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Schroeder Diffuser

July 24, 2017

Many an animated cartoon has used the "echo canyon" joke in which a person shouting a word, or a short phrase, first hears an echo of his voice, then hears a response to a later phrase. There's an Echo Canyon State Park near Pioche, Nevada, and various other echo canyons in the United States, including one in Zion National Park near Springdale Utah. The Simpsons, when aired in short segments on The Tracey Ullman Show, had an Echo Canyon short, March 12, 1989.[1-2]

There's even a mathematics-themed echo canyon joke, as follows.[3] The interesting thing about this joke is that "string theorist," or a few other scientific professions, could be used in place of "mathematician."
Three men are in a hot-air balloon, and they found themselves lost in a canyon, somewhere. One of the men says, "I've got an idea. We can call for help in this canyon, and the echo will carry our voices far." So, he leans over the basket and yells out, "Helllloooooo! Where are we?"

(They hear the echo several times).

Fifteen minutes later, they hear this echoing voice: "Helllloooooo! You're lost!!"

One of the men says, "That must have been a mathematician." Puzzled, one of the other men asks, "Why do you say that?"

"Three reasons. (1) he took a long time to answer, (2) he was absolutely correct, and (3) his answer was absolutely useless."

Recorded music was increasing in popularity in the 1950s, and that gave rise to many garage recording studios having primitive equipment and a single ribbon microphone as a centerpiece. To prevent recordings from sounding like they were made in an empty closet, the audio engineers needed some inexpensive ways to prevent echos from the walls and ceilings. One technique was to hang blankets. Another was to use inexpensive acoustic absorbers, as shown in the photo.

Figure caption

The left part of this cardboard egg carton would make a fine acoustic absorber. Quite a few of these would be needed to cover the walls and ceilings of a recording studio. Unless treated, these would be a fire hazard.

(photo by Dfrg.msc (Nicolas), modified, via Wikimedia Commons.

Most of us have witnessed how the echo in rooms of an empty house or apartment is extinguished once the furniture is in place. The soft fabrics of furniture and carpets, and the irregular surfaces of the furniture are good at preventing echos. Some rooms, such as auditoriums, are necessarily large, unfilled volumess. Surface appliances have been developed to prevent echos in such rooms (see photo). The general idea is to makes these from a sound-absorbing material, and form the surfaces in a way that residual reflections are diffused.

Figure caption

Left, an anechoic chamber at Yokosuka Research Park, Japan. Right a radar absorbing surface. The principles of radar wave absorption and acoustic wave absorption are similar. (Left image by Jpatokal; right image by Togabi; both via Wikimedia Commons.

In 1975, noted German physicist, Manfred R. Schroeder (1926-2009), published a design for an acoustic diffuse reflector.[4] The Schroeder diffuser has been used quite often since that time for both architectural and technical applications. One requirement for such structures that depend on modification of the phase of the reflected wave, including radio antennas, is the requirement that the size of the elements is on the order of the wavelength, which is large for most audio frequencies. There are many YouTube videos showing how to build a Schroeder diffuser.[5-7]

Principle of a Schroeder Diffuser

As they say, "It's all done with mirrors." The varied heights of the reflecting surfaces in a Schroeder diffuser create phase shifts that cause reduction in the intensity of the acoustic wave. (Created with Inkscape.

Now, physicists and engineers at Nanjing University (People's Republic of China) and North Carolina State University (Raleigh, North Carolina) have tackled the problem of the diffuser's size being of the order of a wavelength.[8-9] For a Schroeder diffuser, the element depth must be about half the wavelength of the lowest frequency; and, for a male speaker, this is 85 Hz, for a half-wavelength of eighty inches.[9] They've used a metamaterial concept to make a metasurface that reduces the dimension by a factor of ten.[8-9]

This is the first use of a metasurface as a sound diffuser.[8] The diffuser is an array of evenly spaced square chambers having varied aperture.[9] The apertures are designed to introduce phase shifts into the reflected sound waves, the same principle on which the Schroeder diffuser is based. The research team used numerical simulation, and they created 3D-printed diffusers of an acrylonitrile-butadiene-styrene plastic.[8] Experiment showed that the metasurface diffuser worked as well as a Schroeder diffuser.[8]

NCSU acoustic diffuser

A conventional Schroeder diffuser is shown on the left, and the metasurface diffuser is shown on the right.(NCSU image.

Says Yun Jing, an assistant professor of mechanical and aerospace engineering at NCSU and an author of the paper that describes this work,
"Diffusers are often made out of wood, so our design would use 10 times less wood than the Schroeder diffuser design... That would result in lighter, less expensive diffusers that allow people to make better use of their space... We've built fully functional prototypes using a 3-D printer, and it works... The design should work just as well using wood."[9]

Phase as a function of opening width

Phase shift as a function of aperture dimension for the metasurface acoustic diffuser.

(Data from Ref. 8, graphed using Gnumeric.[8]


  1. Echo Canyon, Simpsons Wiki.
  2. Echo Cannyon, YouTube Video by The Simpsons Shorts, June 27, 2016.
  3. Math Jokes Page of Zachary S. Tseng, Pennsylvania State University Math Department Web Site. The joke can be found in many places, such as here, here, and here.
  4. M. R. Schroeder, "Diffuse sound reflection by maximum−length sequences," The Journal of the Acoustical Society of America, vol. 57, no. 1 (1975), pp. 149 ff., doi: http://dx.doi.org/10.1121/1.380425.
  5. How to build an acoustic diffuser, YouTube Video by TheDSPproject, May 7, 2013.
  6. Calculate Skyline Diffusor, based on research by R. Walker, Engineering Division of the British Broadcasting Corporation.
  7. Do-it-Yourself Acoustical Treatment: How to Build a Diffuser, YouTube Video by FocalOnline (Peter Johnson), May 10, 2012.
  8. Yifan Zhu, Xudong Fan, Bin Liang, Jianchun Cheng, and Yun Jing, "Ultrathin Acoustic Metasurface-Based Schroeder Diffuser," Phys. Rev. X, vol. 7, no. 2 (June, 5, 2017) Article No. 021034, DOI:https://doi.org/10.1103/PhysRevX.7.021034. This is an open access article with a PDF file here.
  9. Matt Shipman, "New Sound Diffuser Is 10 Times Thinner Than Existing Designs," North Carolina State University Press Release, June 6, 2017.

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