This paper has a pretty major typo in Eqn 11, which started giving me flashbacks to undergrad physics where authors would make major leaps in a derivation with a one sentence explanation.
Section II c tries to explain “the relationship between concavity and force” and concludes with:
> Then, adding a proportionality constant
k and using dimensional analysis, we arrive at
d2y/dx2 = v^2 d2y/dt2
> which is the 1D wave equation
.. except dimensional analysis would say that equation has
m^-1 = m^3 / s^4
.. which makes no sense, until you realize they flipped dx and dt, and that isn’t actually the 1D wave equation.
Although to be fair, correcting author’s typos in textbooks was one of the more educational parts of undergrad for me, so maybe that’s secretly the point of the paper.
It's beyond my current ability to evaluate if there's an error, but if so good catch. The contacts for the authors are on the following linked pages, I'm sure they'd appreciate correcting a mistake sooner than later.
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.
If there's one thing that quantum made me understand (and this was after 3 tries, which seems to be how many times it takes me for some things to become intuitive), it was continuous probability distributions (applicable to other parts of engineering/statistics) and how to understand the purpose of all those integral expressions without getting lost in all the symbols (learning to simplify understanding before getting wrapped up in details).
The formula for finding the wave function (i.e., the probability wave), is below:
i ℏ ∂/∂t Ψ(x, t) = Ĥ Ψ(x, t)
where i is the imaginary number, ψ (x,t) is the wave function, ħ is the reduced Planck
constant, t is time, x is position in space, Ĥ is a mathematical object known as the
Hamiltonian operator. The reader will note that the symbol ∂/∂t denotes that the partial
derivative of the wave function is being taken.
I love these kinds of Simple English pages where the author has evidently though that "simple" is supposed to mean "summary".
> 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
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.
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.
Since we are discussing wave equations in a computer science-related forum:
What is the 'best' algorithm for a discrete 1D (or n-D) wave simulation? With 'best' I mean simple but as realistic and stable as possible. The algorithm should operate on a 1 dimensional (or n dimensional) array of floats.
One of the open source codes from their research group is called Clawpack and is a good starting point for understanding things, and for cross-validation if you code something yourself.
But keep in mind, there are many different physical formulations of the wave equations that complicate matters quite a bit, especially beyond 1D. A shallow water wave has different physics from a deep water wave, which is again different from a sound wave. These require different numerical treatment. And e.g. if you go to large scale atmospheric (Rossby) waves, you need to solve on a sphere which is topologically a bit involved. It's a very rich field of study.
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.
The finite-differencing time-domain method [1] (sometimes also called leap-frog [2]) is easy to implement and robust for scalar and electromagnetic waves. This other book by LeVeque [3] is a great introduction on finite-differencing methods for linear equations.
I appreciate the research, but this sentence cracks me up: "At first glance, this equation may look simple since it only consists of two second-order partial derivatives."
Section II c tries to explain “the relationship between concavity and force” and concludes with:
> Then, adding a proportionality constant k and using dimensional analysis, we arrive at
d2y/dx2 = v^2 d2y/dt2
> which is the 1D wave equation
.. except dimensional analysis would say that equation has
m^-1 = m^3 / s^4
.. which makes no sense, until you realize they flipped dx and dt, and that isn’t actually the 1D wave equation.
Although to be fair, correcting author’s typos in textbooks was one of the more educational parts of undergrad for me, so maybe that’s secretly the point of the paper.