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

114 6. Membrane and PlatesBecause the three terms of Equation (6.9) cannot all sum to zero and the first andsecond terms are wholly independent of each other, we separate Equation (6.9)into two separate differential equations, one wholly dependent on x and the otheron y:where kx 2 and k2 y are constants related by1 ∂ 2 XX ∂x + 2 k2 x = 0(6.10)1 ∂ 2 YY ∂y + 2 k2 y = 0k 2 x + k2 y = k2The solutions to the equation set (6.10) consists of sinusoids, with the resultz(x, y, t) = α sin(k x x + φ x ) sin(k y y + φ y )e iωt (6.11)Here α represents the maximum displacement of the membrane in the transversedirection, and φ x and φ y are determined by boundary conditions. With the firstand the third of the boundary condition set (6.7) we find that φ x = φ y = 0 and theremaining conditions necessitate that sin k x L x = 0 and sin k y L y = 0. Therefore,the normal modes occur fromz(x, y, t) = αe iωt sin k x x sin k y y (6.12)for which k x and k y turn out to be discrete values established byk x = nπ/L x n = 1, 2, 3,...k y = mπ/L y m = 1, 2, 3,...The frequencies of the allowed modes of vibrations are found from√ (f nm = ω nm2π = c ) n 2 ( ) m 2+(6.13)2 L x L yEquation (6.13) constitutes a fairly simple extension of the allowable frequenciesof a idealized free vibrating string to two-dimensional status.The fundamental frequency is found by merely setting n = m = 1 in Equation(6.12). The overtones corresponding to m = n > 1 will be harmonics of the fundamentalfrequency, but those in which m ≠ n (with either m or n > 1) may notnecessarily be so. A number of possible modes in a rectangular membrane areillustrated in Figure 6.2. The shaded areas vibrate π radians out of time phase withthe unshaded areas. Each normal mode is designated by an ordered pair (n, m),and the nodal lines are those with zero displacement at all times. In theory, rigidsupports could be placed along these lines without affecting the nodal pattern forthe associated specific frequency.