Directivity and the directivity index

What directivity means

Directivity describes the spatial distribution of a sound source's radiation. A source with high directivity concentrates its output in a narrow angular region; a source with low directivity (omnidirectional) radiates uniformly in all directions.

For a loudspeaker, directivity is frequency-dependent. At low frequencies, where the wavelength is large compared to the cone diameter, the driver is nearly omnidirectional. As frequency rises and the wavelength becomes comparable to the cone diameter, radiation concentrates increasingly on-axis. This narrowing of the polar pattern with increasing frequency is the characteristic behaviour of a piston radiator.

The directivity index

The directivity index (DI) quantifies the ratio of the on-axis intensity to the intensity that would be produced by an omnidirectional source radiating the same total acoustic power:

DI = 10 × log₁₀(4π × I_on-axis / W_total) dB

where I_on-axis is the on-axis intensity at a given distance and W_total is the total radiated acoustic power. Equivalently:

DI = L_on-axis − L_power dB

where L_on-axis is the on-axis SPL and L_power is the power-averaged SPL (the sound power level minus a constant).

For a point source (omnidirectional): DI = 0 dB. For a source radiating into a hemisphere (2π): DI = 3 dB, since the same power is concentrated in half the solid angle. For a piston on an infinite baffle at high frequencies: DI rises with frequency as the beam narrows.

How DI changes with frequency for a piston

A circular piston of radius a mounted in an infinite baffle produces a directivity index that rises with frequency in the piston band. The parameter ka = 2πfa/c describes the ratio of cone circumference to wavelength:

• At ka << 1 (low frequencies, large wavelength): DI ≈ 3 dB — the driver radiates into a hemisphere uniformly.
• As ka increases through 1–3: the beam narrows on-axis and DI rises.
• At ka = 3.83 (the first zero of the piston directivity function): the first off-axis null appears, and the on-axis response begins to show the high sensitivity that accompanies strong directivity.

For a woofer with an effective cone radius of 65 mm (a 165 mm driver), ka = 1 at approximately f = c / (2πa) = 343 / (2π × 0.065) ≈ 840 Hz. Above this frequency, the driver begins to beam significantly. This is why crossover frequency selection must account for driver directivity — not just frequency response flatness.

The power response

The power response (or sound power response) is the total acoustic power radiated across all angles, as a function of frequency. It differs from the on-axis frequency response wherever directivity is not constant with frequency.

A driver with rising DI may have a flat on-axis frequency response but a falling power response. In a room, the listener hears a combination of direct sound (proportional to on-axis response) and reflected sound (proportional to power response). If DI rises steeply with frequency — as it does for a large woofer operating above its beaming frequency — the power response falls sharply at high frequencies. The reverberant field becomes bass-heavy, and the room sounds warm and lifeless at high frequencies even if the on-axis response is flat.

This is the fundamental case for managing directivity in loudspeaker system design: a loudspeaker with smoothly rising DI will sound different in a room from one with constant DI, even if both have flat on-axis responses.

Constant directivity

A loudspeaker system with constant directivity maintains a consistent DI across its operating bandwidth. This means the ratio of on-axis to off-axis output is the same at all frequencies, and the shape of the power response mirrors the on-axis frequency response.

Constant directivity is desirable because:

• The reverberant field has a similar spectral balance to the direct sound, reducing tonal coloration from room reflections
• The off-axis response is a scaled version of the on-axis response, making the loudspeaker more consistent across listening positions
• EQ applied on-axis also corrects the off-axis and power response

Achieving constant directivity across a wide frequency range is technically challenging. Approaches include:

• Waveguides and horns — constrain radiation to a defined angle, maintaining DI across a broad band. Common in professional audio.
• Coaxial drivers — place the tweeter at the acoustic centre of the woofer, minimising the physical offset between drivers at crossover and maintaining a consistent polar pattern.
• Careful crossover design — ensure that the polar pattern of the system at crossover is consistent with the individual driver patterns above and below it.

DI and sensitivity

DI and sensitivity are directly related. A driver with higher DI concentrates its radiated power on-axis more effectively, producing higher on-axis SPL for the same input power. The on-axis sensitivity gain from DI is:

Sensitivity increase = DI − 3 dB (relative to a hemispheric radiator)

A driver with DI = 10 dB has 7 dB more on-axis sensitivity than an omnidirectional source radiating the same total power into half-space. High-sensitivity horn-loaded drivers achieve their efficiency partly through high DI — they radiate a large fraction of their power on-axis rather than distributing it around the room.

Measuring directivity

A complete directivity characterisation requires measuring the frequency response at multiple angles around the loudspeaker — typically in both the horizontal and vertical planes at 10° or 15° intervals. The results are displayed as a polar plot at a given frequency, or as a directivity balloon in three dimensions, or as a directivity map (waterfall or heat map with frequency on one axis and angle on the other).

Directivity measurements require a free-field or gated measurement environment. They are time-consuming but are increasingly standard in serious loudspeaker characterisation, appearing in publications such as those from the ASC and Spinorama datasets.