This is a great question.    In addition to the very helpful answers already given, there is a bit more to add.

Perhaps the followers of this thread will be interested in what is known K-wave theory introduced in

https://www.researchgate.net/publication/220035818_Kondratieff_waves_in_global_invention_activity_1900-2008

The basic idea is to identify up swings and down swings in a long wave phase in social activity (see, e.g., Table 1 on page 1281).

Article Kondratieff waves in global invention activity (1900–2008)

Heraclitus said that “Everything flows and nothing stays”, see https://en.wikiquote.org/wiki/Heraclitus . Wave motion is one of essential motions of matter. Pendulum motion is mechanics is described by the differential equation x’’+ax=0 that has a solution in the form of sine periodic wave. Similar processes exist in continuous media. We have mechanic waves in the air, water and the Earth. We also have electromagnetic waves in all of those media, and their spectrum (range of existing frequencies) is indeed very wide. Are all natural motions described by waves? Probably not; an eruption of a volcano (or land-slide) is a shock-type irreversible process. Both are considered to be natural disasters. Waves are mostly reversible and are not disasters (except for tsunami, but it is different wave having not periodic but solitary structure).

If we look at social phenomena, there are also waves (business or political cycles), but they are not dominating the research landscape. The dominant concept of neoclassical economics is equilibrium, while dynamic research is more concerned with stochastic phenomena.

Why wave approach has penetrated social phenomena much less? What is the reason for that? Non-knowledge of natural sciences and wave theory by social researchers? Or maybe the belief of politicians in irreversibility? Many revolutionaries (and even some kings) start from "we destroy all the past, and create completely new". But in 20-50 years the social life often returns to past structure. In the USA republicans change democrats. Even counter-revolution often follow revolution. Both can be viewed as some kind of wave process. However, peaceful reelection creates less irreversible changes (catastrophes) comparing to revolutions.

Well, first of all, there is no a unique interpretation of the wave funcion - in quantum theory.

That said, I think the main philosophical role consists in making clear that there is no distinction between ontology and epistemology. Vis-à-vis normal science, this recognition is superlative!

Dear Carlos, in quantum physics all is based on the solutions of Schrodinger equation. We have a cloud of probabilities, different for different energy levels. Only at the absolute zero temperature there are no oscillations, while they are present at all positive temperatures, when some quantum levels are excited. Periodic motion is non-dissipative, and thus is reversible. This is more important philosophically than some mystic in interpretation of complex values in wave function.

Dissipative processes are typical in statistical physics, when we deal with growing entropy. Still at negligible friction we have almost periodic processes. The day on the Earth is about the same, although it will be significantly less in a billion of years. But by that time our civilization will be extinct, so for us the Earth's rotation is periodic.

But I do not want to move into philosophy of physics but just want  to discuss more the social applications. Also at more understandable language for the general public.

P.S. Just to clarify for the general public: "epistemology is about the way we know things when ontology is about what things are", see https://www.quora.com/What-is-the-difference-between-ontology-and-epistemology . In natural sciences the difference emerges at micro level, where we cannot see it directly. In social science we can observe all, but there may be hidden (secret) and manipulated information.

Dear Yuri! The philosophy of the wave theory originally presented in a convincing work that I placed in the Attachment.

Vladimir

This is a great question.    In addition to the very helpful answers already given, there is a bit more to add.

Perhaps the followers of this thread will be interested in what is known K-wave theory introduced in

https://www.researchgate.net/publication/220035818_Kondratieff_waves_in_global_invention_activity_1900-2008

The basic idea is to identify up swings and down swings in a long wave phase in social activity (see, e.g., Table 1 on page 1281).

Article Kondratieff waves in global invention activity (1900–2008)

Gear Yuri Yegorov,

I like your classification. I think and then reported to be able to say something more specific, something to add ...

Dear Vladimir, James and Pavel, thank you for the interest to the topic.

James, I am aware about Kondratiev waves and even wrote a conference  text about that: https://www.researchgate.net/publication/292991377_Long_Economic_Waves_as_Innovation_Cycles

However, it was a special conference, while modern microeconomics and macroeconomics textbooks are typically silent about that. In some books for mathematical economics one can find a cobweb model; see https://en.wikipedia.org/wiki/Cobweb_model .

However, waves and periodicity are present virtually in all production. Consider first agricultural cycle: prepare land - throw seeds - put water - gather the crop. This is an annual cycle. There are many daily cycles: production of fresh bread, collecting garbage, etc. Movement of public transport is also periodic with variable frequency.  On the demand side, see the typical profile for electricity demand during a day (attached). It also depends parametrically on season. The source is http://www.testbig.com/ielts-writing-task-i-essays/graph-below-shows-demand-electricity-england-during-typical-days-winte-3 .

