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The velocity vector field given by

$$ \vec{U}(x, y) = \langle u_x, u_y \rangle = \langle a cos(\lambda x), a sin(\lambda x) \rangle,$$

where a is the fixed amplitude, and lambda is the fixed wavelength, does not describe a circle with radius $a$; instead, plotting this vector field in matlab shows a nice, wavy profile.

Where am I mistaken? To me, it looks like the parameterization of a circle.

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  • $\begingroup$ Frequency is measured in # per second. Monthly is 12/year. $\endgroup$ Commented Nov 4, 2017 at 0:08
  • $\begingroup$ @Hurkyl The vector from (3,2) to (0,0)? $\endgroup$ Commented Nov 4, 2017 at 0:16
  • $\begingroup$ How is u a vector field? You have defined it only as one vector, some point in R^2. $\endgroup$ Commented Nov 4, 2017 at 0:21
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    $\begingroup$ I don't have time to write a full answer, so a brief comment: if you gather up all of the vectors drawn and relocate them to the origin, their tips will trace out a circle. The wavy profile is that you are dragging the base of the arrow horizontally as the vector circles around its base. (at least, if what's going on is what I think is going on) $\endgroup$
    – user14972
    Commented Nov 4, 2017 at 0:37

1 Answer 1

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If we look at your equation, assuming that your parameter x is in fact the polar angle $\theta$, then our vector field looks like. $$\vec u=a (cos(n \theta), sin(n \theta))$$ The we have a field of vectors, whose strength doesn't depend on position, but the directions rotates as a multiple of the angle.

It's a little bit like the rotation and revolution of a planet. As we look at various angles around the origin, the direction of the vector field changes.


These diagrams look like

$n=1$

enter image description here


$n=2$

enter image description here


$n=3$

enter image description here


$n=4$

enter image description here


If you really do mean x and not the polar angle $\theta$, then our equation is $$\vec u=a (cos(n x), sin(n x))$$

and you get diagrams that look like this.

enter image description here

And this

enter image description here

As we move from left to right along the x direction, the direction of the vector field rotates. Up and down along the y direction the vector field is constant.

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    $\begingroup$ You're assuming that in the OP, $(a, x)$ refers to the polar coordinates of a point, and you wrote those coordinates as $(a, \theta)$ instead? It's probably worth explicitly mentioning your assumption at the top, since it's not standard. $\endgroup$
    – user14972
    Commented Nov 4, 2017 at 0:19
  • $\begingroup$ I had a small glitch in the code and the diagrams weren't exactly what I wanted. The new diagrams are up, with a constant field strength. $\endgroup$
    – David Elm
    Commented Nov 4, 2017 at 0:26

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