That was actually my question: is there a waveguide sound signature?
Is there a correlation between the issue represented in this measurement/representation and it's potential audibility?
It really depends immensely on the waveguide, the tweeter, and how they are coupled.
I guess it makes sense to discuss it firstly from the point of view of a dome tweeter in a relatively small constant directivity waveguide, which is the kind typically used in a home audio loudspeaker.
In such a case, at lower frequencies (the bottom* of the tweeter's passband), the dimensions of the tweeter and the waveguide are small in comparison to the length of the waves. There is generally no problem here, other than that, below a certain frequency, the waveguide will cease to control the directivity of the sound.
This frequency can be calculated based on Keele's experimentally derived equation:
F = K / (a*w)
where:
F = frequency at which the waveguide ceases to control directivity (Hz)
K = 25,306 (unitless, experimentally derived)
a = waveguide included angle in degrees
w = waveguide mouth width (or height) in metres
So to take a hypothetical small waveguide like that on some of the speakers mentioned above of, say, width 12cm and included angle 120°, we would expect to get good, relatively even 120° directivity control in the horizontal plane down to:
F = 25,306 / (120*0.12) = 1,757Hz
Assuming the mouth of the waveguide terminates abruptly with the baffle at its edge, below F you would expect to see a narrowing of directivity to 2a/3, which in this case = 80°.** This narrowing would be centred at 10^(Log(F) - 0.176), which in this case would be approximately 1,170Hz. Below this frequency, the beamwidth will increase until it is the baffle controlling directivity at 180°, or beyond that until there is no directivity control (4pi radiation).
(As an interesting aside, baffle themselves can be modelled as waveguides of included angle 180° according to the above equation.)
Note that the above applies only to the dimension of the waveguide corresponding to the axis of interest. So on a typical elliptical or rectangular home audio waveguide, which is asymmetrical, you would have differing F(a) and F(2a/3) on each axis. For a waveguide that is wider than it is tall, horizontal directivity control will extend to a lower frequency. Where the included angle also varies with axis, this will of course also affect the frequencies of F(a) and F(2a/3), i.e. as included angle decreases, F increases.
It's been experimentally determined that flaring of the mouth of the horn tends to ameliorate the magnitude of the narrowing of the beamwidth at F(2a/3). The other benefit is of course that it minimises (but does not irradicate) time domain effects resulting from diffraction. These two reasons are why all good constant directivity waveguides have a flared mouth (and why baffles have bevelled edges). The profile of the flare is also important but that's another story.
So that's the lower frequencies. Above this frequency range, we have the middle frequency band, where we tend to have the fewest issues. In this band, the direcitvity of the waveguide is fairly constant and is determined by the included angle of the waveguide.
The higher frequencies are where things get complicated again. Up here, directivity depends on:
- the included angle of the waveguide
- the size (and to a lesser extent shape) of the dome and throat of the waveguide
- the shape of the wavefront presented by the dome to the waveguide throat (which tends not to be uniform with a dome beyond a certain frequency)
- any small discontinuities at the throat of the waveguide, for example caused by the tweeter surround, etc.
If things are functioning optimally, (1) will be the dominant factor.
At some (higher) frequency though, (2) will begin to become significant, and both the tweeter's own (narrow) directivity pattern and the dimension of the waveguide throat will become dominant. "Higher order" modes will also begin to cross the surface of the throat (instead of propogating outward from it), and these will tend to affect both the waveguide's directivity and (in worse cases) the axial response of the waveguide.
The frequencies at which these higher order modes occur will be determined by the dimensions of the throat, and their intensity will depend on the amount of diffraction that occurs at it, which will depend primarily on how smoothly the shape of the wavefront transitions to the shape of the throat, which of course depends on both the shape of the wavefront (see 3) and the angle/shape of the throat.
Also important will be (3): as freuqency increases, the tweeter becomes less and less pistonic, and begins to produce a wavefront that is less and less coherent. This will tend to affect how the wavefront couples to the throat of the waveguide, and may result in diffraction, interference, and inconsistencies in directivity. The effects of (4) will be similar to (3).
In fact, all of (2), (3) and (4) will tend to have similar effects, and to interact. This is why it is very difficult towards the top of the audio band to get smooth axial response and directivity out of such a waveguide, which typically couples to a 1" or 1.1" dome and has a throat slightly larger than that. The wavelength of 20kHz, for example, is around 2/3", which is small in relation to a typical dome/throat.
To summarise, colouration may be caused by any of the following:
- variations in off-axis directivity immediately below F (see Keele's equation above), which will vary with axis if the waveguide is asymmetrical (lower frequencies only)
- if the waveguide is asymmetrical, variations in off-axis directivity as a result of the waveguide having different dimensions and possibly different included angle on different axes (all frequencies)
- variations in off-axis direcitivity and/or axial response plus effects in the time domain resulting from higher order modes (higher frequencies only)
- variations in off-axis direcitivity and/or axial response plus effects in the time domain resulting from diffraction effects as a result of the points described in (2), (3) and (4) (higher frequencies only).
*EDIT: ideally this is actually below the tweeter's passband, not at the bottom of it
**EDIT 2: Re-read my old notes and realised I was slightly incorrect here; directivity should narrow to 2a/3° if the waveguide is in 4pi space. For a waveguide that terminates (abruptly) to a baffle, to my knowledge Keele doesn't give an experimentally derived beamwidth. In another study conducted by Johansen, abrupt widening of the waveguide angle at the mouth was shown to give a similar (but less effective) result to flaring on directivity below F. Therefore I infer that, where a waveguide terminates abruptly to a baffle, the beamwidth at 10^(Log(F) - 0.176) Hz will in fact be greater than 2a/3°, but not as great as if the transition is stepped or flared.
