Some properties of horn equation model of ultrasonic system vibration and of transfer matrix and equivalent circuit methods of its solution

• Transfer matrix method gives well-solvable resonance equations.
• From the structure of transfer matrix some general properties of solutions can be determined.
• Impedance matrix method and equivalent circuit method give incorrect resonances.
• True resonance curves can have jump discontinuities.

Traditional technique of horn equation solved by transfer matrices as a model of vibration of ultrasonic systems consisting of sectional transducer, horn and load is discussed. Expression of vibration modes as a ratio of solutions of two Schrödinger equations gives better insight to the structure of a transfer matrix and properties of amplitudes of displacement and strain, and enables more systematic search for analytic solutions. Incorrectness of impedance matrix method and of equivalent circuit method on one hand and correctness and advantages of transfer matrix method in avoiding numerical artifacts and revealing the real features of the model on the other hand are demonstrated on examples. Discontinuous dependence of the nth resonant value on parameters of ultrasonic system, recently described in Sturm–Liouville theory, and consequently, a jump from half-wave to full-wave mode, is observed in a transducer model.