What are some examples that require the use of Sinusoidal Regression? I am interested in knowing where do regression models of the form:
$$
\phi(x,A,B,C)=A\sin(kx+B)+C
$$
for a given $k\in\mathbb{R}$ are applied in domains such as engineering and science. What example are there that requires the use of this model?
I had encountered this model recently on math stack exchange. I would suspect the use of this model would be in a domain that heavily requires the use of signal processing but I can't seem to find any answer on the web of any application of sine regression.
These questions on MATHSE are mainly about how to transform the nonlinear sine model to a linear one via substitution in order to be solved numerically (such as using the normal system)
https://math.stackexchange.com/questions/3926007/least-squares-regression-of-sine-wave
https://math.stackexchange.com/questions/2368821/find-a-sinusoidal-regression-equation-for-some-given-data

Just an example that I often perform on MATLAB with my students: content of La Baells dam on Llobregat River, in Catalonia, from 2007 to 2017.
First I run this regression:
$$vol=A \cdot sin(T\cdot 2 \cdot pi/365.25)+B\cdot cos(T\cdot 2 \cdot pi/365.25)+K$$
where $T$ is time in days and $vol$ is water content in cubic hectometres:

That reveals the seasonal nature of water content. A closer inspection reveals a maximum in spring and a minimum in fall because in Central Catalonia the summer is the driest season, so precipitation is at minimum and water usage for irrigation and other uses is at maximum. We can also see that in 2007 we had a severe drought and that rainfall is quite irregular across years.
That can be improved by adding an harmonic:
$$vol=A \cdot sin(T\cdot 2 \cdot pi/365.25)+B\cdot cos(T\cdot 2 \cdot pi/365.25) + C \cdot sin(T\cdot 2 \cdot pi/365.25/2) + D\cdot cos(T\cdot 2 \cdot pi/365.25/2)+K$$

And here we can see a secondary maximum in early winter, after fall rains - fall is the second season with most precipitation, after spring.
The next few harmonics don't seem to improve the fit, and therefore they aren't included in this post.
