Based on wave acoustics, focusing of a plane progressive ultrasonic wave by a bi-concave lens which is made of acrylic resin and is submerged in water is investigated theoretically. For simplicity, it is assumed that the wave is normally incident on the lens and there is no energy dissipation such as sound absorption losses due to viscosity in water. Another important assumption is to fulfill a boundary-free requirement in the radial direction of the lens. By imposing the appropriate boundary conditions of both the continuity of vibration velocity and of stress on the water/lens interface, not only longitudinal but also transverse waves are successfully obtained in closed forms, although their expressions are somewhat involved. Several numerical examples demonstrate that the amplitude of the ultrasonic wave passing through the lens increases with propagation in the pre-focal region, takes a maximum gain with a slight shift in position from the geometric focus, and then decreases gradually in the post-focal region. Great oscillations in the amplitude appear in the pre-focal region. There seem, however, no striking peaks and dips in the field away from the focus. Additional important finding is that the transverse wave influences more or less on the formation of the focusing patterns.

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