There is also product cycle. A certain type of product is first invented, then penetrates the market, wins it, but after some time becomes morally non-acceptable. The dynamic of output typically has bell shape; it is a solitary wave. Why a wave? There might be not only temporal but also spatial pattern. Suppose it was invented in country X and reached the maximal level of output at time T1. Country Y copied it with some delay and the peak of production there happened at time T2>T1. Consider production of certain types of cars and mobile phones as examples.

Conference Paper Long Economic Waves as Innovation Cycles

Exploring the economy, Mandelbrot discovered that the seemingly random fluctuations in prices may follow the hidden mathematical order in time, which is not described by the standard curve.

Intrigued by the unexpected with the result, Benoit Mandelbrot began to study the cotton prices over a long period (more than a hundred years old) statistics. Fluctuations during the day seemed to be random, but Mandelbrot was able to find out the trend of their change. He followed the symmetry in the long-term fluctuations in prices and short-term fluctuations. This finding was a surprise to economists.

***

https://en.wikipedia.org/wiki/Benoit_Mandelbrot

Benoit B. Mandelbrot  (20 November 1924 – 14 October 2010) was a Polish-born, French and American mathematician with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life." He referred to himself as a "fractalist".  He is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal'", as well as developing a theory of "roughness and self-similarity" in nature.

Behind the role of wave theory in dynamics of natural and social processes, there are aperiodic dynamics also.

Aperiodic dynamics in a deterministic adaptive network model of attitude formation in social groups

Adaptive network models, in which node states and network topology coevolve, arise naturally in models of social dynamics that incorporate homophily and social influence. Homophily relates the similarity between pairs of nodes’ states to their network coupling strength, whilst social influence causes coupled nodes’ states to convergence. In this paper we propose a deterministic adaptive network model of attitude formation in social groups that includes these effects, and in which the attitudinal dynamics are represented by an activator-inhibitor process. We illustrate that consensus, corresponding to all nodes adopting the same attitudinal state and being fully connected, may destabilise via Turing instability, giving rise to aperiodic dynamics with sensitive dependence on initial conditions. These aperiodic dynamics correspond to the formation and dissolution of sub-groups that adopt contrasting attitudes. We discuss our findings in the context of cultural polarisation phenomena....

Dear Ljubomir, a-periodic structures also belong to deterministic dynamics. Even chaos, mentoined by Prof. Zabrodskii in the works of Mandelbrot, belongs to it; see other discussion on that: https://www.researchgate.net/post/Do_we_understand_the_philosophical_difference_between_stochastic_and_chaotic_phenomena

I would like to attract your attention to biology with its intermediate position between physics and social processes. There are two types of animals - predator and prey. By the way, Lotka-Volterra equation of interaction between them also has periodic solutions; see https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations .

But now I want to touch a different dimension. Vegetarians typically spend most of their time eating and sleeping. For example, panda eats and sleeps more than 90% of the day. The movements and actions of those animals are periodic (for example, antelopes move to drink at 11am in a certain location, while zebra do it at 13:00). Let us look at the dynamics of actions of a predator - lion. It may make several jumps during a day in pretty random time, and one of those jumps will be successful. So if we observe the time series of lion's jumps during hunting, no statistical technique can infer some regularity in this random process. One day it may be at 11:12, and other day at 13:45. At the same time, we can discover a lot of periodic processes in the activity of preys.

I do agree dear @Yuri. It is same with fish. This is fine resource. It is logical! 

This graph shows the cyclical relationship predicted by the model for hypothetical predator and prey populations.

http://www.tiem.utk.edu/~gross/bioed/bealsmodules/predator-prey.html

http://jmahaffy.sdsu.edu/courses/f09/math636/lectures/lotka/qualde2.html

Indeed, cycles are often observed in biology and environment. As for economics, perhaps agriculture has cycles of output, that is highly dependent on weather. We rarely see a pure cycle, but any natural process can be decomposed into spectrum of events with different frequencies.

In economics, however, this is not typical case. Well, long Kondratiev waves were mentioned already. As for business cycles, there are models with response to shocks. But typically there is fast decay, so one can hardly observe several oscillations. In other words, responses may be similar to function f=exp(-at) cos (bt) +eps, with a and b such, that one can detect at most 1 cycle.

Perhaps, decay plays high role of economics because of rationality. Nature is passive and subject to its laws. If rational humans observe cycles, they behave in such a way that cycles disappear very fast. At the same time, we have a lot of chaotic behavior without clear pattern, that one cannot learn and change.

Waves tranlate into cycles, and cycles translate into spirals; the motion of the spiral development can either be progressive or regressive. There is a great affinity (methodical analogy) between frequency processes (energy transduction) of natural and social systems, concerning pattern recognition.