Interesting, but not very physical. The frequencies present in
a waveform are unchanged if you replace the waveform with its
derivative. As such, it makes no difference whether you have a whole
number of cycles before changing frequencies. The derivative of a
sine wave that changes frequency when it crosses zero will change
frequency at a different amplitude.
A sudden change of frequency will splatter all over the spectrum, even
if the change happens at an instantaneous amplitude of zero. For a
two-frequency modem on a standard analog telephone line, that doesn't
matter, since the telephone circuit blocks extreme frequencies, and
since the receiving modem is listening only for those two frequencies.
Frequency shift keying on radio is generally done by simply adding a
radio frequency to the two tones. So instead of 1200 and 2200 Hz,
it might be at 3501200 and 3502200 Hz. Obviously that will change
whether the (unchanged) duration of one bit is a whole number of
waves for one or both frequencies. Fortunately, that doesn't matter.
One way to avoid splattering all over the radio spectrum, which would
of course be a bad thing, is to change the frequency gradually rather
than abruptly. Slide smoothly from 1200 to 2200 or vice versa.
Another way is to gradually decrease the amplitude of the 1200
tone while gradually increasing the amplitude of the 2200 tone or
vice versa. Yet another way is to just run the signal through a
narrow-pass analog filter. None of these depend on whether there
are a whole number of waves, or on when the waveform crosses zero,
or on what the radio frequency is.
Similarly with the even older and simpler method of on-off keying
(e.g. Morse code). The amplitude should gradually increase and
decrease, since suddenly starting or stopping a radio signal will
produce "key click" noise all over the spectrum.
Of course by "gradually" I mean typically on a millisecond time scale.
Though the time scale can be anything. The shorter it is, the more
bandwidth the signal uses, and the more bits can be sent per second.
The link between channel capacity (bits per second) and bandwidth
(difference between the highest and lowest frequency used) is so
strong that most people say "bandwidth" when they mean "channel
capacity."
The link between channel capacity and bandwidth is due to the
Heisenberg uncertainty principle. The more precisely you know the
frequency of a signal, the less precisely you can know its time of
arrival, and vice versa.
What frequencies are present in a signal depends as strongly on the
receiver as on the transmitter. Alternate once per second between
1000000-1 Hz and 1000000+1 Hz. Is there any energy at exactly 1000000
Hz? If you have a narrow band receiver, yes, nearly all the signal
power is there. If you have a wide band receiver, no, none is.
You can see this effect for yourself at sdr.hu, a website that lets
you see and hear, in real time, radio signals in the HF band and below
as picked up at various places all over the world. You can select the
receiver location, frequency, bandwidth, and mode. It's one of the
truly wonderful websites, right up there with Wikipedia, Google,
IMDB, and OEIS.
(Similarly with the signal's direction versus where on your antenna
it landed. If you know very precisely where it landed, you know its
position on the axis at right angles to its direction of propagation
to a high precision, which means you know its momentum on that axis,
i.e. the direction it came from, to a very low precision. This is
why directional antennas have to be large.)
A few nitpicks:
The first digital modem in widespread use was the Bell 103 (and
compatibles), dating to 1962, and still in widespread use through the
1980s. It worked only up to 300 bits per second. It wasn't as fast
as the later Bell 202, but unlike the 202 it was full duplex, meaning
that signals could be sent in both directions at once. In one
direction, it switched between 1070 and 1270 Hz, in the other
direction, it switched between 2025 and 2225 Hz. It was always more
common than the 202, though perhaps not as common as the later Bell
212, which was a full duplex 1200 bits per second modem.
I don't think ham radio operators ever used Bell 202, at least in the
US. For one thing, ASCII wasn't allowed on the air until 1980, by
which time Bell 202 was long obsolete. Hams have been using the older
five bit ITA2 code (often wrongly called Baudot) since 1946. Hams
typically use 45 bits per second and a 170 Hz frequency difference
between mark (1 bit) and space (0 bit) when sending ITA2 ("RTTY").
I think deaf people also still use ITA2 over the telephone.
