Directivity Match: Horns, Waveguides, and Woofer Harmony

Demystifying Directivity Match

In the world of loudspeaker design, achieving a seamless transition between different driver components is crucial for optimal sound quality.

This article delves into the intricacies of directivity matching between horns or waveguides, and woofers, a process that ensures a cohesive and natural listening experience.

The Importance of Directivity Match

When designing a loudspeaker that combines a woofer with a horn-loaded or waveguided high-frequency driver, one critical aspect is matching their directivity patterns around the crossover region.

Each direct radiation driver has its own transition frequency, the frequency at which the loudspeaker’s directivity changes from wide (omnidirectional) dispersion to narrower, more focused dispersion due to the physical size of the source. This transition frequency is roughly estimated by the formula f = c / (2πR), where R is the effective radiating radius.

Understanding Cone Radiation Directivity

There are two main stages of diaphragm – or cone driver – behavior as frequency increases: pistonic motion and non-pistonic motion.

At low frequencies, the diaphragm moves uniformly in pistonic motion, radiating sound broadly and evenly.

As frequency rises, the size of the radiating surface becomes comparable to the wavelength, causing phase differences (delays) between the diaphragm center (driven by the voice coil) and its edges. These phase shifts lead to a progressive narrowing of directivity.

This transition from pistonic to non-pistonic motion primarily results from geometric and wave propagation effects across the diaphragm surface, and it occurs even if the diaphragm material were perfectly rigid.

Material properties like stiffness and damping can also influence diaphragm behavior but have a lesser impact on directivity compared to the geometric phase effects.

This narrowing of directivity is the reason why the crossover to the high-frequency driver is usually placed around that frequency, so that both drivers have matching directivity for a smooth sound transition.

The size and shape of the baffle also affect this directivity transition.

This transition frequency f_transit can be approximated by:

f_transit ≈ c / (2 π R)

where c is the speed of wave propagation in the diaphragm material, and R its radius.

Why Directivity Match Matters

At the crossover frequency (close to the transition frequency), the directivity of both drivers should ideally match to ensure a smooth spatial response.
Without this match, response dips appear off-axis, degrading the coherence of the reproduced soundstage.

Even when listening on-axis only, the reverberated energy — shaped by off-axis radiation interacting with room acoustics — plays a significant role in the perceived sound.

This is because, at typical listening distances around the critical distance, the sound reaching the listener is approximately a 50/50 mix of direct sound from the speaker and reverberated sound reflected by the room.

If the woofer’s directivity narrows too much in the crossover region, it emits little energy off-axis in that frequency range. As a result, the room receives less energy in this midrange band, creating an energy dip in the perceived sound and a sensation of an “empty midrange.”

This interplay between directivity and room reflections makes directivity matching crucial for natural tonality, clarity, and spatial coherence.

directivity horn match

Acoustic Center Distance

Another key factor is the physical distance between the acoustic centers of the woofer and the high-frequency driver.
Ideally, this spacing should be less than or equal to 66% of the wavelength at the crossover frequency — a topic we cover in more detail in our vertical lobing article.

Minimizing this distance helps maintain vertical consistency, reducing lobing artifacts and further reinforcing the perception of a seamless transition between drivers.

Empty Midrange Feeling

Some mid-woofer drivers, such as those from PURIFI, have excellent distortion performance. This often makes it tempting to push their usable range higher than usual.
However, as previously discussed, a woofer’s directivity narrows as frequency increases.

For example, an 8-inch woofer becomes too narrow in dispersion after around 1200–1300 Hz to blend properly with a wide-dispersion device like an AMT or a direct-radiating tweeter.
Using such a crossover point leads to a poor directivity match: at crossover, the tweeter still radiates energy widely off-axis, while the woofer no longer does.

The result is a significant mismatch in how sound is distributed in space.
This isn’t visible in the on-axis frequency response but becomes obvious in polar plots and, more importantly, audible in the room.

It can produce what many describe as an “empty midrange” — not because those frequencies are absent on-axis, but because the woofer no longer emits energy off-axis in that range.
As a result, the room does not reflect those frequencies back to the listener as it does for lower or higher bands, creating a dip in perceived energy and listener envelopment in the midrange.

In other words, it’s the interaction between the woofer’s narrowing directivity in the crossover region and the room’s reflective field that creates this perceptual gap.

The Perfect Match: A Myth?

The pursuit of a perfect directivity match at all frequencies, can be a misguided approach. Measurements taken in anechoic chambers (highly sound-absorbent environments) often reveal that achieving this perfect match is practically impossible. Here’s where the crossover filter design comes into play.

The crossover slope, along with meticulously adjusted time delays, plays a vital role in ensuring a smooth transition between the woofer’s coverage and the horn/waveguide’s coverage. This transition should be free of abrupt changes (on-axis or off-axis) to avoid unwanted coloration or distortions in the sound.

Moreover, a common misconception in modern loudspeaker design is prioritizing an exact 120-degree directivity match at crossover and up to 20 kHz on the depends of more important aspects like coverage adapted to listening distance.

This “one-120°-coverage-fits-all” approach leads to completely ignores psyckoacoustics principles and coverage adapted to distance as see in critical distance article.

Improved Directivity with Round-Over Returns

Our horns incorporate a design element called a “round-over return.” This feature enhances directivity control by mitigating the narrowing effect call midrange narrowing/beaming that occur in the midrange frequencies.

The round-over return is a smooth, curved transition that seamlessly follows the horn’s profile until it meets the side of the enclosure. This design minimizes disruptions to the wavefront, preventing unwanted narrowing of the sound dispersion pattern in the midrange.

Benefits of Round-Over Returns:

Directivity Matching Between Horns: A Different Approach

When dealing with multiple horns or waveguides in a loudspeaker design, the crossover approach differs slightly. Here, the ideal crossover point is not where each component loses its directivity, but rather where both maintain a constant directivity pattern. This constant directivity should be consistent not only at the crossover frequency but also one octave below and one octave above the crossover point.

Following these principles of directivity matching ensures a natural and cohesive sonic experience for the listener, free from unwanted artifacts and with a smooth transition between different driver components within a loudspeaker.

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