I liked the explanation of the physics of why there should be a spike. But I don't see the relationship to the original equation. It seems to me that it's just a coincidence that the original equation had a spike.
The explanation is no explanation at all. It is true that water has inertia, and it is true that water has to go somewhere. It is not true that this is sufficient to explain why the water goes up.
In particular, the statement "because of inertia and the fact that water is not compresible, it has to keep moving" is false.
The fact that water is not compressible means that the water moving in one direction is capable of exerting an extremely large force on the water moving in the other direction when they run into each other, which would bring the water to a halt as the hole closes. The interesting question is: why doesn't this happen?
One possibility is that the free surface at the bottom of the hole travels upward due to what amounts to a buoyancy force (think of the hole as a bubble with no top). The water pressure at the bottom of the hole is the same as it is everywhere at that depth, and so the bottom move upwards, and this pressure force creates upward momentum that is not met by any opposing force when the bottom of the hole reaches the water's surface, and therefore creates an upward-moving column of water.
One way of testing this would be by inserting an empty pipe into the water against a flat disk of the same diameter, and then rapidly pulling the pipe out. The flat disk would impede water from moving upward into the hole, which would--on my account--dramatically reduce the size of the spike.
As to the "unreasonable effectiveness of mathematics", the argument is bogus: http://www.tjradcliffe.com/?p=381 Our mathematical descriptions of reality almost all admit of extraneous solutions that have to be discarded based on physical--not mathematical--considerations. This is what you would expect if mathematics was a human-made tool for describing reality. It is not at all what you would expect if mathematics was somehow the pre-existing armature upon which physical reality was constructed.
I think that you're responding to an extreme interpretation of the "unreasonable effectiveness of mathematics". I don't think you have to believe that nature follows perfectly simple mathematical rules to be surprised to the extent that mathematics is effective at describing physical reality.
You make a good point, that mathematics is often designed and invented to describe reality. You need to take into account the fact that a lot of what you "get out" of mathematics has already been "put in". But it is my opinion that even accounting for this, it is quite frequent that you "get out" more than you "put in" to mathematics.
I think a strong set of examples of this sort of behavior is when attempts to address one problem in a mathematical theory often end up addressing other seemingly unrelated problem. The introduction of quantized energy states to explain the blackbody radiation distribution then goes on to explain the photoelectric effect. (I'm not too familiar with what physical phenomena were observed before QM but needed QM to explain them to give more examples).
Now you might say "well of course this theory solves both issues, because they are essentially the same phenomenon, and so there's nothing surprising about your theory explaining what it was designed to explain". But that's precisely the point, that mathematics gives us the tools to see relations between things that on the face of them are different. And that there's no reason that this /has/ to be true (without appealing to mathematics), makes the fact that it is true seem surprising.
Of course, being surprising, or unreasonably effective, etc is a relative statement. Certainly to someone who expected the universe to be described mathematically to precisely the degree that it is in reality, nothing would be surprising. But I think to most people who haven't been introduced to mathematics (kids), the power it gives you certainly is surprising.
The fact that water is not compressible means that the water moving in one direction is capable of exerting an extremely large force on the water moving in the other direction when they run into each other, which would bring the water to a halt as the hole closes. The interesting question is: why doesn't this happen?
Why would you expect that to happen? The hole in the water is an area of low pressure that the water is going to rush in to fill. Now that water has a lot of kinetic energy and momentum. Why would it suddenly lose that momentum? And how does water's incompressible nature help that? Surely, if anything, this contributes to the difference in resistance between the water and the air and help explain the spike that you initially see.
It is probably not even possible to model how a drop hits the water with only a 2D cross-section. You probably need three dimensions to model it correctly.
In its present form, it is more like modeling how an infinitely long cylinder is hitting the surface of the water.
As near as I can tell the graphs seem to be mirrored around the origin - that is, they're graphs of abs(x), not x. Which naturally causes a discontinuity for any function whose slope isn't zero at the origin...
It's discontinuous if the value is undefined at the origin. That happens here because the denominator of (x + t) yields a division by zero. A nonzero slope would give a kink in the curve, but not a discontinuity.
One approach to avoiding the whole question of abs(x) would be to use cosine instead of sine, since cosine is symmetric across x = 0.
> One approach to avoiding the whole question of abs(x) would be to use cosine instead of sine, since cosine is symmetric across x = 0.
I don't think that this would work, since the division by an odd function would just make it odd again (that is, `sin(x)/x` is even precisely because `sin` is odd). Note, however, that blaming `abs` for discontinuity (or non-differentiability) is a bit of a red herring; any even function on the real line, no matter how smooth, can be written as a composition with `abs`.
Hurmph. I don't see how it makes much difference whether he uses sine or cosine, since he's also adding t to the operand and then animating over it, causing the waveform to move along the x-axis. He can use one and start from t=0 or use the other and start from t=pi/2, as you like.
My point was simply that the "spike" the article goes on about (which I earlier called a discontinuity, oops) shows up because the author is reflecting his plot around the Y axis. The article makes it sound like the spike was some physically significant feature of the function being graphed.
I agree that it doesn't make any difference when you allow phase shifts (I was intentionally focussing on the `t = 0` case), so I'm puzzled by your suggestion that the author should have used cosine instead.
My point was more that there's nothing inherently 'spiky' about even functions; as your own example of cosine shows, they can be just as smooth (or spiky) as any other function.
Yes, you're right—I thought that you had made the post that was actually made by T-hawk (https://news.ycombinator.com/item?id=9022743 )—but I think that it doesn't change my point.
> As near as I can tell the graphs seem to be mirrored around the origin - that is, they're graphs of abs(x), not x. Which naturally causes a discontinuity for any function whose slope isn't zero at the origin...
I think that you mean they are graphs of x ↦ f(abs(x)), not just f; and that this causes a point of non-differentiability, not necessarily a discontinuity. Of course, the even extension x ↦ f(abs(x)) of any continuous function `f` on the non-negative real numbers is again continuous (because `abs` is).
