** Update: A downloadable version of the calculator to do this work is available (courtesy of Daniel Lundberg). Go to the bottom of this post for preview and instructions.
When a coupled array is assembled, its operating range is limited primarily by its power capability. Even very large arrays will congeal fairly quickly and once they have joined together let no phase tear them asunder. Wow! Not often that we can work hard-core religion language into speaker array theory (not to say there is not a lot of mysticism out there in line array theory land). So coupled arrays, once joined, once fully formed will maintain there shape over distance, finally either running (literally) out of air, or into the wall.
Uncoupled arrays are quite the opposite. They can’t wait to destroy themselves. The battle begins with each speaker owning its piece or real estate close by, in front of it. As we continue forward we have a happy meeting with the neighboring speaker’s response. They greet with an in-phase handshake and we have a crossover, known as the unity line. At this point the speakers are working together and the line that runs from speaker center to center (through the crossover) is approximately unity gain. This is exactly what we want to happen – an extended line of unity gain, wider than a single speaker. Ideal for frontfills, underbalconies, parade routes, racetracks and more. This is both a happy beginning AND a happy ending. How so? The beginning part is obvious, but the ending part…………well what I mean here is that this beginning is the best response we will get. It is all downhill from here as the more distant areas directly in front of each speaker no longer have sole ownership of the coverage. The others speakers are spilling in and they are arriving late. VERY late in acoustic terms. The displacement between the speakers (a factor that is large in an uncoupled array) now creates a very rapidly changing variation of time offsets between the elements. The result is combing that moves rapidly down in frequency and becomes stronger with each step we go deeper into coverage.
How far can we go before we throw up the white flag and surrender? One could evoke a variety of subjective answers such as: until it sucks, or until I can afford another set of speakers to take over etc., but these are not very satisfying to me. There is a verifiable milestone: three’s company. When we reach the point where the entire length of the coverage line is within the pattern of three sources we have reached full immersion into the combing. Three is a magic number. With three sources arrayed along a line, or an arc it is impossible to find a location that is equidistant to all three. This guarantees two or three arrival times from speakers operating within their coverage angle. That is the fight I was talking about before. The only way to stop the fight is to drown it out with another much louder speaker – like a mains to take over for your frontfill, or stop it – like a back wall for your underbalconies.
In my book I go through a set of design calculations for uncoupled line source and point source arrays. The variables are the coverage shape of the speaker (The Forward Aspect Ratio/FAR), the spacing, and the splay angle. From these we can determine where the coverage will start (D unity) and where the coverage should end (D limit). If you know the speaker and where your audience starts, you can determine the spacing, and where you will need to connect to the mains. If you have fixed positions you can get the right speaker model etc.
An example reference chart using a 80 degree speaker in an uncoupled line source is in the book. This shows nicely how to solve for this particular model and then one can refer back to the FAR chart to get the angle/FAR conversion for other speakers.
Another example reference chart uses a 90 degree speaker in an uncoupled point source source in the book. In this case the splay anglwe variable is added to the equation.
It is not possible to put an XL file into the book and not practical to give a separate chart for each speaker angle/spacing etc. but folks that bought the book don’t have a working calculator/spreadsheet that they can go to on their computer so I was in the process of making one for the blog and then Daniel Lundberg contacted me with his calculator based on this same concept. Whereas mine was derived from observing the trends and behiavior of many, many, array interactions, Daniel’s goes to the heart of the trigonometry involved.
So over the past few days I ran through some different models of speakers athdifferent angles and spacing to check for consistency between a) my published values derived through observation of other speakers at other angles
b) Daniel Lundberg’s values derived through trigonometry and geometry
c) what we can see on the MAPP plots now
The good news is they are all in very close agreement. The largest discrepancy is in the limit values for the longest range, and even these are relatively close.
Here is what the downloadable version of the calculator to do this work looks like (courtesy of Daniel Lundberg). You can have a copy of it. Free.
HOWEVER, the security rules of this blog host prohibit me from posting an XL file.
Therefore, if you want a working copy of this calculator, you will need to send me an email request to firstname.lastname@example.org. If you think this is just a trick to get you on my mailing list…………