  We study the propagation of a compactly supported high-frequency wave through
a semi-linear wave equation with a null structure. We prove that the
self-interaction of the wave creates harmonics which remain close to the
light-cone in the evolution. By defining a well-chosen ansatz, we describe
precisely those harmonics. Moreover, by applying the vector field method to the
equation for the remainder in the ansatz, we prove that the solution exists
globally. The interaction between the dispersive decay of waves and their
high-frequency behaviour is the main difficulty, and the latter is not
compensated by smallness of the initial data, allowing us to consider the
high-frequency limit where the wavelength tends to 0.
