Physicists talk about vibrational modes of a string, for example the superposition of `sin(3x)` and `sin(5x)`. And the maths on a line segment or a disc can apply to electromagnetic fields, superstrings, sound waves, seasonal or multi-year or multi-century weather patterns, or the ODE’s for a particle wavefunction.

$E(x,y,z)= sum_{k=1}^M {(a_k e^{(i beta_k z)}+ b_k e^{(-i beta_k z)})E_k(x,y)}$

$H(x,y,z)= sum_{k=1}^M {(a_k e^{(i beta_k z)}+ b_k e^{(-i beta_k z)})H_k(x,y)}$

But if you take the perspective that an eigenbasis is constructed out of elements of a set, you can talk in the language of mathematics about a diverse range of topics that feel like “adding separable, independent pieces of little modes” together.

• paths that I walk to work
• mood patterns. Like a basis functioncould be
• that you always get more depressed in the winter
• and that is added in varying ratios to your workweek basis function, where Friday is the best
• and added to another “vibrational” mode which is the whips and wiggles of a particular client’s project
• If you want to talk about a non-scalar codomain, the things that I buy go through cycles. It’s autumn now and I am definitely going to be purchasing pumpkin-flavoured beer, coffee, pie, fuggineverythingIcangetapumpkinof.
• I usually smoke about 12 packs of cigarettes per year. However it’s not like I smoke one kretek every 2.53 days. Rather about once a month I hanker for a pack and then I do the whole pack within a day or two. So that’s a little “wavelet tooth” mode added onto the string of my purchasing behaviour, at irregular yet regular intervals. It’s not quite an addiction but it is a habit, or at least a repeated activity.
• Internet traffic patterns on a macro scale: at the holidays every epicyclic year people go home, get bored of their families, and surf Facebook. Likewise the work-week instantiates its own 7-day epicycle of time, as does the work-day. But here again if you want to use a non-scalar codomain and talk about not only numbers of traffic but geotag it to where it came from, then the work-day epicycles are actually bound to specific spots on the globe and it’s these eigenmoditas that are added up to get the geotagged traffic pattern.

(So on my regular rag about dividing groups only into homogenous equivalence classes, the eigenmodal components have something to add. The Sydney people have a warm-weather mode affecting them when the New York people have a cold-weather mode. But the Christmas eigenmode is affecting them both simultaneously, however not all of the people in either place necessarily celebrate it, yet probably everyone is affected by the tradition.)

• the Mauna Loa CO₂ data. In robust & exploratory data analysis class, Karen Kafadar showed us how to decompose this time series of carbon dioxide measurements at a Hawai’ian volcano into seasonal effects, last-day’s lags, and perhaps a different “mode” that’s picking up on local cause of a greenhouse effect.
• Crime waves. Like, literally, if they’re waves. Perhaps in the summer it’s easier to be outside so whenever the temperature waves up, so does the crime rate. Or perhaps a standing soliton of copycat murders propagates throughout a region. Or in Christian countries, maybe a sufficient number of people in rich neighbourhoods go on vacation that it makes sense to criminals to watch who’s left for the holidays and systematically bust into people’s houses whilst they’re away.
• Some theories of culture posit that “Everything goes in cycles”. Not just the Elliott wave theory, but I remember a Mormon friend explaining me his theory (and I’m not sure if this is generally held in Mormon doctrine or not) that the USA was getting too sinful and profligate and irreverent and secular, and that in time, like Sodom and Gomorrah, G-d would lay his country low.
• (I’ve heard other Estadounidenses draw parallels to “the latter days of Rome”, without realising that Rome’s fall can’t be pinpointed specifically because it didn’t Dirac down but rather “crumbled” or slowly changed, which is not their meaning.)