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

6.4 Freely Vibrating Circular Membrane with Fixed Rim 115Figure 6.2. Four modes of a vibrating membrane. Lines located within the borders of themembranes constitute nodal loci where displacements are zero for these respective modes.6.4 Freely Vibrating Circular Membrane with Fixed RimAs mentioned in the foregoing it is preferable to adopt the polar coordinate version(6.4) of the wave equation (6.3) to treat the case of a circular membrane that isfixed at its rim. Accordingly, the zero displacement of the membrane’s boundaryat radius r = a gives the boundary conditionz(a,θ,t) = 0The harmonic solution to Equation (6.4) can be represented as the product of threeterms, each of which is functions of only one variable:z(r,θ,t) = R(r)(θ)e iωt (6.14)The boundary condition stipulated at the rim of the circular membrane now becomesR(a) = 0and insertion into Equation (6.4) yields the polar coordinate version of theHelmholtz equation: d2 Rdr + dR2 r dr + R d 2 r 2 dθ + 2 k2 R = 0 (6.15)where k = ω/c, as before. Equation (6.15) is then rearranged to effect the separationof two variables, so we obtainr 2 ( d 2 RR dr + 1 )dR+ (kr) 2 =− 1 d 2 (6.16)2 r dr dθ 2