I often get the question "wavenumber..... never heard of that one before!". It does seem like an odd choice of words for a name. Let's start off by saying that the goal was to not have another "Acme Corp." or "Amalgamated Services Inc." in the world. Personally, I am also tired of acronyms that carry little to no information and have no inclination for a generic "Joe Smith Services". Names of organizations, and to a large degree the products/services they provide, should be more fundamental. So, let's start with "Wavenumber".
First, we will start with the concept of "waves". Everyone has some intuition about what a wave is. There are sound waves, radio waves, shock waves, waves in ocean.... I think you get the point. As my background is in electronics and acoustics, I really like waves and deal with waves a lot! Regardless of the type of wave, there are a handful of useful terms that we can use to describe a wave. One term that is ubiquitously used in modern language is frequency. "Woah man, that song has some low frequencies in it". So, let's see if we can understand what this really means. Figure 1 is an illustration that will help us as it shows the concept of a wave in time.
Figure 1: Waves in Time.
We can see a nice looking wave in Figure 1. It has peaks and valleys that occur over time. This wave could represent the signal from a microphone, the temperature of your bedroom over 48 hours or even the price of rice in China over the past 100 years. For the sake of clarity I chose not 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 time in between the crests of the wave. We call this the period of the wave an usually assign it the variable . Period has the units of time (seconds, minutes, hours, etc.) and is just the measurement of length of the wave in time. While picking the crests of the wave is convenient, we could have picked the starting point of the measurement anywhere. The crests are nice to use as it's easy to see where one cycle stops and the next one starts. The next important characteristic of a wave is its frequency
. Frequency is measured in cycles per second and the unit is called Hertz (or Hz). Frequency is simply defined as 1 divided the period
.
Before we get to wavenumber, lets talk about frequency a little bit more. Thinking about frequency in cycles per second makes intuitive sense in that it is the number of cycles, or repititions, of the wave in one second. To make things a little more complex, engineers some times refer to something call radian frequency. We need to make this distinction as it will be important for the concept of wavenumber. See Figure 2 for an explanation.
Figure 2: Radian vs. Cyclic Frequency.
In Figure 2 we can see that radian frequency is simply. This yields a unit of radians per second. There are
radians around one circle (360 degrees). One cycle of a wave is the same saying the wave has traversed
radians. Also note the unit of cyclic frequency. It is cycles per second. When we take the reciprocal of the period (which is in seconds) you would think that we would a unit of
. That would be the mathematically correct way of doing it but we like to insert the "junk" unit of cycles to give it some physical meaning. Calling the unit of frequency "inverse seconds" doesn't really sound meaningful in my opinion.
Now that we understand frequency, let's modify Figure 1 a little bit. What we will do is change the time axis to be a space axis. Instead of thinking about the wave wiggling in time, let's think about it wiggling in space.
Figure 3: Waves in Space.
First, we simply apply a unit of distance to our axis. Now when we measure the distance between the crests of the waves we call it wavelength. Wavelength is universally represented with the Greek symbol lambda. Wavelength is the same as period except that we are now talking about waves in space instead of waves in time. We define wavenumber as
. If we measure wavelength in meters, the unit of wavenumber is radians per meter. Notice how I put the radians in parenthesis. I really like calling the unit radians per meter as it means something physical to me. Most refer to the unit of wavenumber as "inverse meters" or
as they claim the the radian part is a junk unit. Well, I feel that if we can add a junk unit of cycles to cyclical frequency I should be able to use the unit radians per meter when talking about wavenumber. It is just being consistent. That, and "inverse meters" has absolutely no physical meaning to me. I can visualize radians per meter easily. Lastly we usually assign the symbol
to represent wavenumber.
So, wavenumber is simply a measure of frequency in space! But where does the logo come from? It's really just an obscure of writing the expression for wavenumber. See Figure 4.
Figure 4: Obfuscating Wavenumber.
Well, the algebra looks fine. is just a obfuscated wave of expressing wavenumber. The last question is why the funny arrow over the expression? Well, wavenumber is much cooler than frequency as it is multi-dimensional! Figure 5 shows a simple wave in 2d space. The red lines represent the peaks, or crests of a wave. The wavelength is just the distance between these lines. The x and y axis represent space. The z (if you could see it) is coming out of the screen and would represent the amplitude of the wave. This is a common wave of drawing 2d waves. By showing only the crests, or wavefronts, it's easier to visualize the 2d shape.
Figure 5: Waves in 2d space.
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 number can be broken down into subcomponents and is a vector. This means that waves can have different "spatial frequencies" in each axis. In fact, in the real world wavenumber is actually 3d! So, as it is a vector, wavenumber has a little arrow over it to indicate that it has both magnitude and direction. That is why the obfusacted expression in the logo has an arrow as well.
I hope this clears things up a bit! Wavenumber is both cool to visualize and it very fundamental to our world! Because of this I feel it is a great name for a company and wanted to put my flag in the soil. Chances are that you probably had intuition about waves in space but never had to the proper words to describe the phenomemon. When you are listening to your friend fancy new audio system and says something about frequency adjustments you can say "no, no these are waves in space. We need to adjust the wavenumbers!"
In case you are wondering "can't waves exist in both time and space simultaneously?". The answer is yes! Wavenumber and frequency are linked by a simple variable: velocity of propagation! For acoustics this is the speed of sound. For electro-magnetics its the speed of light!
-Eli Hughes