Encouraging students to understand the 1D wave equation

Dimensional analysis just means checking that the units work out. You can do it!

x, dx, y and dy have units of length. dt has units of time. v, I’m guessing is velocity, so length/time.

length/length^2 = (length/time)^2 * length/time^2

->

1/length = length^3/time^4

These are different types of things, they can’t be equal to each other.

It is essentially like static types in programming, but for physicists, and it is a similarly simple but dramatically powerful idea.

There wouldn't if the computer algebra step was integrated in the typesetting. Rather you first derive/verify the equations in a CAS and then write them in TeX for the paper. Even if export (like Mathematica) you may still decide to fix formatting a bit doing a seemingly obvious change which ends up changing the expression.

I can't find the homepage for Maxima?

https://schema.org/speakable :

> Indicates sections of a Web page that are particularly 'speakable' in the sense of being highlighted as being especially appropriate for text-to-speech conversion.

- [ ] [Simple] Wikipedia doesn't yet have a schema:speakable attribute on any of the schema:Articles,

but Simple Wikipedia's is interesting for reference

One of the things that engineering school really drove home for me was the idea that "all models are wrong, some models are useful". You're completely correct that this model is "wrong" because it doesn't account for relativity, but in a lot of cases (plucking a string, waves in water, sound, etc) the relatively component is so small as to be insignificant. We do this all the time across all disciplines: resistors are treated as purely resistive even though their leads have some amount of capacitance and inductance, steel beams are treated as isotropic even though they might have some kind of crystal-grain-induced directionality in strength, the classic F=mg model of gravity works just fine for lots of practical problems. All of these are "wrong" but they're still incredibly useful and give good-enough answers.

To the 1D/2D question, that's more a matter of semantics I think. A more accurate name for it would be the "Wave Equation for a wave propagating in one spatial dimension over time" but that doesn't quite roll of the tongue quite the same way :).

Mathematician here, but I can also speak for the physicists in this regard. When we say “1d”, “2d” or “3d”, we refer to “space”, i.e. space dimensions. Those diagrams you refer to are called “space-time” diagrams, and reflect the situation in space (y-axis) at a given time (t-axis).

A first remark can be made here: the wave equation is not symmetric in space and time; as a consequence, space and time are fundamentally different (this is even more clear on the heat equation, where not only do we have space-time asymmetry, but also irreversibility). A second remark is that the wave equation is hyperbolic, meaning it has underlying geometrical objects called “characteristic curves”.

This “characteristics” are quite special, as are trajectories in space-time where the solution of the wave equation (the wave profile) looks constant. In the simplest of scenarios, this characteristics are straight lines in space-time (i.e. x-ct=const), and have the remarkable property of separating space-time.

For the wave equation, there are two characteristic curves: x-ct=const and x+ct=const. The first one represents a wave traveling forward in space and the second one represents a wave traveling backwards. Together, they form a “light-cone”, and break space-time into two: “space like” space-time (up and down regions of the cone) and “time like” space-time (left and right region of the cone). For every event in space-time, there is a light cone, and nothing “space like” can communicate with something “time like” without sending waves traveling faster than the wave equation’s propagating speed (commonly called “c”), but any wave traveling faster than “c” would break uniqueness of solutions (information paradoxes). Geometrically, this means that no two places, A and B, in space-time where t_A > t_B can communicate without sending signals traveling faster that the speed of propagation “c” and thus breaking uniqueness, hence no point in the future can speak with a point in the past, and this implies causality.

Adding more space brings even more structure: 3d wave equation “averages out” information, and 2d wave equation solutions lead to saturation of information.

It is remarkable that such a “basic” equation (linear, 1d, second order) can have so many properties, and that such properties go along very well with our experienced reality.

Daniel Fleisch has written a series of books "Student Introduction To.." for maxwells equations, waves, tensors. All his books are famous for explaining everything step by step and making it informal but super clear.

Thank you and the sibling comment. These are interesting resources for practical and physical simulations of real phenomena.

I thought more about an idealized wave (which water waves are not) through a homogeneous euclidean medium of idealized single oscillators. Each oscillator can only communicate to its direct neighbors.

I should read through some of your materials as this might be what they talk about in the first chapters. But they focus too much on the actual physics and less about the implementation.