4 ms·

It threw me for a bit that x here refers to the distance from the origin, whether or not that distance is leftwards or rightwards. Anyway, the function's being

by Chinjut 12y ago

It threw me for a bit that x here refers to the distance from the origin, whether or not that distance is leftwards or rightwards. Anyway, the function's being described isn't really the way water waves actually work, is it? Surely, dropping a stone in water does not start out disturbing every portion of the water immediately like "sin(x)/x", but rather, the disturbance only emanates out over time from the origin, with at any moment only so much water disturbed. (Then again, I'm not a physicist, so what do I know?)

jjoonathan 12y ago
You're right, but the inaccuracy is a consequence of the differential equation model's assumptions rather than a math error. You'll see the same unphysical instantaneous propagation in simple models of heat flow and diffusion. Accounting for laggy propagation, although more realistic, is difficult and usually unnecessary in the classical regime with a few major exceptions, most notably that magnetism is actually just the "lag" in electric forces, hence "electromagnetism."
- JadeNB 12y ago
  > most notably that magnetism is actually just the "lag" in electric forces, hence "electromagnetism." I've never heard that before! Is there any place that I can read about it?
lisper 12y ago
> Anyway, the function's being described isn't really the way water waves actually work, is it? That's exactly right. In fact, that is one of the interesting features of math: you can often get the right answer even when the math is completely unjustified! This actually happens more than you might think. Quantum mechanics was essentially discovered this way. Even the classical mechanics literature is chock-full of wrong math that happens to give the right answer. See: http://mitpress.mit.edu/sites/default/files/titles/content/sicm/book-Z-H-5.html#%_chap_Temp_2 http://mitpress.mit.edu/sites/default/files/titles/content/s...
- JadeNB 12y ago
  > In fact, that is one of the interesting features of math: you can often get the right answer even when the math is completely unjustified! As a mild nitpick, I would say that this is a feature of mathematical physics. A mathematician on his or her own would not be satisfied with getting the 'right' answer by wrong methods—in fact, would have no way to judge what is 'right' except by appealing to right methods (there being no outside arbiter who can decide the matter independently, as is the universe itself in mathematical physics).
fenomas 12y ago
> Anyway, the function's being described isn't really the way water waves actually work, is it? No relation at all. They just look similar insofar as all smooth wave functions look sort of similar at small enough amplitudes. But yeah, the sinusoidal equation doesn't show the initial perturbation, since it's not modeling a perturbation at all.
- ifdefdebug 12y ago
  That animation doesn't resemble a water wave at all. In a water wave, every single point of the surface does a circular movement (elliptic in shallow water). This resembles a transverse wave at best. It doesn't even look like the surface of a water wave. Take a look at this (for instance): http://www.acs.psu.edu/drussell/demos/waves/wavemotion.html http://www.acs.psu.edu/drussell/demos/waves/wavemotion.html
  - fenomas 12y ago
    Sure - I said they look similar "insofar as all smooth wave functions look sort of similar", which is to say, only to the extent that they're both waves. But none of the interesting physical characteristics of a water wave are present.

by Chinjut 12y ago

jjoonathan 12y ago

You're right, but the inaccuracy is a consequence of the differential equation model's assumptions rather than a math error. You'll see the same unphysical instantaneous propagation in simple models of heat flow and diffusion. Accounting for laggy propagation, although more realistic, is difficult and usually unnecessary in the classical regime with a few major exceptions, most notably that magnetism is actually just the "lag" in electric forces, hence "electromagnetism."

JadeNB 12y ago
> most notably that magnetism is actually just the "lag" in electric forces, hence "electromagnetism." I've never heard that before! Is there any place that I can read about it?

lisper 12y ago

> Anyway, the function's being described isn't really the way water waves actually work, is it? That's exactly right. In fact, that is one of the interesting features of math: you can often get the right answer even when the math is completely unjustified! This actually happens more than you might think. Quantum mechanics was essentially discovered this way. Even the classical mechanics literature is chock-full of wrong math that happens to give the right answer. See: http://mitpress.mit.edu/sites/default/files/titles/content/sicm/book-Z-H-5.html#%_chap_Temp_2 http://mitpress.mit.edu/sites/default/files/titles/content/s...

JadeNB 12y ago
> In fact, that is one of the interesting features of math: you can often get the right answer even when the math is completely unjustified! As a mild nitpick, I would say that this is a feature of mathematical physics. A mathematician on his or her own would not be satisfied with getting the 'right' answer by wrong methods—in fact, would have no way to judge what is 'right' except by appealing to right methods (there being no outside arbiter who can decide the matter independently, as is the universe itself in mathematical physics).

fenomas 12y ago

> Anyway, the function's being described isn't really the way water waves actually work, is it? No relation at all. They just look similar insofar as all smooth wave functions look sort of similar at small enough amplitudes. But yeah, the sinusoidal equation doesn't show the initial perturbation, since it's not modeling a perturbation at all.

ifdefdebug 12y ago
That animation doesn't resemble a water wave at all. In a water wave, every single point of the surface does a circular movement (elliptic in shallow water). This resembles a transverse wave at best. It doesn't even look like the surface of a water wave. Take a look at this (for instance): http://www.acs.psu.edu/drussell/demos/waves/wavemotion.html http://www.acs.psu.edu/drussell/demos/waves/wavemotion.html
- fenomas 12y ago
  Sure - I said they look similar "insofar as all smooth wave functions look sort of similar", which is to say, only to the extent that they're both waves. But none of the interesting physical characteristics of a water wave are present.