Eigenfrequency spacing analysis and eigenmode breakdown for semi-stadium-type 2-D fields

This paper describes eigenfrequency statistics including modal patterns and degrees of freedom for a semi-stadium-type 2-D field. The authors numerically investigated sound fields surrounded by 2-D semi-stadium-type boundaries as examples of boundaries where chaotic properties are hidden, in order to understand the characteristics of complex sound fields and to gain new insight into sound-field design. One limit of the semi-stadium boundaries is a rectangular boundary that gives a regular field, while another limit is a stadium boundary in which 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 can be interpreted as the degree of freedom of the sound field. For the regular limit case, i.e., a rectangular case, the distribution is the exponential distribution with the freedom of unity, while in the chaotic case, i.e., the stadium case, it is the Wigner distribution with a degree of freedom of two. The authors, however, analyze the semi-stadium sound fields. The analysis of the fluctuations in the distribution function for the eigenfrequencies showed that the skewness decreases as the boundary approaches the stadium condition. Modal patterns also clearly showed breaks of the regular pattern of nodal lines in a rectangular case as the boundary was deformed from the rectangular to the stadium condition. The breaks of the modal pattern could be also confirmed by decreasing of the skewness for the sound pressure distribution.