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

6.4 Freely Vibrating Circular Membrane with Fixed Rim 117A number of values of q mn that yield zeros of the Bessel functions are given inAppendix B. We can therefore write the normal modes of vibration asz mn (r,θ,t) = A mn J n (k mn r) cos(mθ + ε mn )e iω mnt(6.19)with k mn a = q mn and the natural frequencies found fromf mn = 1 q mn c(6.20)2π aThe real part of Equation (6.19) describes the physical displacement in thenormal mode (m, n) as follows:z mn (r,θ,t) = A mn J m (k mn r) cos(mθ + ε mn ) cos(ω mn t + φ mn )where A mn = A mn = e iφmn . The arbitrary constant ε mn is an azimuthal phase angle.For each normal mode, this constant of integration defines the directions alongwhich the radial nodal lines of zero displacement occur, but the value of ε mndepends on the value of the azimuthal angle at which the membrane is excited att = 0. A number of the first few (and simpler) modes of vibration with ε mn = 0 areshown in Figure 6.3. Each mode is designated by the ordered pair of integers (m, n).Integer m governs the number of radial nodal lines, whereas integer n determinesthe number of azimuthal nodal circles. Since mode (0, 0) would obviously be trivial,Figure 6.3. Vibration modes in a circular membrane fixed at its perimeter. A number ofsimpler modes are shown here.