Marconi's experiments from 1895 to 1899 to devise instruments for the transmitting and recording of messages without the use of wires, resulted in a number of successes which astonished the scientific world, but when he so perfected the apparatus that he was enabled on December 12, 1901, to transmit across the Atlantic Ocean from Newfoundland to England, this crowning success was accepted by one and all as one of the greatest achievements of modern times.
Though wireless telegraphy was looked upon by many as depending upon a mysterious phenomena, far too deep for the amateur to understand, it was not long after articles treating at length upon the subject began to appear in the newspapers and technical journals that boys started to study into it.
It sounds a lot like CB radio or ham radio. Of course I wouldn't expect kids to make this today to really communicate across a city, given a telephone or cell phone would be convenient. But it would be fun to make just to understand how it works and see if they can transmit.
Here's a nice example of the type of error that you need to watch for when using older texts for reference:
The Fundamental Principles of Wireless Telegraphy.
Throughout all space a substance is supposed to exist,
though nothing definite is known about it, and this substance,
which is colorless, odorless, and without weight,
and called ether, is supposed to have remarkable qualities
for transmitting vibrations through space.
Of course we know now that the energy is not transmitted through an ether of any sort.