Theory of sound field in a room

In the normal-mode theory of Morse, it gives a series of normal modes as the solution of forced vibration in a room. But actually there is always the direct radiation besides the normal modes which represent the reverbrant sound field only. The reason is that the normal modes were assumed only in the source, and naturally normal modes only are obtained in the solution. A theory of double source is proposed, that the sound source is both the source of the direct radiation as if in free space before the boundary surfaces were reached by the direct radiation, and after the first reflection from the boundary surfaces, the source of the reflected wavelets, randomly distributed both in space an in time on the boundary surfaces that build up the normal modes after further reflections. The wave equation is formed accordingly, and the solution of the wave equation, the sound field in a room, contains explicitly both the direct radiation and the reverberant sound formed of normal modes. The approximate mean square sound pressure is found to be the dircet sound determined by the sound power of the source,and reverberant sound determined by the sound power reduced by a factor of π/2, different slightly from the result obtained from energy consideration, if the source is pure tone. There is essentially no difference for a source of band noise.