Understanding the Propagation of Spherical Waves in Complex Quantities

[TheArun]

I have two questions which has been troubling me:
1. How can we say that meaning of e^(jkR) is a spherical wave traveling in negative R direction. It can be viewed as polar form of vector with magnitude 1, but how a spherical wave?

2. When we take instantaneous value of a complex quantity , why is real part only considered?

 

[blue_leaf77]

Spherical wave is the one with magnitude 1/R, in addition to the phase factor you have there. It's called a spherical wave because the surface of constant phase corresponding to this wave forms a sphere with radius R.

It can be viewed as polar form of vector with magnitude 1, but how a spherical wave?

Every wave expressed as such is indeed based on the polar representation of complex number, the same way a plane wave is written as $\exp (ikz)$.

2. When we take instantaneous value of a complex quantity , why is real part only considered?

I believe that's only the case when the original quantity being expressed as complex number are physically observable quantity, like electric or magnetic fields and displacement. Them being represented as complex number is usually done to make the computation easier. Other complex quantity like projection coefficient in quantum mechanics should stay complex.

 

[TheArun]

So this is what I understood based on your reply and my knowledge. Please correct me if I'm wrong.
1. Any wave is treated as a vector(may be becoz wave is energy propagated in a particular direction; magnitude is present; obeys vector law of addition)

2. A plane wave propogates in a plane whose axes can be considered as real and imaginary axes of a complex plane. It's position vector here is represented by the complex number z.

3. Since wave is a vector and vector a complex quantity, a wave can be represented in polar form as we do for a vector in complex plane. I.e.,
Wave vector= |magnitude|exp(jkz)

4. A spherical wave is a wave emanating from a point,such that all points in a sphere around that wave has equal value of the wave vector. It is of the form
Wave = (1/R)exp(jkR)
I.e the value of this vector decreases inversely with distance from emanating point and phase changes accd to relation kR(as phase must obviously change with distance as wave travels)
But still can't see how that will mean a negative propagating wave. And does
Wave = exp(jkR) simply mean magnitude is 1 ?

Thanks for reading. Hope you will answer!

 

[TheArun]

I believe that's only the case when the original quantity being expressed as complex number are physically observable quantity, like electric or magnetic fields and displacement. Them being represented as complex number is usually done to make the computation easier. Other complex quantity like projection coefficient in quantum mechanics should stay complex.

Yes. I was indeed considering a wave related to E field. Scalar potential(electric) 'V'.
If you are actually removing imaginary portion, you will be neglecting some information in that wave. Which and why?

 

[blue_leaf77]

1. Any wave is treated as a vector(may be becoz wave is energy propagated in a particular direction; magnitude is present; obeys vector law of addition)

It depends on which "vector" you are referring to. If it's the vector in the sense of complex number (which I believe better called "phasor" for a reason of disambiguity), a sinusoidal wave $\cos \omega t$ can be replaced with $e^{i\omega t}$ by virtue of Euler's formula $e^{i\theta} = \cos \theta + i\sin\theta$.

2. A plane wave propogates in a plane whose axes can be considered as real and imaginary axes of a complex plane. It's position vector here is represented by the complex number z.
3. Since wave is a vector and vector a complex quantity, a wave can be represented in polar form as we do for a vector in complex plane. I.e.,
Wave vector= |magnitude|exp(jkz)

No. A plane wave of the form $e^{ikz}$ propagates along z axis and this z axis is real axis, it's not a complex number. On complex plane, the real and imaginary parts of a phasor $e^{ikz}$ are $\cos kz$ and $i\sin kz$, respectively.

But still can't see how that will mean a negative propagating wave. And does
Wave = exp(jkR) simply mean magnitude is 1 ?

Actually, all references I have read till now always refer $(1/R) \exp(jkR)$ as diverging spherical wave and $(1/R) \exp(-jkR)$ as converging spherical wave. If $k$ is of real value, the magnitude is 1.

If you are actually removing imaginary portion, you will be neglecting some information in that wave. Which and why?

Removing the imaginary part may indeed discard some information, but if the original quantity being represented by a complex number was real to begin with, who cares about the imaginary part.

 

[nasu]

The wave representations are incomplete. These expression do not represent waves. They don't propagate as propagation means something happening in time and there is no time in them.
You need to add the time part to see if the wave is diverging or converging.
For example, consider a term like $e^{i (kr -\omega t)}$
For a surface of constant phase, when t increases r must increase as well, in order to keep the phase unchanged. So the surfaces of equal phase are spheres of increasing radius and the wave is divergent.
For $e^{i (kr + \omega t)}$ the same argument tells you that as the time increases r should decrease, so the wave is convergent.

However you could write the wave as $e^{i (-kr +\omega t)}$ and have a convergent wave again.
So just knowing the "r" part of the wave you cannot tell.