Hydrostatic shock
The term hydrostatic shock describes the theory that a penetrating projectile produces remote wounding and incapacitating effects in living targets, in addition to local effects in tissue caused by direct impact, through a hydraulic effect in liquid filled tissues.  There is scientific evidence that hydrostatic shock" can produce remote neural damage and produce incapacitation more quickly than blood loss effects.  The debate between proponents of bullets that are "light and fast" versus bullets that are "slow and heavy" often refers to this phenomenon.

# Theory's introduction
It is unclear when the term "hydrostatic shock" first came into use to describe remote effects of penetrating projectiles, but Frank Chamberlin, a World War II trauma surgeon and ballistics researcher, noted remote pressure wave effects.  Col. Chamberlin described what he called explosive effects and hydraulic reaction of bullets in tissue.
...liquids are put in motion by shock waves or hydraulic effects... with liquid filled tissues, the effects and destruction of tissues extend in all directions far beyond the wound axis. 
He avoided the ambiguous use of the term shock because it can refer to either a specific kind of pressure wave associated with explosions and supersonic projectiles or to a medical condition in the body.
Col. Chamberlin recognized that many theories have been advanced in wound ballistics.  During World War II, he commanded an 8,500 bed hospital center that treated over 67,000 patients during the fourteen months that he operated it. P.O. Ackley estimates that 85% of the patients were suffering from gun shot wounds.   Col. Chamberlin spent many hours interviewing patients as to their reactions to bullet wounds.  He also conducted many live animal experiments after his tour of duty.  On the subject of wound ballistics theories, he wrote:
If I had to pick one of these theories as gospel, Id still go along with the Hydraulic Reaction of the Body Fluids plus the reactions on the Central Nervous System.
Other World War II era scientists noted remote pressure wave effects in the peripheral nerves.  There was support for the idea of remote neural effects of ballistic pressure waves in the medical and scientific communities, but the phrase "hydrostatic shock" and similar phrases including shock were used mainly by gunwriters (such as Jack O'Conner) and the small arms industry (such as Roy Weatherby, and Federal Hydrashock.)

# Dr. Fackler's contra-claim
Dr. Martin Fackler, a Vietnam-era trauma surgeon and wound ballistics researcher, claimed that hydrostatic shock had been disproved and that the assertion that a pressure wave plays a role in injury or incapacitation is a myth.   Others expressed similar views. 
Dr. Fackler based his argument on the lithotriptor, a tool commonly used to break up kidney stones. The lithotriptor uses sonic pressure waves which are stronger than those caused by most handgun bullets, yet it produces no damage to soft tissues whatsoever. Hence, Fackler argued, ballistic pressure waves cannot damage tissue either.
However, tissue damage due to lithotriptors has been documented.
Dr. Fackler also claimed that a study of rifle bullet wounds in Vietnam (Wound Data and Munitions Effectiveness Team) found no cases of bones being broken, or major vessels torn, that were not hit by the penetrating bullet.  In only two cases, an organ that was not hit (but was within a few cm of the projectile path), suffered some disruption.  Dr. Fackler cited a personal communication with R. F. Bellamy.  However, Bellamys published findings the following year estimated that 10% of fractures in the data set might be due to indirect injuries, and one specific case is described in detail (pp. 153-154).  In addition, the published analysis documents five instances of abdominal wounding in cases where the bullet did not penetrate the abdominal cavity (pp. 149-152), a case of lung contusion resulting from a hit to the shoulder (pp. 146-149), and a case of indirect effects on the central nervous system (p. 155).  Rather than contradict distant injuries, as Fackler claimed, the WDMET data from Vietnam actually provides supporting evidence.

# Inferences from Blast Pressure Wave Observations
A shock wave can be created when fluid is rapidly displaced by an explosive or projectile. Tissue behaves similarly enough to water that a sonic pressure wave can be created by a bullet impact, generating pressures in excess of 100 atmospheres (1500 PSI).
Duncan McPherson, a former member of the International Wound Ballistics Association and author of the book, Bullet Penetration, claimed that shock waves cannot result from bullet impacts with tissue.   In contrast, Brad Sturtevant, a leading researcher in shock wave physics at Caltech for many decades, found that shock waves can result from handgun bullet impacts in tissue.  Other sources also indicate that ballistic impacts can create shock waves in tissue. 
Blast and ballistic pressure waves have physical similarities.  Prior to wave reflection, they both are characterized by a steep wave front followed by a nearly exponential decay at close distances.  They also have similarities in how they cause neural effects in the brain. In tissue, both types of pressure waves have similar magnitudes, duration, and frequency characteristics. Both have been shown to cause damage in the hippocampus.  It has been hypothesized that both reach the brain from the thoracic cavity via major blood vessels.
For example, Ibolja Cernak, a leading researcher in blast wave injury at the Applied Physics Laboratory at Johns Hopkins University, hypothesized, "alterations in brain function following blast exposure are induced by kinetic energy transfer of blast overpressure via great blood vessels in abdomen and thorax to the central nervous system."    This hypothesis is supported by observations of neural effects in the brain from localized blast exposure focused on the lungs in experiments in animals. 
Hydrostatic shock expresses the idea that organs can be damaged by the pressure wave in addition to damage from direct contact with the penetrating projectile.  If one interprets the "shock" in the term "hydrostatic shock" to refer to the physiological effects rather than the physical wave characteristics, the question of whether the pressure waves satisfy the definition of shock wave is unimportant, and one can consider the weight of scientific evidence and various claims regarding the possibility of a ballistic pressure wave to create tissue damage and incapacitation in living targets.

