Simple electromagnetic wave including delay

[a1titude]

TL;DR

$E(r,t)$ and $H(r,t)$ around $I*\sin(2*pi*f*t)$

I want to derive the electric and magnetic field at a point around an infinitely long wire which carries a sinusoidal alternating current. And I want the answer includes the phase delay according to the limited speed of propagation of the wave. Surely the answer must satisfy the Maxwell's equations. I have never seen such answers from anywhere yet. Please, someone derive them for me or recommend me some references.

 

[marcusl]

Look at the sections 8.2-8.3 on retarded potentials in Stratton, Electromagnetic Theory.

 

[Baluncore]

And I want the answer includes the phase delay according to the limited speed of propagation of the wave.

That is a type of travelling wave antenna, but with an infinite length wire.

The velocity of the sinusoid propagating along the wire is critical in determining the field pattern.
For sub-luminal conductor propagation, (with insulation or oxide), most of the energy is launched in the direction the sinusoid is propagating along the conductor, the conductor forward of there, can often be discounted.

The problem was solved over 100 years ago. You have obviously been reading the wrong books and papers.

This paper, by Beverage, Rice and Kellogg, is well worth reading at breakfast.
https://www.crosscountrywireless.net/The Wave Antenna.pdf