[Draft]

#### The linear venue

Some grocery stores still have spring scales in the fresh produce section. Let’s say, just to get the raw weight, you use the scale to weigh some oranges – 2 lb 6 oz. Then you weigh some apples – 3 lb 5 oz. Putting them together in a plastic bag from a nearby dispenser, you check the combined weight. The scale shows 5 lb 11 oz. A linear model, you see.

Imagine rigging an elastic string (or thin elastic cord) horizontally between two poles (like a long clothes line). First, you hang a small weight (which may be released remotely) from the center of the span. Then setup your smartphone camera for super slo-mo, start recording, and release the weight. Second, you hang together that weight and an identical one, and slo-mo again. How does the motion of the string in each case compare?

In high school, I built a small water tank (containing a water tunnel) to analyze the relative performance of foils for hydrofoils. Various shapes. It also might have been useful to model how waves interact (interfere) and combine (overlap). With waves originating from one or both ends of the tank, and demonstrating the principle of superposition – and the related mathematics. [1]

In particular, the mathematics of linearity – systems of linear equations, which came later in college. But those mathematics classes did not connect with my experience in fluid dynamics courses. (My recollection is that the professor for linear algebra did research in traffic analysis.)

As noted in a recent video (below) by Parth G, “Luckily, the universe seems to behave linearly very often …” That is, our linear mathematical models accurately predict the interactions of waves – in systems which are wave-like.

Wiki: “… many physical systems can be modeled as linear systems.” [2]

#### The wave equation – natural things and strings

• Wiki > Wave equation

The wave equation is a second-order linear partial differential equation for the description of waves – as they occur in classical physics – such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.

Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d’Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.

Note that differentiability – as in a linear partial differential equation – assumes a continuous model space. Smoothness. (As in calculus for those infinitesimals.)

#### Superposition principle – for linear systems

Wiki has some animations here.

The math: “… if input A produces response X and input B produces response Y then input (A + B) produces response (X + Y).”

Additivity: F(x1 + x2) = F(x1) + F(x2)

Homogeneity: F(ax) = aF(x)

• Wiki > Superposition principle

The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually.

#### A technical overview by a science communicator

Here’s another video by [Patreon] science communicator Parth G on a foundational mathematical concept in physics, both in classical and quantum physics. An introduction to the principle of superposition for waves, waves which can be modeled as (differential) linear equations.

• YouTube > Parth G > “How Waves Overlap, and Why Common Sense Works! Principle of Superposition for Linear Equations” (Nov 25, 2021)

… why does it make sense to

add the displacementof each wave at every point in space and time in order to find the resultant wave? …In this video, we will look at the

Principle of Superposition. It explains why when two waves overlap, we can simply add their displacements at each point to find the resultant wave. This is what we would find reasonable due to our common sense. But common sense isn’t always correct – so why is it accurate here?To understand this, we need to realize that the

wave equation(the classical governing equation for describing all sorts of waves) is linear in wave displacement. In other words, the displacement of a wave (u) only appears as a single factor of u everywhere in the equation. … Thislinearityensures that if we take any two known solutions of the wave equation, then adding them together produces yet another solution to the wave equation – in this case the resultant wave. …Luckily, the universe seems to behave linearly very often, and any linear system can use the principle of superposition to find solutions that are formed by summing other existing solutions. …

##### Notes

[1] So, perhaps an earlier exposure to Fourier-transforms. And later a better understanding of Maxwell’s equations and the Schrödinger equation.

[2] With the caveat that: “Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behavior.”