Resonant Oscillation of a Liquid Metal Column Driven by Electromagnetic Lorentz Force Sources

In this paper, a theoretical study is conducted in order to establish the feasibility of a liquid metal acoustic resonator ͑liquid gallium or liquid aluminum͒ for high-amplitude acoustic oscillations. The fundamental resonant frequency typically lies between 5 and 40 kHz. The oscillations are induced by an alternating Lorentz force density applied directly to the liquid metal volume. Depending on the boundary conditions, two different resonator types ͑open–closed and open–open͒ are theoretically investigated. The analysis incorporates the effects of impedance termination, volume absorption, wall friction, acoustic radiation from the open end, and nonlinear inflow–outflow losses. The actual elasticity of the container, either a ceramic or quartz tube, and the coupled solid–liquid interactions are taken into consideration. Based on this investigation, theoretical predictions are conducted for the quality factor and the pressure level for the liquid metal resonator under various geometric and boundary conditions. They indicate that resonant amplitudes of 10–20 atm can be achieved using commercially available high-current audio amplifiers. LIST OF SYMBOLS a tube radius ͓m͔ B, B magnetic flux density ͓Wb/m 2 ϭTesla͔ B/AϭnϪ1 nonlinearity parameter of the liquid c 0 sound speed in the liquid metal ͓m/s͔ c l longitudinal wave speed in the termination/ tube material ͓m/s͔ c p phase velocity of compressional waves in a elastic plate ͓m/s͔ c p specific heat at constant pressure ͓J/kg K͔ E Young's modulus of the termination/tube material ͓GPa͔ f, f, f ˆ Lorentz force density and its complex amplitude ͓N/m 3 ͔ f n driving force density for nth resonance ͓N/m 3 ͔ F total Lorentz force ͓N͔ F n total driving force for nth resonance ͓N͔ h wall thickness of the container ͓m͔ i,i x current density ͓A/m 2 ͔ I, I ˆ total current ͑not rms!͒ and its complex amplitude ͓A͔ j imaginary unit k 1 ,k 1n wave number of the driving force ͓1/m͔ K circumferential stiffness of the tube wall ͓N/m 3 ͔ L height of the liquid metal column ͓m͔ p, p 0 acoustic pressure and ambient pressure, respectively ͓Pa͔ P pressure amplitude of resonant oscillations ͑at antinode͒ ͓atm, Pa͔ Q quality factor of the resonator r,r 0 observation point and source location point, respectively S tube cross-section ͓m 2 ͔ w velocity of the liquid metal in z direction ͓m/s͔ x,y,z Cartesian coordinates ͓m͔ Z s ϭ␳ s c l termination acoustic impedance ͓kg/m 2 s͔ ␣ semi-empirical loss factor for inflow–outflow losses …