Spectrogram
The spectrogram is the result of calculating the frequency spectrum of windowed frames of a compound signal.  It is a three-dimensional plot of the energy of the frequency content of a signal as it changes over time.
Spectrograms are used to identify phonetic sounds, to analyse the cries of animals, and in the fields of music, sonar/radar, speech processing, etc.  A spectrogram can also be called a spectral waterfall, sonogram, voiceprint, or voicegram.  The instrument that generates a spectrogram is called a sonograph.

# Format
In the most usual format, the horizontal axis represents time, the vertical axis is frequency, and the intensity of each point in the image represents amplitude of a particular frequency at a particular time.  Often the diagram is reduced to two dimensions by indicating the intensity with thicker lines, more intense colors or grey values.
There are many variations of format.  Sometimes the vertical and horizontal axes are switched, so time runs up and down.  Sometimes the amplitude is represented as the height of a 3D surface instead of color or intensity.  The frequency and amplitude axes can be either linear or logarithmic, depending on what the graph is being used for.  For instance, audio would usually be represented with a logarithmic amplitude axis (probably in dB), and frequency would be linear to emphasize harmonic relationships, or logarithmic to emphasize musical, tonal relationships.

# Generation
Spectrograms are usually created in one of two ways; either with a series of bandpass filters, or calculated from the time signal using the short-time Fourier transform (STFT).
The filter method is usually used in the analog, continuous version of measurement.  The frequency range of the signal (an audio signal, for instance, would have frequencies in the range of 20 Hz - 20 kHz) is divided into equal sections, either linearly (0-100, 100-200, 200-300, ...), or logarithmically (10-100, 100-1000, 1000-10000, ...).  The signal is input to a corresponding filter, which removes most of the signal that does not fall within its frequency band (imperfect window functions and limited frequency resolution will cause some "bleeding" between adjacent frequency bands).  The magnitudes of each filter's output are recorded as functions of time.  Each recording then corresponds to a horizontal line in the image; a measurement of magnitude versus time for a specific frequency band.
To calculate the spectrogram using the magnitude of the STFT is usually a digital process.  Digitally sampled data, in the time domain, is broken up into chunks, which usually overlap, and Fourier transformed to calculate the magnitude of the frequency spectrum for each chunk.  Each chunk then corresponds to a vertical line in the image; a measurement of magnitude versus frequency for a specific moment in time.
The spectrums or time plots are then "laid side by side" to form the image or a three-dimensional surface.
The spectrogram is given by the magnitude of the STFT of the function:

# Creating sound from a spectrogram
The above process can be reversed; some programs are available that turn a digital image into sound:
- MetaSynth for Macintosh;
- Coagula for Windows;
- Enscribe for Linux;
- ARSS, The Analysis & Resynthesis Sound Spectrograph, multi-platform;
- JavOICe, a Java applet.
- FL Studio's "BeepMap" additive synthesizer.
This technique allows electronic music artists to "hide" images in their music.  Examples include:
- Aphex Twin hid an image of himself in a spectrogram. The image can be found on Track 2 of the Windowlicker EP as a nine-second sweeping section right at the end. (It is recognizable in an MP3, but the compression changes the spectrogram and it is not as clear as from the CD.) Aphex Twin also hid the image of a spiral shape in his first track from the "Windowlicker" EP. 
- The song "Look" from Venetian Snares' album Songs About My Cats, contains several images of his cats.
- The song "3recurring" from Plaid on their album Rest Proof Clockwork contains the recurring 3s represented as a logo on the cover of the "Not for Threes" album.
- Nine Inch Nails has employed this technique on a few occasions as part of an ongoing Alternate Reality Game for their album Year Zero.
The MP3 version of "My Violent Heart" (deliberately leaked before Year Zero was released) contains a few brief moments of static at the end.  When analyzed in a spectrogram, it reveals the image of a hand reaching down from the sky.  This image echoes a similar theme seen throughout the alternate reality game viral marketing campaign launched for the album.  On the official version of the album, the sound appears at the end of the song "The Warning."
Another new Nine Inch Nails track from Year Zero was also found on a flash drive in a bathroom stall on their current European tour. This time it was "Me, I'm Not." Along with the song, was an MP3 of "cricket chirping", which when put through a spectrogram revealed the number: 216.333.1810.
Using this method again to deliver clues as part of the Year Zero ARG, there were two extra audio files included in the Garageband Multitrack files for the song Capital G.  The images in the sounds are of various avatars of user at the Nine Inch Nails oriented forum Echoing the Sound.  The user names are used to form the title of a new site that is part of the ARG.
- The MP3 version of "My Violent Heart" (deliberately leaked before Year Zero was released) contains a few brief moments of static at the end.  When analyzed in a spectrogram, it reveals the image of a hand reaching down from the sky.  This image echoes a similar theme seen throughout the alternate reality game viral marketing campaign launched for the album.  On the official version of the album, the sound appears at the end of the song "The Warning."
- Another new Nine Inch Nails track from Year Zero was also found on a flash drive in a bathroom stall on their current European tour. This time it was "Me, I'm Not." Along with the song, was an MP3 of "cricket chirping", which when put through a spectrogram revealed the number: 216.333.1810.
- Using this method again to deliver clues as part of the Year Zero ARG, there were two extra audio files included in the Garageband Multitrack files for the song Capital G.  The images in the sounds are of various avatars of user at the Nine Inch Nails oriented forum Echoing the Sound.  The user names are used to form the title of a new site that is part of the ARG.
- An untitled audio track by Treetops McPhoenix to accompany a video for the Stephen Moles Coma ARG on Facebook is made entirely from images of the Mona Lisa.
On spectrograms of the records of Mike Oldfield's song "Tubular Bells" signals from the Rugby VLF transmitter are found.
Some modern music is also created using spectrograms as an intermediate medium; changing the intensity of different frequencies over time, or even creating new ones, by drawing them and then inverse transforming.  See Audio timescale-pitch modification and Phase vocoder.

# VLF-reception with the PC
Using spectrograms generated by audio-band FFT-software is a very convenient way to receive frequencies below 24 kHz. This technique allows wide-range reception of the VLF-range.