Although much progress has already been made in solving problems in step responses of pressure in a linear taperd pipe, new developments are still needed before the two dimennsional wave equation can be solved routinely. This paper describes one such development. A new method of solving one dimensional wave equatin in linear tapered copound pipes with arbitrary cross-sections has been devised. Results which are obtained from this theory are compared with experiments in pipes with square cross-sections. And we obtained the following conclusions: (1) Damping coefficients in an arbitrary cross-section are defined and are expressed analytically for some representative cross-sections. (2) The viscous solution of wave equatin obtained by Laplace transformation explains well distorted pressure histories in a linear taperd pipe with an arbitrary cross-section. (3) Pressure waves in a square cross-section include higher modes of frequensy and damp rapidly more than ones in a circular cross-section. But fundamental characteristics as waves are independent of the cross-section of the pipe except for the small aspect ratio.