Introduction
In Part 1 we learned why we chose to use a hard dome as the radiator: the dome simply yields the best dispersion for a given size. A further advantage of a dome is that has high geometric stiffness which gives a high bandwidth.
In this blog we'll dive deeper into the compromises around choosing the dome dimension and the supporting innovations we developed.
Need to Displace Volumes of Air
Physics dictates that the displaced air volume of the radiator grows inversely with the square of the frequency. This is why woofers need to be large and have a long stroke in order to displace enough air at low frequencies. Even a tweeter is challenged at the low end, e.g., when required to play high SPL down to 1.5kHz or even lower in some 2-way speakers.
The need for high volume displacement requires a combination of a large radiating surface (Sd) and long linear stroke. However, a large radiating area generally conflicts with the need for wide dispersion and high bandwidth. This suggests that a very long stroke is the solution. However, this presents its own challenges, particularly with mechanical stability and motor design. Additionally, another complication factor comes into play: the air we breathe is not linear.
The Sound of Pressure
Sound is pressure variations that propagate as waves in the air. The local pressure applies force on the surrounding air which accelerates away from the pressure. This results in air particles moving and pressure building up further along the wavefront. The air acts as a spring and also has a mass. Normally, all this is assumed to be linear since the sound we perceive represents very small variations of the atmospheric pressure.
For example: 94dB SPL, which is loud music from a speaker, is a pressure amplitude of 1 Pascal which shall be compared to the normal pressure of 101,300 Pascal that we live in at sea level - this corresponds to a whopping SPL of 191dB. Even at this loud sound level of 94dB, the pressure variation is almost 100dB below the average static pressure. Indeed a tiny variation, and for the same reason, we normally consider the propagation of sound in air to be linear.
Higher Pressure at the Source
However, in order to have 94dB SPL hitting your ear from, say, 2m distance, we need a much higher SPL locally at the radiator surface. This is where the radiating area matter a lot: it takes a given volume displacement (or rather volume acceleration, to be precise) to reach 94dB at 2m distance.
For a smaller sized radiator, this requires more acceleration and therefore the local pressure is higher than for a large-area radiator, again for the same 2m distance SPL. For a typical 1" (25mm) tweeter, the pressure ratio from the surface to 2m distance is around 200 times (i.e. 46dB) . This means that the 94dB SPL at 2m requires up to 140dB SPL at the radiator surface, which is only 51dB below the theoretical maximum SPL of our atmosphere. The SPL ratio to the 2m on-axis distance is illustrated for the 25mm disk radiator with the color scale in dB:
a 25mm circular piston radiating in an infinite baffle. Colorscale: SPL in dB relative to 2m on axis
Everyone Cooking with the same Nonlinear Air
Now, at such high SPL, the air is noticeably nonlinear. The gas law states that pressure is proportional to the inverse of the volume, i.e., P=k/(V+dV) where dV is the displaced volume and k is a constant. This is clearly a nonlinear relationship, which gets more pronounced as the pressure variation increases relative to the average pressure.
Such a nonlinear relationship creates harmonic distortion, mainly 2nd harmonics but also a little bit of higher harmonics. In fact, most good tweeters have their 2nd harmonic distortion limited by this fundamental mechanism. Comparing same-sized tweeters at the same SPL reveals a shockingly consistent lower floor of the 2nd harmonic distortion. This is to be blamed on our atmosphere rather than the tweeter it self. For some reason, this has remained a well-kept secret.
Large Area is Desired
The nonlinearity of air means that we cannot just uncritically use long stroke to achieve volume displacement. To lower the 2nd harmonics, we need as large an area as possible since this reduces the surface air sound pressure level for a given sound pressure level at the listener. An increased area obviously conflicts with the desire for wide dispersion. It also conflicts with the goal to maintain pistonic operation across the entire audible band, as a larger radiator structure resonates at a lower frequency. We can push that bandwidth-area trade-off to our advantage by optimising the geometry of the dome. This typically results in a rather tall dome, which also helps widen dispersion.
However, to achieve an even better trade-off, we needed ways to improve dispersion beyond what a direct radiating surface allows.
Inventing the Coherer
This led to the development of the coherer: hard objects, smaller than the shortest wavelength reproduced, that are optimised to scatter the sound to achieve the desired dispersion pattern. The coherer elements are placed close to the radiating surface and work together with an integral waveguide to perfect the radiation pattern. These coherers camouflage as a dome protection grill but are the result of intensive computer optimisation and do wonders for the dispersion pattern.
Our dome tweeter with coherer and integral waveguide (104mm version)
With this coherer technology, we can achieve our goal of combining low distortion, high volume displacement, and area with super-wide and smooth dispersion up to the highest audible frequency.
Time to Decide
Based on intensive optimisation work in Comsol and Matlab, we decided for a 33mm hard dome with coherer technology and integral waveguide. This allows an impressive 140deg beam width up to approximately 20kHz, combined with a high radiating area approaching 10cm2.
Directivity pattern from the datasheet
A circular disk piston radiator would need a diameter as small as 14mm to achieve the same 140deg beam width at 20kHz. However, this small disk has an area of only 1.5cm2, which is many times smaller than the radiating area of our large dome. Such a small radiator would produce much higher distortion due to the much higher local sound pressure and would run out of of excursion before a midrange or midwoofer could take over.
Summary
The nonlinearity of atmospheric air limits the distortion performance of all drivers, but it particularly affects tweeters due to their small size (radiating area). The PURIFI solution combines a large hard dome for ultra-low distortion with the coherer and waveguide for very wide dispersion.