A comprehensive linear mathematical model is constructed to address the open problem of the radiated wave for the distensible tube wave energy converter. This device, full of sea water and located just below the surface of the sea, undergoes a complex interaction with the waves running along its length. The result is a bulge wave in the tube which, providing certain criteria are met, grows in amplitude and captures the wave energy through the power take-off mechanism. Successful optimization of the device means capturing the energy from a much larger width of the sea waves (capture width). To achieve this, the complex interaction between the incident gravity waves, radiated waves and bulge waves is investigated. The new results establish the dependence of the capture width on absorption of the incident wave, energy loss owing to work done on the tube, imperfect tuning and the radiated wave. The new results reveal also that the wave–structure interactions govern the amplitude, phase, attenuation and wavenumber of the transient bulge wave. These predictions compare well with experimental observations.