Phase front of a quantity e^jwt

[TheArun]

Homework Statement

This is a statement i found in one of my textbook of "Antenna analysis".

The phase front of a exponential time varying quantity(charge distribution) 'e^jwt' is a spherical wave front and is represented by e^-jkR.

Homework Equations

1. How can we represent spherical wavefront as e^-jkR for a quantity exponential time variation e^jwt ?
2. Does the term wt have a physical meaning?

The Attempt at a Solution

Please look into my attempt from the beginning and correct me if I am wrong anywhere.

Consider a complex quantity e^jwt = cos wt + j sin wt.
Re{e^jwt} = cos wt
Im{e^jwt} = sin wt

Now, if we plot this quantity with one axis as real axis i.e., cos wt and other axis as imaginary axis i.e., sin wt. Then we obtain a circle for different values of wt.

Is this the circle we that is the constant phase spherical wavefront of e^jwt. But in this circle actually the phase wt does vary which is how we obtain the circle.
How do we relate this circle to e^-jkR?

 

[blue_leaf77]

The phase front of a exponential time varying quantity(charge distribution) 'ejwt' is a spherical wave front and is represented by e-jkR.

Is that exactly what your textbook says? It sounds a little off to me, because the phase front doesn't really depend on the frequency of oscillation. In fact, the spherical phase front of antenna is only justified for distances very large compared to the emitted wavelength, for distances close from the antenna, the phase front is far from being spherical, yet the field still oscillates sinusoidally in time as $e^{j\omega t}$.

1. How can we represent spherical wavefront as e-jkR for a quantity exponential time variation ejwt ?

You may write the emitted field at large distance as $E(\mathbf{R},t) \propto e^{j(\omega t - kR)}$, where $R = |\mathbf{R}|$.

2. Does the term wt have a physical meaning?

$\omega t$ is the phase at the observation point a distance $R$ from the antenna, it's then also related to the phase at the antenna itself. Imagine you take a snapshot of $E(\mathbf{R},t) \propto e^{j(\omega t - kR)}$ (i.e. $t$ is fixed), you will then observe as group of concentric spheres with spacing equal to $\lambda = 2\pi/k$. This is where spherical wave got its name.