Power Response and Directivity Index

Unveiling a Speaker’s Sonic Behavior: Power Response and Directivity

Loudspeaker design is a delicate balance of factors.

Understanding a speaker’s directivity (how sound radiates in different directions) and its interaction with power response (the speaker’s ability to reproduce sound consistently across the frequency spectrum) is crucial in loudspeaker design, especially when considering listening distance.

While constant directivity waveguides offer distinct advantages, the interplay between a speaker’s coverage pattern (how sound energy is distributed) and the listening room’s acoustics plays a vital role in achieving a balanced sound.

A point about Critical Distance

In sound reproduction, understanding the critical distance is crucial, as it defines the ideal listening distance for achieving a balance between the direct sound from the speakers (direct field) and the reflected sound from the room (diffuse field).

A common target is a 50/50 ratio of direct to reverberated sound, although preferences can vary.

Our dedicated article about critical distance goes deep in this concept, where room acoustics (measured by Sabine absorption) and speaker directivity factor are the two main parameters.

The Challenge Beyond Critical Distance

As the listener moves further away from the speaker, two key challenges arise beyond the critical distance:

• Inadapted Coverage to Listening Distance (Critical Distance): A loudspeaker with a coverage pattern that is inadapted to the listening distance (critical distance) will cause uneven sound reproduction, particularly in high frequencies. This can manifest in two ways:

  • Too Restricted Coverage: This leads to a perceived lack of detail and brightness (a muffled sound). This is because the sound reaching listeners lacks sufficient high-frequency energy due to the limited coverage pattern.

  • Too Wide Coverage: Conversely, a coverage pattern that is too wide relative to the critical distance can deliver an excessive amount of high-frequency energy to distant listeners. This can make the sound seem overly bright or aggressive, leading to listener fatigue.

• Excessive High-Frequency Energy due to Room Acoustics: Even with a well-designed loudspeaker, the reverberant field in the listening room can sometimes add an excessive amount of high-frequency energy. Rooms with reflective surfaces like hard floors and bare walls can exacerbate this issue. This can contribute to a perceived harshness or fatiguing sound, particularly for listeners positioned further away from the speaker.

Power Response

This metric reflects the speaker’s ability to reproduce sound across the frequency spectrum with consistent loudness.

Imagine it as a frequency-dependent Sound Pressure Level (SPL) output. Ideally, a flat power response signifies the speaker outputs all frequencies at the same intensity, resulting in a balanced sound.

Deviations from a flat response, such as peaks or dips, indicate variations in loudness across different frequencies.

Power response is typically measured in an anechoic chamber (a specially designed room that absorbs sound reflections) and visualized as a frequency response curve. This curve plots the sound pressure level (SPL) output of the speaker in decibels (dB) across the frequency range (Hz).

There’s no single formula to generate this curve as it’s a result of the speaker’s physical design and materials.

Directivity Index (DI)

This measurement focuses on how the speaker radiates sound in different directions.

Unlike power response, DI doesn’t consider overall loudness, but rather how efficiently the speaker directs sound energy towards the intended listening area.

A high DI suggests the speaker is more directional, focusing its energy forward, while a low DI indicates a more omnidirectional sound pattern.

Then, a flat DI curve indicates a constant directivity behavior, meaning the speaker emits constant off-axis energy relative to on-axis.

DI calculation is demonstrated in Appendix section.

Optimizing Loudspeaker Design for Balanced Listening

The key lies in carefully considering both power response and directivity characteristics:

• Coverage Pattern Design: A well-designed loudspeaker, regardless of whether it utilizes a constant directivity waveguide or another approach, should provide a constant coverage pattern that effectively disperses energy, particularly high frequencies but by respecting psyckoacoustics principles too, tailored to the intended listening distance, the critical distance. This helps maintain a balanced and detailed sound experience across the listening area.

• Accounting for Reverberant Field: Understanding the impact of the listening room’s acoustics on the reverberant field is crucial. In some cases, acoustic treatments or room correction techniques may be necessary to mitigate excessive high-frequency buildup and ensure a natural listening experience.

Conclusion

By carefully considering power response, directivity characteristics, and the influence of the listening room, loudspeaker designers can create speakers that deliver a natural and detailed listening experience across a wider range of distances.

This comprehensive approach is essential for achieving a balanced sound, regardless of the specific loudspeaker design.

Appendix: DI Formula

The formula for DI using the spherical integral is:

DI(θ) = 10 log ( (∫ P(θ, φ) dΩ ) / P(0, 0)^2 )

• DI(θ): Directivity Index at angle θ (degrees)
• P(θ, φ): Off-axis Sound Pressure Level at a specific angle (θ, φ) in spherical coordinates (θ for horizontal angle, φ for vertical angle)
• dΩ: Differential solid angle element (used in the integration process)
• P(0, 0): On-axis Sound Pressure Level (dB)
• ∫: Integral symbol representing summation over the entire sphere’s surface area

Understanding and calculating dΩ:

The differential solid angle element (dΩ) represents an infinitesimally small piece of the sphere’s surface area. It’s crucial because the integral in the DI formula sums the off-axis sound pressure levels (P(θ, φ)) across the entire sphere. However, areas closer to the poles have a smaller surface area compared to those near the equator. dΩ accounts for this variation:

Formula for dΩ in Spherical Coordinates:

dΩ = sin(φ) dθ dφ

• φ (phi): Vertical angle (ranging from 0° to 180°)
• θ (theta): Horizontal angle (ranging from 0° to 360°)

Explanation of dΩ Formula:

• sin(φ): This term ensures that off-axis sound pressure levels near the poles (where φ approaches 90°) are weighted less due to the smaller surface area in those regions. As φ increases towards the poles, the sine of φ gets smaller, effectively reducing the weight given to those SPL measurements.

• dθ and dφ: These represent infinitesimal changes in the horizontal and vertical angles, respectively. Multiplying them by sin(φ) calculates the area of a tiny rectangle formed by these infinitesimal changes on the sphere’s surface.

Integration and DI Calculation:

1. For each off-axis measurement point defined by specific θ and φ values, we:
   • Calculate the corresponding dΩ using the formula above.
   • Multiply the measured sound pressure level (P(θ, φ)) by this dΩ. This weights the SPL measurement based on the area it represents.
2. The integral (∫) symbol then sums these weighted sound pressure levels (P(θ, φ) * dΩ) across the entire sphere.

By incorporating dΩ, the integration process considers the varying sizes of surface area elements. This ensures an accurate calculation of the total sound power radiated by the speaker, which is then compared to the on-axis SPL to determine the directivity index (DI) for different off-axis angles.