Without assuming that secondary sources actually exist, and without assuming any particular type of wave, one can show that the wave function in a region containing no sources, due to primary sources outside that region, is as if the primary sources had been replaced by a distribution of "secondary" sources on the boundary surface of that region. I have tried to show this from first principles in "Exact derivation of Kirchhoff's integral theorem and diffraction formula using high-school math".

Without assuming that secondary sources actually exist, and without assuming any particular type of wave, one can show that the wave function in a region containing no sources, due to primary sources outside that region, is as if the primary sources had been replaced by a distribution of "secondary" sources on the boundary surface of that region. I have tried to show this from first principles in "Consistent derivation of Kirchhoff's integral theorem and diffraction formula and the Maggi-Rubinowicz transformation using high-school math".