THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

116 6. Membrane and PlatesThe left-hand side of Equation (6.16) is solely a function of r while the rightdepends only on θ. In order for the equality of Equation (6.16) to prevail, bothsides of the equation must be set equal to a constant m 2 . Then we obtain from theright-hand side of Equation (6.16)d 2 dθ + 2 m2 = 0which in turn yields the harmonic solution(θ) = cos(mθ + ε)Here ε is the phase angle. The azimuthal coordinate is of periodic nature, repeatingitself every 2π radians. In order that the displacement z be single-valuedfunction of position, z(r,θ,t) must then equal itself every 2π radians, that is,z(r,θ,t) = z(r,θ + 2π, t), with the result that the constant m is constrained tointegral values m = 1, 2, 3,.... The left-hand side of Equation (6.16) thereforebecomes the Bessel’s differential equation:d 2 Rdr + 1 ( )dR2 r dr + k 2 − m2R = 0 (6.17)r 2The solution of Equation (6.17) isR(r) = AJ m (kr) + BY m (kr) (6.18)where J m (kr) and Y m (kr) are, respectively, the transcendental Bessel functions ofthe first and the second kind, each of the order m. Bessel functions are oscillatingfunctions with diminishing amplitudes for increasing kr. Y m (kr) approachesinfinity as kr → 0. The numerical values of Bessel functions and their propertiesare given in standard tables and advanced mathematical computer programs.An abbreviated set of Bessel formulas and tables art given in Appendix B. Becausethe circular membrane includes the origin r = 0 and the displacement of themembrane must remain finite at that point, it is necessary to set 1 B = 0, reducingEquation (6.18) toR(r) = AJ m (k)Applying the boundary condition R(a) = 0 requires that J m (ka) = 0. Let us designateby q mn those values of the argument ka at which the mth order Bessel functionJ m equals zero. From J m (q mn ) = 0 we can find those discrete values of k whichare given byk mn = J mn(q mn )a= J mna1 On the other hand, if an annular membrane stretched over region a < r < b (thus excluding the originr = 0) is considered, both Bessel functions must be retained in Equation (6.18) to provide the twoarbitrary constants needed for the boundary conditions at the inner and outer borders.