Abstract: The present invention offers methods and apparatus for low-frequency sonic borehole logging with tube waves using discrete frequency measurements. A source transducer emits sonic energy at a location in a borehole of at least one discrete frequency which induces propagation of tube waves in the borehole; sonic energy of the tube waves is received at multiple receiver locations in the borehole, the multiple receiver locations being spaced apart from one another and from the source transducer location; and at each said receiver location, the complex pressure response (amplitude and phase, or real and imaginary parts) of a receiver to the received sonic energy of the tube waves is detected relative to a phase reference.