When a tube or duct flow is subjected to a streamwise periodic variation of wall temperature or wall heat flux, a periodic thermally developed regime is established at sufficient distances from the inlet. In the developed regime, similar heat-transfer characteristics occur in successive streamwise modules whose length is equal to the period of the imposed variation. A methodology was developed for obtaining universal solutions for the periodic fully developed regime. The end result of the development enables the wall heat flux distribution corresponding to any given wall temperature variation to be determined via the summing of a simple series. A similar series enables the wall temperature to be determined when the heat flux distribution is given. The series contains influence coefficients which represent the solution for an array of pulses. A procedure for determining these influence coefficients is presented which circumvents the need to solve the differential energy equation. The use of the methodology was illustrated by application to sine wave wall temperature and heat flux distributions.