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

5.4 Other Boundary Conditions 93The allowed modes of vibration possess the radial frequencies ω n and the correspondingcyclic frequencies f n given byω n = nπc/L or f n = nc/2L, n = 1, 2, 3,...In simplifying Equation (5.11), the complex displacement ξ for the nth mode ofvibration isThe real part of Equation (5.12) isξ n = iA n e iω nt sin k n x (5.12)ξ n = sin k n x(A n cos ω n t + B n sin ω n t) (5.13)where the real amplitude constants A n and B n are related to the complex constantA n as follows:2A n = B n + iA nThe full solution to Equation (5.7) consists of the sum of all of the individualharmonic solutions, i.e.,∞∑ξ = sin k n x(A n cos ω n t + B n sin ω n t) (5.14)n=1The constants A n and B n can be evaluated by using the Fourier analysis describedin the last chapter, provided the initial conditions are known with respect to thedisplacement and the velocity of the bar.5.4 Other Boundary ConditionsIt should be understood that the boundary conditions corresponding to rigid supportsare difficult to realize in practice. The free-end condition, on the other hand,can be simulated by supporting the bar on extremely pliant supports placed somedistance inward from the ends. The end of the bar can now move freely and nointernal elastic force exists at that location. We now apply Equation (5.3), settingF x = 0; this gives rise to the condition ∂ξ/∂x = 0 at the free end. If the bar is freeto move at both ends (this is termed the free–free bar), the condition ∂ξ/∂x = 0applied to x = 0 in the wave equation, solution (5.10) yieldswith the resultA = Bξ = Ae iωt (e −ikx + e ikx ) (5.15)Inserting the condition ∂ξ/∂x = 0 into the above Equation (5.15) for the locationx = L yields−e −ikL + e ikL = 0 or sin kL = 0