Eigenmode analysis and degree of freedom in chaotic semi-stadium sound fields

This paper describes eigenfrequency spacing statistics, including modal patterns and degrees of freedom, in a semi-stadium type 2D field. The authors investigated numerically the sound fields surrounded by 2D semi-stadium type of boundaries as examples of boundaries where chaotic properties are hidden. One limit of the semi-stadium boundaries is a rectangle which gives a regular field, while another limit is a stadium boundary where the chaotic property emerges. The numerical results show that eigenfrequency spacing in all the cases can be expressed as a family of Γ distributions extended to a non-integer degree of freedom. This fractal degree of freedom might be interpreted as the degree of freedom of the sound field. For the regular limit case, that is, a rectangular case, the distribution is the exponential distribution with a degree of freedom of unity, while in the chaotic case, that is, the stadium case, it is the Wigner distribution with a degree of freedom of two. Moreover, modal patterns clearly show breaks of the regular pattern of nodal lines seen in a rectangular case as the boundary is deformed from the rectangular to the stadium condition