Need 2D plot of plane wave, cylindrical & spherical wave

In summary, plane wave is represented as exp (ik.r) and the cylindrical wave as 1/sqrt(r) *exp(ik.r) and the spherical wave as 1/r*exp(ik.r)
  • #1
allamsetty
6
0
plane wave is represented as exp (ik.r) and the
cylindrical wave as 1/sqrt(r) *exp(ik.r) and the
spherical wave as 1/r*exp(ik.r)

Have anyone tried to plot these waves? How to do it?

Attempt: in Matlab assuming k=1

>> x=linspace(-1,1,100);
>> for(ii=1:100)
fp(ii)=exp(i*x(ii));
end
>> [x,fp]=meshgrid(x,fp);
>> mesh(x,real(fp))

gave me plane surface that is inclined in the graph however I still need to create Cylindrical wave and spherical wave. I am confused on how to use " r " for cylindrical and spherical wavefronts.
 
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  • #2
Hello Allam, welcome to PF :)

I understand the planar wave expression, ##{\bf k}## and ##{\bf r}## are vectors and the inner product ##{\bf k}\cdot{\bf r}## makes it a planar wave.
What you then render seems one-dimensional to me (not familiar with MATLAB :( ), k has disappeared (probably ok: k = 1,0,0 or so).

##i## in Matlab ? is ##i^2 = -1## ?

Is 1/sqrt(r) *exp(ik.r) really the expression for a cylindrical wave (yes, Bessel function for large ##|{\bf r}|## -- you confused me because I read it as ##{\bf k}\cdot{\bf r}## again, but you mean ## kr##)
You could render this similarly one-dimensionally along a ray. Same for the spherical guy.


I suppose you did google planar wave images and cylindrical wave images already ? All struggle with the problem that you need too many dimensions if you want a full picture.
 
  • #3
I assumed k=1 and ignored the time dependence.
By the By k is not vector it is scalar and it is exp(ik(n.r)) where n is unit normal vector to r

Iam interested in the variation of amplitude in the space. I have developed Matlab code for plotting and in the plot from the color code one can see that amplitude
planewave.png
sphericalwave.png
cylindricalwave.png
varies differently for different waves.
 
  • #4
Strange. Amplitude for planar wave should be a constant.
Looking at the MATLAB code, I suppose the plot shows the real part only.

And if ## {\bf n} \perp {\bf r} ##, then ##{\bf n} \cdot {\bf r} = 0##.

I wonder what I see in the pictures. Top one shows about 4.5 periods of a plane wave in the horizontal direction. Vertically ?
Lower two pics ?
 
  • #5
True , Thank You for correcting me in that n is parallel to r. n is unit vector in the direction of propagation and r is the position vector. I have written another MATLAB code which goes like this for plane wave generation:
x=linspace(0,99);
y=linspace(0,99);
r=sqrt((x.*x)+(y.*y));
for j=1:100
pw(j)=exp(i*(x(j)+y(j)));
end
[r,pw]=meshgrid(r,pw);
mesh(real(pw))
All the 3 pictures show propagation horizontally. In the first one as you rightly said there are 4.5 periods and beam does not diverge.it carries forward its intensity which is constant. In the second and the third the spherical and the cylindrical diverge and intensity diminishes rapidly in case of spherical than cylindrical. the plot actually gives the intensity information and not amplitude.since intensity is square(amplitude). it goes in line. Yeah! In MATLAB "i" means imaginary number of value sqrt(-1).
 
  • #6
please be informed its not the intensity! i guess it is just the real part of complex amplitude !
 
  • #7
pw(j)=exp(i*(x(j)+y(j)));
This doesn't look good, unless you actually mean a planar wave with a k vector (1,1,0). And you are plotting the real part of the wave function along the line x=y anyway, so it shouldn't give a different picture ?
 
  • #8
yes ! iam plotting along x=y only. i can choose any values of x and y however i need to ensure that they contain same number of elements for MATLAB computation.
remember it is actually dot product of (x i +y j) and (i +j) that is giving me x +y that is why it is looking like pw(j)=exp(i*(x(j)+y(j)));
I need to understand some more fundamentals (maths related) before taking up this plotting work. kindly forward me some notes if you have on this topic. your active participation is appreciated. Thank You.
 

1. What is a 2D plot of a plane wave?

A 2D plot of a plane wave is a graphical representation of a wave traveling in a single direction, with constant amplitude and wavelength. It is often depicted as a sine or cosine function with a horizontal axis representing distance and a vertical axis representing amplitude.

2. How is a cylindrical wave different from a plane wave?

A cylindrical wave is a type of wave that propagates outward from a point source in all directions, creating a cylindrical shape. In contrast, a plane wave travels in a single direction, creating a flat wavefront. Cylindrical waves are commonly seen in sound waves, while plane waves are often used to describe electromagnetic waves.

3. What do the different components of a 2D plot of a spherical wave represent?

In a 2D plot of a spherical wave, the horizontal axis represents distance from the source, while the vertical axis represents the amplitude of the wave. The shape of the wave, which is often depicted as a circle, represents the wavefront.

4. How can I use a 2D plot of a plane wave, cylindrical wave, or spherical wave in my research?

2D plots of different types of waves are commonly used in various fields of science, including physics, engineering, and mathematics. They can be used to analyze and understand the behavior of different types of waves, and can also be used to model and predict wave propagation in different mediums.

5. Are there any limitations to using 2D plots to represent waves?

While 2D plots are a useful tool for visualizing waves, they do have limitations. For example, they only show a snapshot of a wave at a specific point in time, and do not take into account factors such as interference or diffraction. Additionally, 2D plots may not accurately represent more complex wave patterns, such as those found in turbulent flow or non-linear systems.

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