The effect on the response is a shelving transition (see Figure 1), increasing to 6 dB above ka = 2, where k is the Wave Number (2π/λ). And, a is the effective radius (i.e., the radius of a flat circular piston with the same surface area). The Directivity Index effectively increases from 0 to 3 dB as the device transitions from radiating spherically (into a full space) to radiating hemispherically (into a half space). Note that rise in frequency response is 6 dB at high frequencies.
So what happens to a loudspeaker’s free-field response when it is placed near one or more boundaries? At very low frequencies, if a sound source is very close to a solid plane boundary (e.g., a speaker near a wall), sound radiation will be over a hemisphere (half space) instead of free field (full space), as sound energy is reflected from the boundary.
This halving of the radiation impedance doubles the apparent level, an increase of 6 dB. The addition of another perpendicular boundary (e.g., a speaker near two walls), again halves the radiation impedance, doubling the apparent level (+12 dB compared to free field). The addition of a third mutually perpendicular boundary (e.g., a speaker on the floor and near two walls), halves the radiation impedance and doubles the apparent level again (+18 dB compared to free field).
This is illustrated in Figure 2, which is a reconstruction of Figure 8 from Roy Allison’s article, “The Influence of Room Boundaries on Loudspeaker Power Output.”. Note that the sound pressure level goes up by 6 dB for each additional boundary. Alternatively, this can be viewed as a progressive increase in the directivity of the source, as the source radiates into a progressively smaller solid angle. Also note that there is a response dip when the distance to the boundary approaches one quarter, which can be mitigated by careful placement and crossover design. VC
 H. F. Olson, “Direct Radiator Loudspeaker Enclosures,” Journal of the Audio Engineering Society, Vol. 17, No. 1,
 R. F. Allison, “The Influence of Room Boundaries on Loudspeaker Power Output,” Journal of the Audio Engineering Society, Vol. 22, No. 6, June, 1974.
 F. E. Toole, Sound Reproduction—Loudspeakers and Rooms, Focal Press, 2008.
 R. V. Waterhouse, “Interference Patterns in Reverberant Sound Fields,” Journal of the Acoustical Society of America, Vol. 27, No. 2, March, 1955.
This article was published originally in Voice Coil, July 2016.