In order to gain insight into the physics of modulated and localized electron heating, like the generation of heat waves by electron cyclotron heating, we solve the one-dimensional linearized electron heat diffusion equation in a homogeneous infinite plasma including a source term with a Gaussian profile. The amplitude and phase of the temperature oscillation in the centre of the deposition can then be analytically expressed as a product of two terms. Only one of them, a normalized complex function, describes the influence of heat diffusion. The analytical results are compared to experimental ones, and all characteristic dependences on modulation frequency, on heat diffusivity and on heating profile width are recovered.