A broadband solid impedance transformer for acoustic transmission between water and air
Total acoustic transmission between air and water was shown in our recent paper to be attainable with a solid interface comprising two parallel thin elastic plates connected by rigid ribs, although the transmissivity is a narrow-band effect. We demonstrate here that broadband transmission can be obtained by introducing a third, central plate. A theoretical analysis combined with numerical optimization shows that the optimal 3-plate impedance transformer has a central plate far thicker than the others. This implies a simpler interpretation of the optimal 3-plate impedance transformer as two elastic plates separated by a mass-like impedance. The characteristics of the broadband transformer may then be understood using results for the previously studied 2-plate system and asymptotic approximations using the small air-to-water impedance ratio. Optimal systems with water and air-side plates of similar material have relative thicknesses of approximately three to one, respectively, with the central mass having areal density approximately 17 times the water side plate. Further identities relate the frequency of total transmission to the plate thicknesses and to the rib separation length. The impedance transformer is compared to an ideal two layer quarter wavelength model, allowing us to identify a minimal attainable Q-factor of about 5.5, which is achieved in examples presented. The formulas for approximately optimized parameters also serve as the initial population for numerical optimization, greatly accelerating the process. Together, the theoretical and numerical results point to a remarkably simple class of purely solid impedance transformers, with system parameters well defined by the asymptotically small parameter: the ratio of air-to-water acoustic impedances.