The term “wavenumber” seems like an odd concatenation, especially when used for a company name. Wavenumber is something fundamental to the physics we experience every day. Most people know what wavenumber is intuitively, but have little reason to assign a word to it. My observation is that even people with science and engineering background have either never heard about it or have a “substrate level” understanding of it. In this paper, you will learn about wavenumber using its trendier cousin frequency.
As for the company name, let us start off by saying that I did not think that yet another "Acme LLC" or "Amalgamated Services Inc." was necessary in the world. I did not want my own name in the company nor to use an “office space” style tech company designation. Names of organizations, and to a large degree the products/services they provide, should be connected to some fundamental purpose, ideal or concept.
Waves in time
As a starting point, let us start with the concept of a wave. Everyone has a level intuition about what a wave is. There are sound waves, radio waves, light waves, waves in ocean.... You get the point. Regardless of the type of wave, there are a handful of useful terms used to describe a wave. One that you might hear often is “frequency.” To gain understanding of frequency, let us consider the concept of a wave in time.
Figure 1: Waves in time.
This sine wave has peaks and valleys that occur over time. This wave could represent the signal from a guitar, the temperature of your bedroom over the past 48 hours or even the price of rice in China over the past one hundred years. For the sake of clarity, I chose not to assign any meaning to the vertical scale. The only thing we care about here is that something is repeating over time.
The first important feature is the space in between the peaks, or crests, of the wave. We call this the period of the wave and usually assign it the variable T. Period has the units of time (seconds, minutes, hours, etc.) and is the length of one cycle of the wave in time. While picking the crests of the wave is convenient, we could have chosen the starting point anywhere in the waveform. The next important characteristic of a wave is its frequency. Frequency is measured in cycles per second and the unit is Hertz (or Hz). Frequency is defined as the reciprocal of period or (1/T).
Thinking about frequency as cycles per second makes intuitive sense as it is the number of cycles, or repetitions, of the wave in one second. Engineers sometimes use radian frequency denoted with the lowercase Greek symbol omega ω. We need to make this distinction as it will be important for the concept of wavenumber.
Figure 2: Cyclic vs. radian frequency.
Radian frequency is computed as (2π/T) which yields a unit of radianspersecond. There are 2π radians around one circle (360 degrees). Moving through one cycle of a wave is the same saying the wave has traversed 2π radians. Also note the unit of cyclic frequency. It is cyclespersecond. When we take the reciprocal of the period (measured in seconds) you would think that we would a unit of (1/seconds), or inverse seconds. This is mathematically correct, but we typically insert the junk, or invisible, unit of cycles to assign physical meaning. Referring to the unit of frequency inverse seconds never sat well with me. Most people have agreed with this sentiment and use cycles per second or Hz.
Waves in space
Now that we understand frequency, let us modify Figure 1 a little bit. What we will do is change the time axis to a space axis. Instead of thinking about the wave wiggling in time, think about it wiggling over space or distance.
Figure 3: Waves in space.
All that we did was change the x-axis to a unit of space. This could be meters, parsecs, furlongs… your choice. When we measure the distance between the crests of the wave, we use the term wavelength. Wavelength is the same thing as period from Figure 1. All we did was replace the time axis with a unit of space. Wavelength is commonly represented with the lower-case Greek symbol lambda λ.
Wavenumber is similar to radian frequency in time and is defined as (2π/λ). If we measure wavelength in meters, the unit of wavenumber is (radians) per meter. Notice how I put the “radians” in parenthesis. It is common in the physical sciences to refer to the unit of wavenumber as inverse meters or (1/m). The rationale is that the radian component is a junk or invisible unit. I strongly feel that we should be consistent with the convention used with period and frequency.
Radians per meter indicates the physical significance. If we can add a junk unit of cycles or radians to cyclical or radian frequency, we should do the same with wavenumber. It is just being consistent and inverse meters has absolutely no intuitive meaning to me.
Notice that I started with the definition of wavenumber as (2π/λ), not (1/λ). Some of the physical sciences use (1/λ). It is the same as using (1/T) for cyclic frequency. In this case, the units of wavenumber would be cycles per meter, which has intuitive physical meaning. It is the number wave oscillations that occur over one meter of distance. A common convention is the symbol k to represent wavenumber.
Wavenumber is simply frequency in space.
The esoteric representation
How do we get to the company logo? It is just an esoteric method of expressing wavenumber.
Figure 4: Getting to the esoteric representation.
There is one last import detail in the esoteric logo representation. Why the funny arrow over the expression? If you are thinking ahead, the consideration of waves in space implies a multi-dimensional concept.
Multi-dimensional thinking
Most humans experience their daily lives in the R3 vector space using an orthonormal basis in cartesian coordinates. More intuitively stated as we think about “up and down”, “left and right”, and “back and forth“ (X, Y, Z). Wavenumber is much cooler than frequency as it is multi-dimensional! One of the first two-dimensional waves you learn about in the field of acoustics is the plane wave.
Figure 5: A two-dimensional plane wave.
Because the wave is in 2d space, we can think of it moving a little it in the x direction and a little bit in the y direction. This means that wavenumber has subcomponents and is a vector. Waves can have different "spatial frequencies" in each axis. If you have spent any time near a body of water, chances are that you have observed a two-dimensional wave. The “Ripples in the water” is a useful metaphor.
Figure 6: Visualizing spatial waves using ripples in the water.
In fact, in the real-world wavenumber is three dimensional. Since wavenumber can be a vector, we add the arrow as it is common convention in mathematics and physics. The esoteric representation includes the arrow. In case you are wondering "can waves exist in both time and space simultaneously?". The answer is yes! Wavenumber and frequency are linked by another variable, the velocity of wave propagation. In the realm of Acoustics, this is the speed of sound. For electromagnetics, it is the speed of light!
Figure 7: The Linkage between wavenumber and frequency.
Planting the flag
Wavenumber is both interesting to visualize and it is very fundamental the physical world. Because of this, I thought it was is a great name for a company and wanted to plant my flag in the ground.
The flag in the ground.
The next time you are listening to your fancy new audio system and a friend asks to make frequency adjustments via an equalizer, you can raise the bar a bit with:
"These are waves in space. We need to adjust the wavenumbers."
