I have been thinking of how to measure the speed of the longitudinal waves. Pardon me if my comments below have already been foreshadowed elsewhere.
One simple way is to have a "clock," analogous to the clock signal in a computer, typically generated by an oscillating crystal; or another analogy is the timing "beat" of an orchestra.
I picture three points far apart on the surface of the Earth: One is the transmitter, one is the receiver, and the third, equidistant from the other two points, is the clock signal generator. An isosceles triangle with the transmitter and receiver at the base vertices.
And since there are already national, radio master clocks (that are commonly used by, for example, wall clocks) then one could save resources by using one of those master clocks as the clock signal and arranging the transmitter and receiver to be at the other vertices of the triangle.
The clock tells both sides, at the exact same time, when a signal is to be sent by the transmitter.
A big virtue of this arrangement, as compared to the transmitter and receiver being in the same place, with the signal being sent on a long round-trip such as reflection off the Moon, is that it doesn't matter if there are rogue reflections from nearby objects: all you need to detect is the first evidence (i.e., the leading edge) of the transmitted signal. That is, rogue reflections cannot be used as an objection to the results, because any reflections of part of the signal on the way from transmitter to receiver will necessarily take a longer path than the straight line between transmitter and receiver.
You also don't need the large amount of power to fire a signal through interplanetary space and no exotic antennae.
Lastly, we already know that such terrestrial transmission of longitudinal waves is possible, because it has been done by both Tesla and Dollard, using modest equipment. But, as far as I know, neither of those gentlemen ever set out to measure the speed of transmission.