# Where am I going wrong in visualizing this simple vector field?

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.

• Frequency is measured in # per second. Monthly is 12/year. Commented Nov 4, 2017 at 0:08
• @Hurkyl The vector from (3,2) to (0,0)? Commented Nov 4, 2017 at 0:16
• How is u a vector field? You have defined it only as one vector, some point in R^2. Commented Nov 4, 2017 at 0:21
• 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)
– user14972
Commented Nov 4, 2017 at 0:37

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

$n=1$

$n=2$

$n=3$

$n=4$

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.

And this

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.

• 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.
– user14972
Commented Nov 4, 2017 at 0:19
• 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. Commented Nov 4, 2017 at 0:26