A loudspeaker can sound clear directly in front of its diaphragm and noticeably darker when heard from the side. This can happen even when the amplifier is linear and the diaphragm moves without distortion. The finite radiating area itself creates an angular response. A circular-piston model provides a useful baseline for estimating this effect, provided its assumptions remain visible. The dimensions and curves below are calculated teaching examples, not specifications for Suyang products.

1. Frequency Response Is Only One View

A frequency response compares output across frequency at a fixed position. A directional response compares positions at the same frequency and distance. An on-axis curve alone therefore cannot establish whether tonal balance remains consistent as the listener moves. Moving a microphone sideways while also changing its distance or the room reflections mixes several effects into one result.

Imagine dividing the diaphragm into many small radiating areas. At a distant point on the axis, their path lengths are nearly equal and their pressures add almost in phase. At an angle, the paths from opposite sides differ. Increasing frequency increases the phase difference associated with that path difference, so parts of the radiation can cancel. This is a linear interference effect; it is not inherently harmonic distortion or a resistive loss in the drive circuit.

Here θ is measured from the forward normal: zero degrees is on axis and 90 degrees lies in the baffle plane. The model is a uniformly moving circular piston in an infinite rigid baffle, evaluated in the front hemisphere. A finite enclosure can diffract sound around its edges, so the model should not be read as a prediction of silence behind a real cabinet.

2. Compare Diameter with Wavelength

Let a be the effective piston radius, d = 2a its diameter, and k = 2π/λ the acoustic wavenumber. The useful dimensionless size is ka:

Wavelength λ and diameter d are in metres, frequency f is in hertz, and sound speed c is in metres per second. We use an illustrative value of 343 m/s throughout. Doubling frequency at a fixed diameter doubles ka; doubling diameter at a fixed frequency does exactly the same. This scaling allows one family of angular curves to describe many combinations of size and frequency.

Small ka gives relatively little variation across the front hemisphere. Larger ka narrows the main lobe and can introduce nulls and side lobes. The value ka = 1 is a useful scale for discussing this trend, not a universal pass/fail boundary. A design decision needs a listening angle and an acceptable response variation before a diameter can be judged suitable.

The frame diameter is not the effective radiating diameter. If effective diaphragm area S_d is known, the equal-area circular estimate is d = 2√(S_d/π). This is only a starting approximation for a rectangular or racetrack diaphragm: its long and short dimensions can produce different angular responses in the two principal planes.

3. The Far-Field Piston Formula

Normalize pressure to the on-axis value at the same frequency and distance. The directional function is then:

J₁ is the first-order Bessel function of the first kind, x is dimensionless, H is a signed pressure ratio, and ΔL is a relative level in decibels. At x = 0 the limiting value is H = 1, so an implementation must avoid division by zero. Taking the magnitude is appropriate for a level plot, but phase must be retained when combining coherent sources.

The first null occurs at x approximately 3.8317 and enters the front hemisphere only when ka is large enough. A deep null is cancellation in a particular direction, not loss of all acoustic output. Real diaphragm modes, edge diffraction and measurement noise can fill it in. The graph clips values below -40 dB only for readability; that floor is neither a product limit nor a claim about measurement capability.

4. An 80 mm Worked Example

The following values use an effective diameter of 80 mm. Every frequency is independently normalized to its own on-axis level of 0 dB. The table therefore isolates angular variation; it does not include the driver's underlying axial response, heating or maximum output. Compare the angles within each row before drawing conclusions about any measured loudspeaker.

The example illustrates why one driver can have little angular variation at low frequency yet a substantial loss higher in its band. Equalization raises output at a frequency for every direction. It cannot remove the geometric difference between those directions. Correcting a single off-axis seat can consequently overcorrect an axial position while consuming additional amplifier and thermal headroom. Reconsider orientation, effective diameter or the crossover arrangement before deciding how much equalization to apply.

5. Check the Crossover and the Installation

If a midwoofer has already narrowed near the crossover while the tweeter remains broad, a flat axial sum may conceal an abrupt change in the side response. Compare both drivers at the intended listening angles, then measure the filtered system. The piston calculation identifies a region worth investigating; it does not replace the effects of filter phase, acoustic-centre offset or spacing between drivers.

Single-diaphragm directivity must also be distinguished from interference between separate drivers. Two vertically spaced radiators can develop vertical lobes because their propagation distances differ away from the reference axis. Good horizontal measurements do not establish good vertical coverage. A coaxial assembly still has geometry to consider, including the tweeter location, surrounding cone and acoustic passages.

For a miniature loudspeaker, the grille, enclosure opening, sound channel and the user's grip can change the installed response. A bare-driver polar measurement describes its particular mounting condition. Repeat it in the finished device. In a vehicle, door-panel orientation and ear position create different listening angles for the two channels; reflections from glass and trim require checks in the actual cabin.

6. A Repeatable Measurement Procedure

1. Fix input and boundaries

   Record drive voltage, mounting baffle, grille configuration and frequency range. Confirm linear operation and keep amplifier gain unchanged throughout the angle sweep.

2. Define the rotation geometry

   Rotate around a documented reference point while maintaining microphone distance to that point. Record the reference axis, angle sign and whether the sweep is horizontal or vertical.

3. Check distance and reflections

   Use an appropriate anechoic environment or a suitable time window. Assess far-field validity against size and frequency, and check whether normalized patterns remain stable at a greater distance. Shorter time windows restrict low-frequency resolution; curves obtained with different windows are not automatically comparable.

4. Keep absolute and normalized data

   Save every raw angular frequency response as well as its ratio to the axial response. An axial dip can exaggerate a normalized ratio, so inspect the raw levels and signal-to-noise ratio before interpreting a peak.

5. Verify the intended use

   Repeat the assessment in the target installation and listening region. Specify the frequency band, angular range and permitted variation. Report the measurement conditions and uncertainty rather than treating a coverage angle at one frequency as a broadband guarantee.