Motion and Boundary Condition Equations

On the other hand, the value of I for a rectangular cross section with dimensions comparable to those of the lateral faces is given by 

By integrating by parts and taking into account the boundary conditions [Eq. (17.250)], the motion equation [Eq. (17.247)] can alternatively be written as 

For the solution of this system of integrodifferential equations, we can adopt a method of separation of variables and develop the solution in terms of a series that depends on the eigenvalues and the corresponding eigenfunctions of the characteristic equation. By assuming 

Comparison of the right-hand sides of Eqs, (17.255a) and (17.255b) for z = I together with further rearrangements, give 

This is the secular or frequency equation, which, for convenience, can be expressed as 

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