The wave equation is a beautiful object. It gives rise to a transparent geometrical solution and within its simplicity it holds the secret to causality, even in 1d.
context: Non-physicist here; and I struggled with maths from calculus onwards despite "liking" it.
Causality requires time - something (A) causing something else (B) means A and B are separated by time (or maybe distance). If they're not, then aren't they "the same thing", and the event is a single system and no energy or information has changed? In fact, the idea of _change_ also requires time. All the graphs in the abstract of the paper show a value Y and time axes.
How is any of this "1D"? Why isn't it 2D, with one of the Ds measuring time? Do physicists just ignore time because it's always there? How can that work if time is relative? Surely that _forces_ us to always pay attention to it? (and if time is relative, doesn't that mean you also need a 3rd D against which to observe the relativity, all just to measure that one dimension you were interested in?)
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.
Causality requires time - something (A) causing something else (B) means A and B are separated by time (or maybe distance). If they're not, then aren't they "the same thing", and the event is a single system and no energy or information has changed? In fact, the idea of _change_ also requires time. All the graphs in the abstract of the paper show a value Y and time axes.
How is any of this "1D"? Why isn't it 2D, with one of the Ds measuring time? Do physicists just ignore time because it's always there? How can that work if time is relative? Surely that _forces_ us to always pay attention to it? (and if time is relative, doesn't that mean you also need a 3rd D against which to observe the relativity, all just to measure that one dimension you were interested in?)