Directivity Factor (Q(θ, ϕ))

In general, a directivity factor Q_θ,ϕ may be defined as the ratio of the radial intensity 〈I_θ, ϕ〉_t (at angles θ and ϕ and distance r from the source) to the radial intensity 〈I_s〉_t at the same distance r radiated from an omnidirectional source of the same total sound power (Figure 3.12). Thus

(3.53)

For a directional source, the mean square sound pressure measured at distance r and angles θ and ϕ is p²_rms (θ,ϕ).

In the far field of this source (r ≫ λ), then

(3.54)

But if the source were omnidirectional of the same power W, then

(3.55)

where p²_rms is a constant, independent of angles θ and ϕ.

We may therefore write:

(3.56)

and

(3.57)

where  is the space‐averaged mean‐square sound pressure.

We define the directivity factor Q as

(3.58)

the ratio of the mean‐square sound pressure at distance r to the space‐averaged mean‐square pressure at r, or equivalently the directivity Q may be defined as the ratio of the mean‐square sound pressure at r divided by the mean‐square sound pressure at r for an omnidirectional sound source of the same sound power W, watts.