1. Wave Field Synthesis
Roger Vargas
PHYS 536

2. Sound Reproduction

3. Stereo and Other reproduction techniques
Sound reproduction techniques seek to mimic the behavior of the original source
Most common techniques today still have some problems:
Phantom Source is often created by speakers
Phantom source can lack a defined position
Has a small ideal listening area, the Sweet Spot
These problems are addressed in Wave Field Synthesis
Sweet spots refer to the tendency of multiple speaker systems to have create a focal point where the audio mixing is heard as originally intended, at this point the acoustic waves from all sources arrive simultaneously
The first and second point have to do with the amplitude that each speaker broadcasts and their position relative to the listener. If the listener moves closer to one speaker the sound will appear to move towards that speaker, similarly if one speaker has a much higher amplitude then the sound will appear to come from that speaker. This has to do a with a concept called amplitude panning
Credit: dsprelated.com

4. Huygens Principle
Every point on a wave front may be considered as a source of secondary spherical wavelets which spreads out in the direction of propagations of the wave front
Interpreted mathematically as the Helmholtz-Kirchhoff integral:
P 𝐫, ω = − 1 4π S [G 𝐫 s 𝐫,ω ∇ 𝐫 s P 𝐫 s ,ω −P 𝐫 s ,ω ∇ 𝐫 s G 𝐫 s 𝐫,ω ]∙ 𝐧 𝐫 s dS
Huygens principle makes no mention of the kind of wave front and as a result even plane waves can be created
Result: Almost any kind of source can be recreated in a “natural” way with very high fidelity
Huygens principle forms the basis of wave field synthesis, suppose I have two sources, the first is some arbitrary primary source, if the sound from that source then enters a boundary of volume V then theoretically if I know exactly what the wave fronts look like I can construct the second source that mimics these wave fronts as they enter the boundary in which case a listener should not be able to tell the difference between my two sources, creating a virtual version of the original source
The Helmholtz Kirchhoff integral is hard to interpret, in an article by Alessandro Lapini et al he describes the terms as follows, P 𝐫 s ,ω is the pressure found at the surface of the volume, G is the greens function and can be thought of as sources created by the pressure disturbances on the surface. That is G corresponds to the secondary sources in Huygens principle. Note G behaves like a monopole source and the gradient of G acts like a dipole source
Plane waves are particularly useful because their pressure level does not decrease as they propagate in space

5. Wave Field Synthesis

6. Wave Field Synthesis
Helmholtz-Kirchhoff Integral while powerful is impractical as is.(Very Expensive, Very computationally intensive)
Make some simplifications:
Secondary sources are discrete not continuous
Approximate field using only monopole sources (No Greens function gradient)
Sources will exist on a linear array instead of covering the entire surface
Result: Linear array composed of independent channels
The consequence of the integral is that it requires you to model every single point on the surface the wave front that passes on it at a given time
These simplifications will of course come with errors that can affect the resultant amplitude, mixing, and frequencies that are produced
The sound played through the WFS has special considerations, for example where was the original source, was it moving, what direction was it facing.
Also note calling it is not always a linear array, some examples online show W arrays, or arrays composed of multiple layers. These configurations are meant to widen the ideal listening area of the WFS.
The independent channels are designed so that each speaker will mimic the wave front passing through the area

7. Limitations of Linear Arrays
The geometry of line array does not allow for the same kind of waves as a plane
Speakers used are not perfect simple sources
Linear array are discretely spaced meaning that spatial aliasing occurs
Confines listener to a horizontal plane that coincides with the array position
Huygens principle states that the secondary sources are of the form of spherical wavelets but line sources emit cylindrical waves
Similarly Huygens principle states that every point on the wave front acts as a simple source but this means that if we go from continuous to discrete space we no longer represent the wave front exactly
Another consequence of moving to a discretized space is that spatial aliasing is introduced. Essentially for short wavelengths they will alias as a lower frequency at a given point and as you move through the resultant field different wavelengths will alias giving a frequency comb. This is actually not too much of a problem as apparently it was found that humans don’t really detect these effects, but for other applications such as noise cancellation, anti-aliasing techniques are required
Finally a 2D source can fully account for the incoming wave front which is a 2D object, a 1D source cannot and as a result the audio is focused on a plane at the same height instead of the whole room. You can still recreate the audio but it won’t be as accurate as a planar array
Linear array acting on horizontal plane at 𝑧= 𝑧 1
Credit: A.J. Berkhout

8. Applications

9. Wave Field Synthesis Applications EMPAC
Uses a linear 496-Channel Array with 5.8cm spacing( 𝜔 𝑚𝑎𝑥 =3.3𝑘𝐻𝑧)
Designed to be easily deployable and can be torn down and set up elsewhere
Has additional mode that runs every other driver
Designed to allow for sound experimentation and to demonstrate WFS to listeners
WFS has many uses in sound reproduction due to its high fidelity even when in a line array, for example it can also be used to reconstruct room acoustics in other areas
The additional mode is there simply so they can show people what difference the spacing makes to the fidelity
Unrelated to the concept of WFS there is also a minimum frequency which relates to the geometry of the design and the speakers design it is around 140Hz
The array in total spans a distance of 33 meters, it is split into 16 units composed of 31 loudspeakers each

10. WFS Application: Active Noise Control (Highly Non Stationary)
Normally ANC can only be done effectively when noise is stationary or slowly changing
Traditional techniques do not work with highly non stationary noise, e.g. gunshot
ANC is based on creating counterwaves that destructively add to the source
Adaptive filter determines error, and adjusts parameters of array
WFS’s high fidelity allows for high accuracy control
This is accomplished using a WFS array and microphones, the microphones will pick up the incoming signal and feed it to the WFS were it will produce a wave front of opposite value
The remaining signal is passed to an error microphone which detects the error, this then leads to a change in the value of a filter used on the WFS speakers, this change allows for better cancellation
WFS is very effective and can cancel out sounds whose duration is at less then 2c𝜏 where 𝜏 is given by the time it takes for a signal to travel from the speaker array to the reference mics, this allows you to then turn off the mics that pick up the signal to prevent the detection of your own countersignal
Highly non stationary noise is difficult to control so it should be noted that WFS allows for better cancelling then other techniques but due to the different possible paths the source can take full cancellation is very difficult
Example of ANC setup
Credit: Alessandro Lapini

11. Works Cited
Berkhout, A. J., et al. “Acoustic Control by Wave Field Synthesis.” The Journal of the Acoustical Society of America, vol. 93, no. 5, 1993, pp. 2764–2778., doi: /
Brandenburg, Karlheinz. WAVE FIELD SYNTHESIS: FROM RESEARCH TO APPLICATIONS. WAVE FIELD SYNTHESIS: FROM RESEARCH TO APPLICATIONS.
Goebel, Johannes, et al. “Wave Field Synthesis.” Experimental Media and Performing Arts Center (EMPAC), empac.rpi.edu/program/research/wave-field-synthesis.
Lapini, Alessandro, et al. “Application of Wave Field Synthesis to Active Control of Highly Non-Stationary Noise.” Applied Acoustics, vol. 131, 2018, pp. 220–229., doi: /j.apacoust
Montag, Matthew. “Wave Field Synthesis in Three Dimensions By Multiple Line Arrays.”
Walls, Seth Colter. “Surround Sound? You Ain't Heard Nothing Yet.” The New York Times, The New York Times, 14 July 2017, you-aint-heard-nothing-yet.html.