| Abstract: | In this paper, exact vibration frequencies of circular, annular and sector
membranes with a radial power law density are presented for the first time. It is
found that in general, the sequence of modes may not correspond to increasing azimuthal mode number $n$. The normalized frequency increases with the absolute
value of the power index $|ν|$. For a circular membrane, the fundamental frequency
occurs at $n = 0$ where $n$ is the number of nodal diameters. For an annular membrane, the frequency increases with respect to the inner radius $b$. When $b$ is close
to one, the width $1 − b$ is the dominant factor and the differences in frequencies are
small. For a sector membrane, $n − 1$ is the number of internal radial nodes and the
fundamental frequency occurs at $n = 1$. Increased opening angle $β$ increases the
frequency. |
