Abstract: By analyzing the symmetric vibration on a disk and a sphere surface, the curves of the zero-order Bessel function and the Legendre function are compared. It is shown that there are apparent similarities between them. After transformation of amplitude, the curves of the Bessel function agree with those of the Legendre function very well. The reasons for the similarities are explained both in mathematical and physical approaches. The asymptotic relationship between the functions of two kinds and an estimating formula for zeros of the Bessel function are derived.

Key words: standing wave mode, Bessel function, Legendre function, eigenvalue, asymptotic relationship