Analytical solution of parabolically tapered rod flexural vibration equation
Résumé
As it was shown earlier the parabolically tapered end of rod does not reflect the flexural waves with frequency high enough. The velocity propagation of waves tends to zero by approaching to the end crossection of the rod and the propagation time appears to be infinite. As a consequence, the wave, propagating to the tapered end of the rod, never reaches the end of the rod. This result was obtained by WKB-approximation analysis of the tapered rod flexural vibration equation. More thorough investigation displays however, that just for parabolic tapering the exact analytical solutions in the form of the linear combination of power functions exist. The indexes of the functions are zeros of fourth degree polynomial. Input impedance matrix of parabolically tapered rod is calculated and new, modified WKB-approximation for rod with arbitrary smoothly varying cross-section is suggested.
Domaines
Acoustique [physics.class-ph]
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