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Acoustic Shadowing: Unpacking Wave Numbers (k) & Introducing The Wave Ratio Thesis

If you’re truly interested in learning about sound propagation and architectural acoustics, it’s something you should be acquainted with, at least casually.

By Michael Fay September 9, 2019

The kr Axiom

The kr axiom: If the answer you get is 2.00 or more (a 2:1 wave ratio), you will have a serious problem with acoustical shadowing. If answer is 0.25 (a 0.25:1 wave ratio) or less, there will be very little or no noticeable problem.

The 2:1 ratio, or greater, tells us the frequency will be folded, parallel across the boundary, at least two full cycles and will therefore create a noticeable acoustic shadow. The 0.25:1 ratio represents the quarter-wave dimension of the frequency being evaluated. I won’t delve into quarter-wave theory here, but suffice it to say if the obstruction is one-quarter the size of the wave, or smaller, there will be little or no problem.

Based on the finish material and surface shape of the obstruction, everything in between will be partially blocked, absorbed, diffused, and/or diffracted.

Please keep in mind, the shading and blocking that occurs does not render the relevant frequencies silent. It does, however, significantly reduce their amplitude and distort the phase relationship of those frequencies, relative to the direct, unimpeded sound energy.

Let’s apply the kr axiom to a few examples: Using 2 kHz and a 3” pillar: 2,000/1,127 = 1.78 waves that will fit into 1’. Now multiply this by the SigDim of 3” (0.25’) and you get 0.44:1, which is pretty good. But, because the kr is above our 0.25:1 threshold, there will be some discernible shadowing.

What if we use 2 kHz and a SigDim of 18”? 1.78 * 1.5’ = 2.66. This is bad because the wave ratio is much larger than 2.00.

Conversely, if you need to create a sound barrier, having a good understanding of the wave ratio dimensions can be very helpful when defining the size of the barrier and the frequencies that will be most effectively blocked.

kr Applied To Speech Intelligibility

Finally, we return to the initial question. For most applications, 4 kHz is the frequency that’s most important to evaluate. 4 kHz represents the shortest wavelength and most easily blocked frequency of the nominal range of frequencies (500 Hz – 4 kHz) used to evaluate speech intelligibility. This frequency is also high enough that most structural impediments will cause some significant, if not full blockage.

Given those guidelines, is it not reasonable to focus on this frequency to gauge the acceptability of any obstruction? Considering this, couldn’t we simply call a SigDim of 0.84” (or even 1” for simplicity) our “line-in-the-sand”, and forego all the other calculations? Any larger impediment will introduce an unacceptable shadowing problem when striving for optimal speech intelligibility.

Then again, I suppose one could build an argument saying that 7” should be our line-in-the-sand. This idea comes from the fact that, on average, adults have about 7” of space between their ears. If only one ear is blocked, it’s possible the other ear will hear the 4 kHz signal unobstructed. Of course, this idea tends to fall apart when you’re listening to a stereo mix, presuming one ear is being shadowed because of the obstruction.

Try Your Own Listening Test

Set up a rig in your shop using a small loudspeaker, a 6″ to 9” wide box or pole, a sine wave generator, and a listener.

At 2 kHz, you should begin to notice some significant shadowing when the box is placed directly in line between the loudspeaker and listener. Move your head around to evaluate what’s happening with one or both ears being blocked. Try lower and higher frequencies to hear what changes.

At 4 kHz you may only hear reflections off other nearby surfaces. I’m sure results will vary based on the test setup, but I think you’ll get the point.

Final Thoughts

I’m reminded that at some venues “obstructed seating” tickets are sold. What’s implied is a visual obstruction to what’s presented on stage. It’s never an audible obstruction. Nevertheless, if you can’t clearly see the front of the speaker(s) that are supposed to be covering your seat, because of a pole, column, light fixture, wall or other blockage, you too will have an obstructed seat. Need I say more?

Editor’s Note: Michael Lawrence, Steve Liddle, Charlie Hughes, Doug Jones, Richard Honeycutt, and Jay Mitchell also contributed to this paper.

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About Michael

Michael Fay
Michael Fay

Michael Fay is owner/principal at GraceNote Design Studio, an audio, video and acoustic design consultancy; a sustaining member of SynAudCon; a member of AVIXA and the Acoustical Society of America; former Integration Division general manager and senior design consultant with Sound Image; and former editor of Recording Engineer/Producer (RE/P) magazine.

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