I understand that signal delays are not theoretical, having built phasing harneses for RF as well as needing to keep stereo channels in sync so they still sound correct when listening in mono. My comment was in reference to the OP's wondering if an RF signal in free space vs that same RF signal also traveling through a wall would arrive at the receiver at the same time. In that specific situation, while there would be a difference, it would be nearly impossible to measure.
Not impossible to measure at all, in fact probably very measurable, although specifying a frequency range and wall material would help here. If the brick wall is something like 8” thick and the frequency is in maybe the UHF range I would be willing to bet no special equipment needed. Just two phase matched sample antennas / feedlines (and if they are not phase matched then just understanding the phase relationship at the frequency of interest works also) and a dual channel scope with a high enough sample rate if you wanted to directly sample. You could also feed both samples into separate mixers but with a single LO so everything stays coherent, downconvert to a much more usable frequency, and use an older analogue scope.
The velocity of propagation is related to the dielectric constant of the medium. Cured Portland cement has a dielectric constant of roughly 2.5 (
http://www.asiinstr.com/technical/Dielectric Constants.htm ). Using this we can find that the VF for a cement wall (not counting changes for coverings or rebar) is going to be about 0.63. Assume that the wall is 8 inches thick and the signal in question is on 446.000 MHz, the 70 cm FM simplex calling frequency.
Representing the forward motion of the wave front in degrees of rotation for simplicity from now on. Were the wall not there the RF would travel about 109 degrees of its cycle in 8 inches through the air VF of 1.0 (446.0 MHz yields 360 degrees in 26.482 inches at VF 1.0). The RF will travel 173 degrees of its cycle in the 8” concrete wall assuming it is penetrating the wall perpendicular to the face of the wall. This is a difference of 64 degrees between the open air and through wall. 64 degrees of phase shift should be pretty easy to see on a dual channel oscope. All you need is the two phase matched sampling antennas / feedlines and an oscope with a sample rate high enough to see one full cycle or a fraction of the cycle of the RF, pretty easy to do these days with 500 MHz scopes being almost entry level.
Naturally doing this exercise at HF frequencies yields much smaller phase shifts, say on the order of a degree or two. But the test equipment sees smaller changes easier at lower frequencies.
A hollow cinderblock wall with its air voids would also reduce the phase delta, as would a wooden structure with drywall.
As has been said many times in this thread, you will not hear such a delay by ear, but you most certainly can measure it. In fact, you might get some interesting phase cancellation affects using an FM receiver that could be heard by ear. Not as delays, naturally, but possibly as signal degradation.
Feeding a signal directly to a 'scope on one channel and through a few thousand feet of cable on another channel it's easy to see the delay on the 'scope display.
It will be easy to see only as long as you don’t fall near a 360 degree length on that long run. Why do I say this? Because I have seen a person grab a random long chunk of coax, many wavelengths, and throw it in line, thinking they were going to be shifting phase, only to have the length be near an electrical wavelength, and the actual phase shift change, in degrees of a single cycle and on the scope, being below their ability to discern.
You don’t need thousands of feet, all you need is a portion of a wavelength at the desired frequency. For example, if talking about a 14 MHz 20 meter signal and a coax with .68 VF, the phase shift observed would be the same for a 23 foot, 10.85 inch coax as it would be for a 4804 foot 6.69 inch coax. Or better yet, no coax at all, directly connected to the sample source, on one channel and the other connected via 4780 feet 7.83 inches of .68 VF coax. Phase shift the same, no apparent delay, but in reality 100 wavelengths of delay and you are looking at the zero crossing 7.143 microseconds later but coincident on a dual channel scope, as long as we are talking about a carrier with no frequency or phase modulation present.
T!