The Physics
Opus in profectus


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Oscilloscope trace
Oscilloscope trace
A beat pattern at two different magnifications

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  x = A sin 2π(f1t) + A sin 2π(f2t)  
x = 2A sin 2π

f1+f2  t

cos 2π

f1f2  t

2 2

Beats are interference in time.Fringes are interference in space.


Why do phones ring?

Spectral analysis of a dial tone (US). Note the peaks at 350 and 440 Hz. The two tones together beat at 90 Hz, which sounds something like a ringing phone.

Dual-tone multi-frequency signaling, call progress tones, telephone event tones,

Telephone Event Tones (US except where indicated)
event tone
(press to play)
frequency (Hz) duration (s)
low high on off
480 620 0.5 0.5
480 620 0.25 0.25
350 440 continuous
0.1 0.1
440 480 2.0 4.0
384 416 1 2
400 450 0.4

Touch-Tone was a registered service mark of the American Telephone and Telegraph Company of New York (now called AT&T) from 1960 to 1984. Since the service mark expired, it is no longer necessary to capitalize the word touch-tone. AT&T's rights to the words may have expired, but the technology behind the words lives on.

Touch-Tone service mark

The touch-tone system is an example of a signaling technique known as dual tone multi frequency (DTMF). In the touch-tone system, pairs of tones are used to represent the digits 0 through 9 and the symbols * (star or asterisk) and # (pound, number, hash, or octothorp). The tones assigned correspond to the location of the button on a telephone keypad. The row determines the low tone and the column the high tone. The numeral 5, for example, lies on the keypad row that plays 770 Hz and the column that plays 1336 Hz. The full set of eight frequencies were chosen to avoid harmonics (no frequency is a multiple of any other) and beats (no frequency is the difference between any two frequencies). While it is possible to use a touch-tone keypad as a musical instrument, the touch-tone frequencies do not correspond to the notes of any known musical scale. Pairs of tones played together produce unique sounds that are unlikely to be generated by accident.

DTMF Touch Tone keypad (Press to play)
1209 Hz
1336 Hz
1477 Hz
1633 Hz
697 Hz
770 Hz
852 Hz
941 Hz

The A, B, C, D tones were used by the US Department of Defense as a part of their Automatic Voice Network or AUTOVON — a military phone network that operated parallel to the civilian phone network from 1963 to 1982. These buttons and the corresponding tones were used to assign priority to a call in the event that the network was operating at capacity.

  1. Flash Override (FO)
  2. Flash (F)
  3. Immediate (I)
  4. Priority (P)

Lower priority calls (no priority, priority, or immediate) would be disconnected if a higher priority call (flash or flash override) needed the line. Thus urgent messages could make it through the system even if it was reporting "all circuits busy". Presumably if a general or admiral needed to call the president, they'd press the A button before dialing. The A, B, C, D tones are still used by some phone networks as internal commands. They are also used by amateur radio operators to control repeaters (devices that rebroadcast radio signals at higher power) and phone patches (devices that connect two way radios to the phone network).


US Civil Defense logo

The Emergency Broadcast System was a broadcast protocol in effect in the United States from 1963–1997. It has since been replaced with the Emergency Alert System. Radio and television stations across the nation were required to transmit a special audio signal whenever the local or national authorities needed to alert the general population of an emergency situation. Although designed to warn of a military attack, the system was normally used to warn of natural emergencies like tornadoes or flash floods. The EBS system is most famous, however, for its frequent tests. These consisted of the EBS alert tone followed by an announcement that the sound you just heard was only a test. (Had this been an actual emergency….) The EBS alert tone is actually two tones of similar pitch that form a particularly spooky beat frequency. Since the EBS was a Cold War civil defense initiative many people associate this tone with the approach of nuclear attack and the end of the world.

This is a test. This station is conducting a test of the Emergency Broadcast System. This is only a test.

flow = 853 Hz,fhigh = 960 Hz,t = 22.5 s

This is a test of the Emergency Broadcast System. This station, in voluntary cooperation with federal, state, and local authorities, has developed this system to keep you informed in the event of an emergency. Had this been an actual emergency, the attention signal you just heard would have been followed by official information, news, or instructions.

This concludes this test of the Emergency Broadcast System.

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