u x u03c9 r u03c9 cos k e x s u03c9 sin k e x 8 where r and s are wave propagation constants applying boundary conditions 6 and 7 to equation 8 and writing the solution as a transfer function in the form of a ratio between the displacement at both ends produces u ue8a0 l u03c9 u 0 1 cos ue8a0 k e ue89e l u03bc ue8a0 k e ue89e l ue89e sin ue8a0 k e ue89e l 9 where u03bc is the ratio of the mass of the mass 14 to the rod 10 mass and is equal to the transfer function in equation 9 represents data and is a function of unknown wavenumber k e the inversion of two of these transfer functions using different attached masses will allow the experimental data to be combined and yield a closed form solution of k e and then e as a function of u03c9 the theoretical form of these transfer functions is u ue8a0 l u03c9 u 0 t 1 ue8a0 u03c9 1 cos ue8a0 k e ue89e l u03bc 1 ue8a0 k e ue89e l ue89e sin ue8a0 k e ue89e l ue89e and 12 u ue8a0 l u03c9 u 0 t 2 ue8a0 u03c9 1 cos ue8a0 k e ue89e l u03bc 2 ue8a0 k e ue89e l ue89e sin ue8a0 k e ue89e l 13 where the subscript 1 denotes the first attached mass and the subscript 2 denotes the second attached mass writing equations 12 and 13 as a function of k e l sin k e l and then equating them yields cos ue8a0 k e ue89e l t 2 ue89e u03bc 2 t 1 ue89e u03bc 1 t 1 ue89e t 2 ue8a0 u03bc 2.