Reversible data hiding is a technique in which hidden data are embedded in host data such that the consistency of the host is perfectly preserved and its data are restored during extraction of the hidden data. In this paper, a linear prediction technique for reversible data hiding of audio waveforms is improved. The proposed variable expansion method is able to control the payload size through varying the expansion factor. The proposed technique is combined with the prediction error expansion method. Reversible embedding, perfect payload detection, and perfect recovery of the host signal are achieved for a framed audio signal. A smaller expansion factor results in a smaller payload size and less degradation in the stego audio quality. Computer simulations reveal that embedding a random-bit payload of less than 0.4 bits per sample into CD-format music signals provide stego audio with acceptable objective quality. The method is also applied to G.711 µ-law-coded speech signals. Computer simulations reveal that embedding a random-bit payload of less than 0.1 bits per sample into speech signals provide stego speech with good objective quality.