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Prove of group velocity orthogonality to slowness surface based on properties of fourier transformтезисы доклада
- Авторы: Trushin A.S., Chekalina V.A.
- Сборник: Abstr. 16 Int. Conf. for Young Researchers "Wave Electronics and Its Applications in the Information and Telecommunication Systems"
- Тезисы
- Год издания: 2013
- Место издания: Saint-Petersburg University of Aerospace Instrumentation Saint-Petersburg
- Первая страница: 35
- Последняя страница: 35
- Аннотация: The concept of acoustic slowness surface is widely used in acousto-optics. It is very convenient to determine, on base of the slowness surface, directions of acoustic beams propagation and also a general view of Schaefer-Bergmann diagrams. In order to construct the slowness surface it is necessary to solve Christoffel equation for every direction of a wave-normal vector and then to plot the inverse velocity in this direction [1]. Most often, perpendicularity of a group velocity to the slowness surface is considered either as an evident fact or as a consequence of complicated mathematical calculations. In the present work, we derive this result from properties of the Fourier transform. An acoustic field in an anisotropic media corresponds to a set of points in the Fourier-space [2], while these points belong to the slowness surface. The Fourier transform of density of force distribution determines distribution of plane wave amplitude. Relative directions of the acting force and polarizations of the acoustic plane wave determine ratio of different modes in the generated field. Thus we obtain fragments of the slowness surface. These fragments are not plane and their radius-vector does not coincide with a normal-vector. We obtain distribution of the field corresponding to such fragments using the following quantitative reasoning. A constrain of the Fourier-transform for some direction in a frequency domain leads to lack of localization of the function for this direction in a space domain. An infinitely narrow patch in the frequency domain corresponds to the infinite beam in the space domain. Normal to this patch is a direction of a corresponding beam. Shift of the patch to k leads to a harmonic modulation of the beam in space domain because of the shift-theorem for the Fourier-transform. The vector k determines direction and frequency of modulation. A bended patch in the frequency domain can be considered as two patches with an angle between them. Every patch corresponds to a beam in the space domain. Therefore a composed patch corresponds to a pair of beams. We can increase the number of patches. With an infinite number of the patches, we obtain a continuously bended patch in the frequency domain and a divergent beam in the space domain. Consequently it is possible to obtain qualitatively a direct proportion of the beam divergence to the slowness surface curvature in addition to the perpendicularity between the group velocity and the slowness surface. This research was supported by grants RFBR 12-02-31036, RFBR 12-02-33122, RFBR 12- 02-01302, RFBR 12-07-0633.
- Добавил в систему: Трушин Арсений Сергеевич