• Support PF! Buy your school textbooks, materials and every day products Here!

Plane wave in cartesian coordinates

  • Thread starter nmsurobert
  • Start date
  • #1
257
28

Homework Statement


Provide an expression in Cartesian coordinates for a plane wave of amplitude 1 [V/m] and wavelength 700 nm propagating in u = cosθx + sinθy direction, where x and y are unit vectors along the x and y axis and θ is the measured angle from the x axis.

Homework Equations



ψ{x,y,z,t) = Aei(kx+ky+kz ± ωt)
k = 2π/λ

The Attempt at a Solution


im not finding many good examples on this but using the plug and chug method i came up with

ψ = Aei(.008(cosθ +sinθ) -ωt)
 

Answers and Replies

  • #2
Simon Bridge
Science Advisor
Homework Helper
17,841
1,642
Notice how there is no space variation in your wave?

The general expression you want is: $$\psi(\vec r) = Ae^{i(\vec k\cdot\vec r \pm \omega t)}$$ ... for Cartesian coordinates, ##\vec r = (x,y,z)## and ##\vec k = (k_x,k_y,k_z)##.
 
  • #3
257
28
i dont see the difference in what i posted and what you posted. you posted the dot product of the propagation vector and the unit vector. isnt that i what i did?
 
  • #4
Simon Bridge
Science Advisor
Homework Helper
17,841
1,642
i dont see the difference in what i posted and what you posted. you posted the dot product of the propagation vector and the unit vector. isnt that i what i did?
Maybe I missed it? You wrote:
##\psi = Ae^{i(.008(\cos\theta +\sin\theta) -\omega t)}##
Where is the x-y-z dependence? If you had done the dot product, wouldn't there be one?

Please write out what you got for the wave-vector ##\vec k##
 
  • #5
257
28
thats where my mistake is. im not sure what my k vector should be. im looking through the text right now trying to figure it out.
 
  • #6
Simon Bridge
Science Advisor
Homework Helper
17,841
1,642
Your wave vector should have magnitude ##2\pi/\lambda## and should point in the direction of propagation.
 
  • #7
257
28
i did that. thats the .008 in my solution. 2pi/700
 
  • #8
Simon Bridge
Science Advisor
Homework Helper
17,841
1,642
0.008 is the magnitude (in nm-1) - what about the direction?
 
  • #9
257
28
well if there is no z component then its headed in the x,y direction. isnt that what the initial u tells me?
 
  • #10
257
28
should there be an x and a y in front of the cos and sin, respectively.
 
  • #11
Simon Bridge
Science Advisor
Homework Helper
17,841
1,642
That's right - the direction is the same as the direction of ##\vec u## ... since ##|\vec u|=1## you can write: ##\vec k = (2\pi / \lambda )\vec u## ...
Since ##\vec u = (\cos\theta, \sin\theta, 0)## you can write: ##\vec k = \frac{2\pi}{\lambda}(\cos\theta, \sin\theta, 0)##

##\vec k\cdot\vec r = \frac{2\pi}{\lambda}(\cos\theta, \sin\theta, 0)\cdot (x,y,z) = \cdots## ... carry out the dot product.
 
  • #12
257
28
ahh ok. what i did was (cosθx, sinθy) ⋅ (x,y)

so my x and y turned to 1's.

thank you!!!
 
  • #13
Simon Bridge
Science Advisor
Homework Helper
17,841
1,642
Ah - then there was a notation mixup:
If we define x = (1,0,0) etc, then r = xx + yy + zz while u = cosθ x + sinθ y and the dot product proceeds correctly.
You may be used to using i-j-k for unit vectors but you can see why you don't want to do that here.

[If you were thinking that x = (x,0,0) then that's a different kind of mixup and r = x + y + z ]
 
Last edited:

Related Threads for: Plane wave in cartesian coordinates

Replies
1
Views
4K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
2
Views
2K
Replies
1
Views
8K
Replies
1
Views
1K
  • Last Post
Replies
3
Views
583
Top