribed on the boundary of the vibrating obstacle and a suitable radiation condition at infinity. We solve this exterior boundary value problem depending on a parameter perturbatively, that is we reduce the solution of the boundary value problem depending on a parameter with the boundary condition prescribed on the boundary of the vibrating obstacle to the solution of a sequence of exterior boundary value problems for the Helmholtz equation with the boundary condition prescribed on the boundary of the obstacle at rest. That is at each order in perturbation theory we solve an exterior problem for the Helmholtz equation with boundary data depending on a parameter. The solution of the boundary value problem associated to the first order correction in perturbation theory due to the vibrations allows us to characterize in a simple case the type of "vibration" of the obstacle (see Section 4 Figs. 4.5-4.7). We propose some elementary observations interesting from the physical point of view. In particular we show that the Doppler spectrum associated to the far field patterns of the scattered acoustic fields depends mainly from the incoming wave and from the excited vibrational modes (Figs. 4.9-4.13). To show this we consider the simplest case when the incoming wave is a plane wave and the mechanical vibration is characterized by only one time frequency. We study how the interaction between the acoustic and the mechanical waves depends on the shape of the obstacle at rest, on the propagation direction of the incoming wave, and on the observation direction of the scattered wave and we observe that this interaction depends essentially only on the acoustic and the mechanical vibrational frequency of the observed phenomenon. The study of these vibrations starting from the scattered waves can be seen as a first attempt to solve inverse acoustic problems concerning the reconstruction of the shape and the vibrations of an obstacle. The exterior boundary value problems coming from perturbation theory are solved using the so called "operator expansion" method in the form proposed in [1]. This method makes possible to obtain highly parallelizable algorithms able to compute the solution of the problem considered order by order in perturbation theory, and able to obtain the required solution of the scattering problem summing up the perturbation series. Really impressive speed up factors are observed and reported when the algorithm is executed on the Chiba Cluster, a parallel machine of the Argonne National Laboratory - USA. We validate the mathematical model and the numerical method proposed solving some test problems. The quantitative character of the numerical results obtained is established. The results obtained on the test problems are discussed both from the numerical and the physical point of view.
In the following animations and Applets we show the real part of the incident and scattered acoustic fields on the colour scale show in the figure below.
"i" is the imaginary unit, c=1532.8 (sound velocity in the sea water in the MKS units)
Vibrating obstacle (epsilon=0.5). Vibration in the north-south direction ( File GIF )
"i" is the imaginary unit, c=1532.8 (sound velocity in the sea water in the MKS units)
Vibrating obstacle (epsilon=0.5). Vibration in the equatorial plane ( File GIF )
Remark: The sounds attached to the movies of the Applets have been processed to emphasize the difference between them.