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Encouraging students to understand the 1D wave equation

• dustingetz 3y ago
  see also A Students Guide to Waves, loved it, read it on the couch (with a lap desk near at hand) https://www.danfleisch.com/sgw/ https://www.danfleisch.com/sgw/ https://www.amazon.com/Students-Guide-Waves-Guides/dp/110764 https://www.amazon.com/Students-Guide-Waves-Guides/dp/110764...
  • 3abiton 3y ago
    What's your elevator pitch for the book?
    • havercosine 3y ago
      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.
  • snake_plissken 3y ago
    And a non-Amazon link =^_^= https://www.cambridge.org/ve/universitypress/subjects/physics/general-and-classical-physics/students-guide-waves?format=PB https://www.cambridge.org/ve/universitypress/subjects/physic...
  • djmips 3y ago
    Amazon link doesn't work (for me) But this does. Weird. https://www.amazon.com/Students-Guide-Waves-Guides/dp/1107643260/ https://www.amazon.com/Students-Guide-Waves-Guides/dp/110764... Oh yours seems to be missing 3260/ at the end
• plaidfuji 3y ago
  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.
  • kwhitefoot 3y ago
    There is no excuse for that sort of thing these days when computational tools such as Maxima exist to check the dimensional consistency of equations.
    • cinntaile 3y ago
      I can't find the homepage for Maxima?
      • jxf 3y ago
        It's here: https://maxima.sourceforge.io/project.html https://maxima.sourceforge.io/project.html
    • forgotpwd16 3y ago
      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.
      • kwhitefoot 3y ago
        A seemingly obvious change like moving a term from one side of the equation to the other?
  • Modified3019 3y ago
    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. https://www.ind.ku.dk/english/staff-auto-list/?pure=en%2Fpersons%2Fricardo-avelar-sotomaior-karam(df4e0a2f-5798-413b-a535-c0c10067b769)%2Fpublications.html https://www.ind.ku.dk/english/staff-auto-list/?pure=en%2Fper... https://research.ku.dk/search/result/?pure=en%2Fpersons%2Fmuhammad-aswin-rangkuti(37594e39-cb99-4910-9b19-538e67f26723)%2Fpublications.html https://research.ku.dk/search/result/?pure=en%2Fpersons%2Fmu...
• dupertrooper 3y ago
  [dead]
• dzolob 3y ago
  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.
  • the_other 3y ago
    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?)
    • tonyarkles 3y ago
      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 :).
    • dzolob 3y ago
• supernova87a 3y ago
  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).
• dr_kiszonka 3y ago
  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."
  • deleted 3y ago
    [deleted]
• aj7 3y ago
  Concavity? Then you pull the wave equation out of a hat? I pity the students tortured by this approach.
• westurner 3y ago
  Wave equation: https://simple.wikipedia.org/wiki/Wave_equation https://simple.wikipedia.org/wiki/Wave_equation https://en.wikipedia.org/wiki/Wave_equation https://en.wikipedia.org/wiki/Wave_equation - "second order PDE partial differential equation in physics" - Range: [-1,1] Q12 Wave function: https://simple.wikipedia.org/wiki/Wave_function https://simple.wikipedia.org/wiki/Wave_function https://en.wikipedia.org/wiki/Wave_function https://en.wikipedia.org/wiki/Wave_function - quantum probability CDF Cumulative Distribution Function - Range: [0,1] + [0,1]i Bloch sphere / 'unit sphere': https://en.wikipedia.org/wiki/Bloch_sphere https://en.wikipedia.org/wiki/Bloch_sphere : - Range: [0,1]x + [0,1]y + [0,1]i
  • Sharlin 3y ago
    > Wave function: https://simple.wikipedia.org/wiki/Wave_function https://simple.wikipedia.org/wiki/Wave_function 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".
    • westurner 3y ago
      https://schema.org/speakable 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
• tomthe 3y ago
  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. Are there textbooks for this or related stuff?
  • semi-extrinsic 3y ago
    The book of Leveque is a classic [1]. 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. [1] https://www.cambridge.org/core/books/finite-volume-methods-for-hyperbolic-problems/97D5D1ACB1926DA1D4D52EAD6909E2B9 https://www.cambridge.org/core/books/finite-volume-methods-f...
    • tomthe 3y ago
      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.
      • dsqrt 3y ago
        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. -- [1] https://en.wikipedia.org/wiki/Finite-difference_time-domain_method https://en.wikipedia.org/wiki/Finite-difference_time-domain_... [2] https://math.mit.edu/classes/18.086/2006/am53.pdf https://math.mit.edu/classes/18.086/2006/am53.pdf [3] https://epubs.siam.org/doi/book/10.1137/1.9780898717839 https://epubs.siam.org/doi/book/10.1137/1.9780898717839