Another question about complex potential

[Omri]

Hi,

For those of you who remember my last thread here, I'm doing a high school graduation project about conformal mapping and the complex potential applied to fluid flow problems. The last time I asked for help you were incredibly helpful so I thought I'd share another issue with you.

I thought of trying to apply the complex potential method to one of the most basic problem I studied in wave theory: interference, for example from two point-like sources. This looks natural if I recall that for a single source at point a we have the complex potential
$$\frac{Q}{2\pi} \mathrm{Ln} (z-a)$$
and superposition holds, so if we have two sources of equal strength at points d,-d (d is real), the overall potential should be
$$\frac{Q}{2\pi} (\mathrm{Ln} (z-d) + \mathrm{Ln} (z+d) )$$
So I tried to work it out algebrically and found that the streamlines should be the curves that give rise so
$$x^2 - y^2 - \frac{2xy}{c} = d^2$$
where c are the constants that represent the different streamlines. Since I don't recognize this algebric form of a curve, I just put it in Maxima and it gave me these graphs:
http://img383.imageshack.us/img383/5893/complex1qq5.png"
(d=0)
http://img374.imageshack.us/img374/7287/complex2wx5.png"
(d=0.1)
http://img376.imageshack.us/img376/4890/complex3xu8.png"
(d=1)

Is that consistent with the "usual" interference analysis done in wave theory? And if so, how is it related to the classical picture, such as:
http://www.paulfriedlander.com/images/timetravel/interference%20-1.jpg"
I realize that the streamlines are the maximum lines of the pictures above, but are they really the same as what I got?

Thanks a lot!

 

[Omri]

Please people... my thread is getting down :-)

 

[Omri]

Just one more jump and I'll stop annoying you guys...

 

[vanesch]

Eh, you shouldn't compare *waves* with the electrostatic picture you calculate. But at first glance, your electrostatic calculation looks right (didn't do the real expansion myself).

What you are calculating is the electric field lines of a static distribution, while the 3rd picture is one of waves. There are of course some similarities in both, but you are dealing with different situations nevertheless.

Doesn't the electrostatic distribution look intuitive to you ? Maybe you should also find the equipotential lines, and superpose them. That will give you a more intuitive picture with both field lines and equipotential lines.

 

[Omri]

Actually I'm working with fluids rather than electostatics. I guess it's the same but my terminology is more fluids-related.

The thing is that I was wondering, if the complex potential let's you model fluid flow situations, will it describe interference? I mean, two sources are supposed to create an interference pattern, and I don't know if this is what the calculation above yields.

Thanks!

 

[vanesch]

Actually I'm working with fluids rather than electostatics. I guess it's the same but my terminology is more fluids-related.

The thing is that I was wondering, if the complex potential let's you model fluid flow situations, will it describe interference? I mean, two sources are supposed to create an interference pattern, and I don't know if this is what the calculation above yields.

Thanks!

Well, interference is the result of the fact that there are phase differences at a certain point, between the contribution of the field at that point from different sources (and that that phase difference varies from point to point). Now, phases are properties of oscillatory phenomena, while we are describing a static situation here. So in a static situation you won't have any interference. In a static situation, a positive source has a positive contribution everywhere, and always, and a negative source has a negative contribution everywhere, and always. So it is not that the SIGN of their contributions will alter from point to point.