Sonar
The display of most passive sonars used to be a two-dimensional waterfall display. The horizontal direction of the display is bearing. The vertical is frequency, or sometimes time. Another display technique is to color-code frequency-time information for bearing. More recent displays are generated by the computers, and mimic radar-type plan position indicator displays.

Performance prediction: Unlike active sonar, only one-way propagation is involved.  Because of the different signal processing used, the minimal detectable signal-to-noise ratio will be different.  The equation for determining the performance of a passive sonar is

SL − PL = NL − AG + DT,
where SL is the source level, PL is the propagation loss, NL is the noise level, AG is the array gain and DT is the detection threshold.  The figure of merit of a passive sonar is

FOM = SL + AG − (NL + DT).

Performance factors: The detection, classification and localisation performance of a sonar depends on the environment and the receiving equipment, as well as the transmitting equipment in an active sonar or the target radiated noise in a passive sonar.

Sound propagation: Sonar operation is affected by variations in sound speed, particularly in the vertical plane. Sound travels more slowly in fresh water than in sea water, though the difference is small.  The speed is determined by the water's bulk modulus and mass density.  The bulk modulus is affected by temperature, dissolved impurities (usually salinity), and pressure. The density effect is small.  The speed of sound (in feet per second) is approximately:

4388 + (11.25 × temperature (in °F)) + (0.0182 × depth (in feet)) + salinity (in parts-per-thousand ).
This empirically derived approximation equation is reasonably accurate for normal temperatures, concentrations of salinity and the range of most ocean depths. Ocean temperature varies with depth, but at between 30 and 100 meters there is often a marked change, called the thermocline, dividing the warmer surface water from the cold, still waters that make up the rest of the ocean. This can frustrate sonar, because a sound originating on one side of the thermocline tends to be bent, or refracted, through the thermocline. The thermocline may be present in shallower coastal waters. However, wave action will often mix the water column and eliminate the thermocline. Water pressure also affects sound propagation: higher pressure increases the sound speed, which causes the sound waves to refract away from the area of higher sound speed. The mathematical model of refraction is called Snell's law.
If the sound source is deep and the conditions are right, propagation may occur in the 'deep sound channel'. This provides extremely low propagation loss to a receiver in the channel.  This is because of sound trapping in the channel with no losses at the boundaries.  Similar propagation can occur in the 'surface duct' under suitable conditions.  However, in this case there are reflection losses at the surface.
In shallow water propagation is generally by repeated reflection at the surface and bottom, where considerable losses can occur.
Sound propagation is affected by absorption in the water itself as well as at the surface and bottom.  This absorption depends upon frequency, with several different mechanisms in sea water.  Long-range sonar uses low frequencies to minimise absorption effects.
The sea contains many sources of noise that interfere with the desired target echo or signature.  The main noise sources are waves and shipping.  The motion of the receiver through the water can also cause speed-dependent low frequency noise.

Scattering: When active sonar is used, scattering occurs from small objects in the sea as well as from the bottom and surface.  This can be a major source of interference. This acoustic scattering is analogous to the scattering of the light from a car's headlights in fog: a high-intensity pencil beam will penetrate the fog to some extent, but broader-beam headlights emit much light in unwanted directions, much of which is scattered back to the observer, overwhelming that reflected from the target ("white-out").  For analogous reasons active sonar needs to transmit in a narrow beam to minimize scattering.

 
The scattering of sonar from objects (mines, pipelines, zooplankton, geological features, fish etc.) is how active sonar detects them, but this ability can be masked by strong scattering from false targets, or 'clutter'. Where they occur (under breaking waves; in ship wakes; in gas emitted from seabed seeps and leaks etc.), gas bubbles are powerful sources of clutter, and can readily hide targets. TWIPS (Twin Inverted Pulse Sonar) is currently the only sonar that can overcome this clutter problem. This is important as many recent conflicts have occurred in coastal waters, and the inability to detect whether mines are present or not present hazards and delays to military vessels, and also to aid convoys and merchant shipping trying to support the region long after the conflict has ceased.

