I find it hard to believe that because the volume is either a square, a star, a pyramid or a circle, it is going to change anything to the absorption. A volume is a volume. regardless of its shape...
It's not the volume that matters, but rather the depth of the air gap. That's what defines the resonant frequency. In fact, if you look at the equations that Everest gives, you'll see that volume is not even part of the equation at all!
This is the same principle as for perforated panels, slot walls, and other tuned devices that rely on resonance for their effect.
Think of it this way: This is all about weights bobbing up and down on springs. It is all about simple one-degree-of-freedom resonance: A mass on the end of a spring. It works like this:
1) Air is a spring. That's not obvious or intuitive to us people, since air doesn't seem "springy" to us humans as we go about our daily lives. But to a sound wave traveling at 343 meters per second, air sure is springy. Very springy. It compresses and expands just like any other spring.
2) The sound wave sees the greatest "springiness" in the direction it is traveling, and nothing at all in the direction at 90° to that. If you think about it, this is logical: If you have two springs at right angles to each other against the front wall of your garage, then as your car runs into that wall, it only feels the springiness of the one that is
facing you, perpendicular to the wall, not the other one going across sideways, parallel to the wall. That one is not a spring at all, since it is facing in the wrong direction. Only the spring facing you is any use.
3) Instead of a volume of air behind the panel, think of it as an array of dozens of springs running perpendicular to the front panel, directly back to the wall behind. Now, since the panel is angled across the corner, you will have springs of different lengths. Some, near the edges of the panel, will be very short. Others, near the middle of the panel, will be very long. So each section of the panel only "sees" the spring that is directly behind it. That's what sets the resonance for that part of the panel. If the panel were parallel to the wall, all the springs would be the same length, and the entire panel would vibrate at one single frequency, but that is not the case if the panel is angled. With an angled panel, each vertical "slice" of the panel has a different resonant frequency, set only by the surface density of that section, and the depth behind that section. The rest of the volume is irrelevant. This is exactly the same as for slot resonators and perforated panel: Each hole only sees the air "spring" directly behind that slot or hole, not the other "springs" in the rest of the volume.
OK, if you try to analyze that analogy in more detail, it turns out to not be very good, and falls apart mathematically, since the entire panel is rigid, each section is joined to the next, they can't vibrate independently at different frequencies, etc. But the basic idea is there: Each part of the panel only reacts to the air depth behind that part. And the overall effect is that the panel acts as if it were tuned to all of the frequencies, somewhat, instead of being tuned to only one, very well. so the entire panel will react to any of those frequencies, since part of it wants to vibrate at that frequency while the other parts don't. But resonance is a powerful force, so the part that does want to resonate can overcome the "resistance" of the parts that don't, to a certain extent.
Thus, the overall effect is a low Q, broadband resonant that affects a range of frequencies centered around the one defined by the average depth.
- Stuart -
