While sitting around in lockdown, I decided to take an in depth look at toe in angle of dipole speakers. The goal was to determine the optimum toe in angle for a generic rectangular room with lateral symmetry. Consistent with this type of room, the primary listening position is located on the centerline of the room.
The work was based on Richard Taylor's 2013 paper "Delaying Room Reflections: Constraints on Room Size and Loudspeaker Placement". This paper is linked in the " LX521 Reference" web page on
www.linkwitz.lab.com. A diagram showing room setup from this page is attached below.
Linkwitz suggested spacing dipole speakers a minimum of 1 meter from room side wall to allow the reflected sound to reach the listener position 6 ms after the direct sound to create phantom source and realistic sound sources. For the case where the speakers and listener form an equilateral triangle with the speakers 1 meter from the side walls, I found that the reflected sound level is about 45 dB below the direct path sound level. However, I discovered that the 1 meter spacing does NOT delay the arrival time of the reflected sound by his recommended 6 milliseconds. Linkwitz also suggested using speaker toe in an alternate method to improve the phantom sources and the sound spaces in front of the listener by minimizing side wall reflections sound levels.
In Taylor's paper, he shows the basic geometry of a side wall reflection (diagram below) As a dipole radiates sound minimally at 90 degrees from the speaker axis, the optimum speaker toe in angle is achieved at an angle parallel to angle ACB in the Taylor diagram. This points the minimum sound radiation from the dipole directly at point O. Therefore, Angle ACB becomes the optimum toe in angle for the speaker axis relative to the side wall. This angle can be found by the equation below, written in an Excel type of equation format.
Angle ACB = Toe in angle = SQRT(v^2+(x+x-u)^2)
where: v = distance parallel to side wall from speaker to listener
x= distance from listener to side wall
x-u = distance from speaker to side wall
If you have a Linkwitz design dipole system, you might want to give this a try and evaluate it for yourself.
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