The Lambert diffuse reflection model revisited

The Lambert diffuse reflection model is used widely in computerized prediction of sound in rooms as well as for outdoor scenarios. One seemingly surprising consequence of the model was pointed out by Borish [J. Audio Eng. Soc. 34, 539-545 (1986)]: A diffusely reflecting, non-absorbing wall seems to give a 3 dB stronger reflection than a specularly reflecting wall for a source and receiver along the same plane normal. Similar observations have been made by others, and it is usually commented that the two reflection types distribute the reflected energy in different directions. The aspect of energy conservation does not seem to have been sorted out entirely. It is shown here that the difference between an omnidirectional receiver, like a microphone, and a surface element receiver, which can give the total reflected power, explains the claim. Analytic solutions and numerical evaluations of the well-known integrals for a single infinite wall confirm that energy conservation is indeed maintained and also lead to a spatial distribution of the Lambert reflection strength, which differs substantially from the previously published values. The special case can serve as a useful benchmark test of implementations of diffuse reflections, which follow Lambert's law.