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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

Wavelength Architecture

Throughout this discourse there are three ways sine-wave frequencies are defined and/or measured: by wavelength, radian, and time.

1) Wavelength – For any given frequency, the linear distance between any two successive points having the same phase, such as at corresponding zero crossing points, or peaks of positive amplitude, is its wavelength (Figure 1).

Figure 1: 1,127 Hz has a wavelength of 1 foot and a quarter wavelength of 3”. Note: the quarter wavelength equals 1.57 radians, the half wavelength is 3.14 radians, and the “k” or wave number is 6.28.

 

While wavelengths are a key element in this commentary, we generally think and communicate using frequency. Therefore, it’s important to understand that every frequency has a corresponding wavelength, which is derived by including the speed of sound, using this formula: λ = v/f

The symbol used to represent wavelength is λ. This is the lower case Greek letter lambda. v represents the speed of sound, and f represents the frequency.

2) Radian – Because sine waves are represented as circular motion, and we need to understand and sometimes use their curvilinear dimensions, the radian is used to calculate and express the total, uncoiled length of a given frequency’s wavelength (Figure 2). Radians can also be used to calculate and define phase angles when needed.

Figure 2: 1,127 Hz shown in Radians (Note the phase angle of one Radian is 51.31º)

 

Any wavelength can be used to define the radius of a circle. That radius is equal to one radian of the circle being characterized. This website can help in better visualizing how 2D sign wave motion translates into 3D circular motion.

3) Time – When traveling through air the speed of sound exhibits an approximate constant, which is implicit throughout this paper as 1,127 feet per second (fps). Based on temperature and humidity, this constant may shift up or down a little, which is why we often see the speed of sound stated anywhere between 1,125 and 1,130 fps.

The scientific term for the time it takes for one full cycle to pass a given point in space, regardless of frequency, is the “period” (Figure 3).

Figure 3: One full cycle, regardless of where the measurement is taken, is the wavelength’s period.

 

Examples: A 1,000 Hz oscillation has a wavelength of 1.127’, and a period of 1 milliseconds (ms). A 125 Hz tone has a 9.02’ wavelength, and a period of 8 ms.

Don’t confuse the different periods with a difference in the speed of sound based on frequency. The leading edge of all audio frequencies travel at the same rate. It’s the trailing edge of each wave cycle that requires more or less time to finish.

This website can help in calculating the wavelength of any frequency, using any speed of sound reference you choose to use: click on the “Solve” button below the formula. You can change all the units from metric to imperial by typing in the measurement units you want to use.


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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.
https://www.gracenoteds.com/

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