Chaotic properties and eigenfrequency distribution of Semi-Stadium 2-D Fields

Abstract

This paper describes the eigenfrequency distribution of semi-stadium 2-D fields whose geometrical figure is formed by rectangular and hyper-circular segments. The sound ray propagation in the fields exhibits chaotic properties when order n of the hyper-circular parts defined by xn+yn=rn decreases to 2. The chaotic behavior can be estimated according to the degree of freedom of the gamma distribution that represents the eigenfrequency spacing statistics and the correlation dimensions of the chaotic structure. The experimental and simulation results of the 2-D-membrane and plate vibration analysis show that the degree of freedom of the gamma distribution changes from 1 to 2 including nonintegers as the boundary of the 2-D space changes from rectangular (n: infinity, ‘‘regular field’’) to stadium (n: 2, ‘‘irregular field’’), which corresponds to the change in the correlation dimensions from 1 to 2 for the sound ray propagation in the field. The family of gamma distributions includes the Wigner distribution, which Lyon [J. Acoust. Soc. Am. 45, 545–565 (1969)] assumed for irregularly shaped boundaries as the case where the degree of freedom is 2.