TECHNOLOGY FOCUS - CHEMICAL ENGINEERING - Preferred System: In the wave-powered desalination system, the pressure of the input seawater flow is increased in a first step powered by wave energy converted by the wave-energy-converter subsystem and a second step powered by power recovered from the pressurized brine flow by the energy-recovery subsystem. The energy-recovery subsystem comprises a fluid motor configured to convert power carried by the pressurized brine flow into power configured to drive a pump, the pump configured to assist in a pressurization of the input seawater flow; and a pressure-transfer device configured to transfer a pressure of the pressurized brine flow directly to the input seawater flow, without conversion into, and back out of, mechanical power. The pressure of the input seawater flow is increased in a first step powered by power recovered from the pressurized brine flow by the energy-recovery subsystem and a second step powered by wave energy converted by the wave-energy-converter subsystem; or increased in a single step where the wave-energy-converter subsystem and the energy-recovery subsystem are configured to convert fluid-motion power into mechanical power that is delivered by a common transmission shaft to a pump configured to pressurize the input seawater flow. The mechanical power is translational mechanical power or rotational mechanical power. The pressure of the input seawater flow is increased in a single step where power carried by the pressurized brine flow is converted to translational mechanical power carried by a translationally oscillating drive shaft configured to assist in powering a linear-displacement pump configured to pressurize the input seawater flow. Also, the pressure of the input seawater flow is increased in a single step where a pressure of the pressurized brine flow is transferred directly to the input seawater flow by a pressure-transfer device, and pressurization of the input seawater flow by the pressure-transfer device is amplified by an addition of wave power to a translationally oscillating motion of the pressure-transfer device. The ratio of the pressurized brine flow to the input seawater flow is fixed by a ratio of a diameter of a connection between two pistons of the pressure-transfer device to a diameter of a cylinder in which the two pistons move, or fixed by a ratio of diameters of two coaxial cylinders in which two pistons of the pressure-transfer device possess different diameters, and move in cylinders possessing different diameters. The conversion of fluid- motion power into oscillatory rotational power and/or a conversion of oscillatory rotational power into fluid-motion power are effected by a vane pump-motor pair, each vane pump in the vane pump-motor pair comprising a single rotor-stator pair. The ratio of the pressurized brine flow to the input seawater flow is fixed by a ratio of diameters of two coaxial single-rotor-stator-pair vane pumps. The conversion of fluid- motion power into oscillatory rotational power or a conversion of oscillatory rotational power into fluid-motion power are effected by at least one vane pump, the at least one vane pump comprising two rotor-stator pairs. The ratio of the pressurized brine flow to the input seawater flow is fixed by a ratio of heights of rotor vanes that move in the pressurized brine flow and the input seawater flow, and a diameter of a connection between two pistons of a pressure-transfer device to a diameter of a cylinder in which the two pistons move. Preferred Method: In desalinating seawater, the pressurizing step involves a first step powered by wave energy converted by a wave-energy-converter subsystem and a second step powered by power recovered from the pressurized brine flow by an energy-recovery subsystem, or a first step powered by power recovered from the pressurized brine flow by an energy-recovery subsystem and a second step powered by wave energy converted by a wave-energy-converter subsystem. The pressurizing step involves a single step in which a wave-energy-converter subsystem and an energy-recovery subsystem convert fluid- motion power into mechanical power that is delivered by a common transmission shaft to a pump configured to pressurize the input seawater flow.