Longuet-Higgins (1964) originally recognised that the energy flux defined by pressure work from the equations of motion was not the same as the mean energy density times the group velocity for planetary waves on a beta-plane. This paper addresses a similar paradox for linear, long period edge waves on an arbitrary shaped (in the offshore direction) straight continental shelf. The approach is to first examine a wavetrain solution to the problem and then to use a multiple scale argument which results in a solution as a group of waves modulated about a central frequency [sgrave] and wavenumberk. The paradox is resolved in both instances by noting that a divergence free quantityJcan be included in the energy conservation equation to establish an equivalence between the two definitions of mean energy flux. For the wavetrain solutionwherexis the offshore direction,h(x) is the depth,A(k,x) is the complex wave displacement, [sgrave] is the frequency andkis the wavenumber. For the modulated group, the quantity J is given bywhereB=B(Y, T) is part of the edge wave complex amplitudeA(k, x)B(Y, T) andY, Tare the long longshore and time variables respectively. We discuss which energy flux definition is preferable in a given situation.