Target characteristics: The sound reflection characteristics of the target of an active sonar, such as a submarine, are known as its target strength.  A complication is that echoes are also obtained from other objects in the sea such as whales, wakes, schools of fish and rocks.
Passive sonar detects the target's radiated noise characteristics.  The radiated spectrum comprises a continuous spectrum of noise with peaks at certain frequencies which can be used for classification.

Countermeasures: Active (powered) countermeasures may be launched by a vessel under attack to raise the noise level, provide a large false target, and obscure the signature of the vessel itself.
Passive (i.e., non-powered) countermeasures include:

Mounting noise-generating devices on isolating devices.
Sound-absorbent coatings on the hulls of submarines, for example anechoic tiles.

Military applications: Modern naval warfare makes extensive use of both passive and active sonar from water-borne vessels, aircraft and fixed installations. Although active sonar was used by surface craft in World War II, submarines avoided the use of active sonar due to the potential for revealing their presence and position to enemy forces. However, the advent of modern signal-processing enabled the use of passive sonar as a primary means for search and detection operations. In 1987 a division of Japanese company Toshiba reportedly sold machinery to the Soviet Union that allowed their submarine propeller blades to be milled so that they became radically quieter, making the newer generation of submarines more difficult to detect.
The use of active sonar by a submarine to determine bearing is extremely rare and will not necessarily give high quality bearing or range information to the submarines fire control team. However, use of active sonar on surface ships is very common and is used by submarines when the tactical situation dictates that it is more important to determine the position of a hostile submarine than conceal their own position. With surface ships, it might be assumed that the threat is already tracking the ship with satellite data as any vessel around the emitting sonar will detect the emission. Having heard the signal, it is easy to identify the sonar equipment used (usually with its frequency) and its position (with the sound wave's energy). Active sonar is similar to radar in that, while it allows detection of targets at a certain range, it also enables the emitter to be detected at a far greater range, which is undesirable.
Since active sonar reveals the presence and position of the operator, and does not allow exact classification of targets, it is used by fast (planes, helicopters) and by noisy platforms (most surface ships) but rarely by submarines. When active sonar is used by surface ships or submarines, it is typically activated very briefly at intermittent periods to minimize the risk of detection. Consequently, active sonar is normally considered a backup to passive sonar. In aircraft, active sonar is used in the form of disposable sonobuoys that are dropped in the aircraft's patrol area or in the vicinity of possible enemy sonar contacts.
Passive sonar has several advantages, most importantly that it is silent. If the target radiated noise level is high enough, it can have a greater range than active sonar, and allows the target to be identified. Since any motorized object makes some noise, it may in principle be detected, depending on the level of noise emitted and the ambient noise level in the area, as well as the technology used. To simplify, passive sonar "sees" around the ship using it. On a submarine, nose-mounted passive sonar detects in directions of about 270°, centered on the ship's alignment, the hull-mounted array of about 160° on each side, and the towed array of a full 360°. The invisible areas are due to the ship's own interference. Once a signal is detected in a certain direction (which means that something makes sound in that direction, this is called broadband detection) it is possible to zoom in and analyze the signal received (narrowband analysis). This is generally done using a Fourier transform to show the different frequencies making up the sound. Since every engine makes a specific sound, it is straightforward to identify the object. Databases of unique engine sounds are part of what is known as acoustic intelligence or ACINT.
Another use of passive sonar is to determine the target's trajectory. This process is called target motion analysis (TMA), and the resultant "solution" is the target's range, course, and speed. TMA is done by marking from which direction the sound comes at different times, and comparing the motion with that of the operator's own ship. Changes in relative motion are analyzed using standard geometrical techniques along with some assumptions about limiting cases.
Passive sonar is stealthy and very useful.