# Physics of Ballistic Pressure Waves
A number of papers describe the physics of ballistic pressure waves created when a high-speed projectile enters a viscous medium.
  These results show that ballistic impacts produce pressure waves that propagate at close to the speed of sound.
Lee et al. present an analytical model showing that unreflected ballistic pressure waves are well approximated by an exponential decay, which is similar to blast pressure waves.  Lee et al. also note the importance of the energy transfer:
As would be expected, an accurate estimation of the kinetic energy loss by a projectile is always important in determining the ballistic waves.
The rigorous calculations of Lee et al. require knowing the drag coefficient and frontal area of the penetrating projectile at every instant of the penetration.  Since this is not generally possible with expanding handgun bullets, Courtney and Courtney developed a model for estimating the peak pressure waves of handgun bullets from the impact energy and penetration depth in ballistic gelatin. This model agrees with the more rigorous approach of Lee et al. for projectiles where they can both be applied.  For expanding handgun bullets, the peak pressure wave magnitude is proportional to the bullets kinetic energy divided by the penetration depth.

# Remote Cerebral Effects of Ballistic Pressure Waves
Goransson et al. were the first contemporary researchers to present compelling evidence for remote cerebral effects of extremity bullet impact.  They observed changes in EEG readings from pigs shot in the thigh.  A follow-up experiment by Suneson et al. implanted high-speed pressure transducers into the brain of pigs and demonstrated that a significant pressure wave reaches the brain of pigs shot in the thigh.    These scientists observed apnea, depressed EEG readings, and neural damage in the brain caused by the distant effects of the ballistic pressure wave originating in the thigh.
The results of Suneson et al. were confirmed and expanded upon by a later experiment in dogs  
which "confirmed that distant effect exists in the central nervous system after a high-energy missile impact to an extremity. A high-frequency oscillating pressure wave with large amplitude and short duration was found in the brain after the extremity impact of a high-energy missile . . ."  Wang et al. observed significant damage in both the hypothalamus and hippocampus regions of the brain due to remote effects of the ballistic pressure wave.

# Remote Pressure Wave Effects in the Spine and Internal Organs
In a study of a handgun injury, Sturtevant found that pressure waves from a bullet impact in the torso can reach the spine and that a focusing effect from concave surfaces can concentrate the pressure wave on the spinal cord producing significant injury.  This is consistent with other work showing remote spinal cord injuries from ballistic impacts.    
Roberts et al. present both experimental work and finite element modeling showing that there can be considerable pressure wave magnitudes in the thoracic cavity for handgun projectiles stopped by a Kevlar vest. For example, an 8 gram projectile at 360 m/s impacting a NIJ level II vest over the sternum can produce an estimated pressure wave level of nearly 2.0 MPa (280 PSI) in the heart and a pressure wave level of nearly 1.5 MPa (210 PSI) in the lungs.  Impacting over the liver can also produce an estimated pressure wave level of 2.0 MPa (280 PSI) in the liver.

# Energy Transfer Required for Remote Neural Effects
The work of Courtney et al. also supports the role of a ballistic pressure wave in incapacitation and injury. 
   The work of Suneson et al.  and Courtney et al. suggest that remote neural effects can occur with levels of energy transfer possible with handguns (roughly 500 ftlbf/700 joules).  Using sensitive biochemical techniques, the work of Wang et al. suggests even lower impact energy thresholds for remote neural injury to the brain.  In analysis of experiments of dogs shot in the thigh they report highly significant (p < 0.01), easily detectable neural effects in the hypothalamus and  hippocampus with energy transfer levels close to  550 ftlbf (740 Joules).  Wang et al. reports less significant (p < 0.05) remote effects in the hypothalamus with energy transfer just under 100 ftlbf (130 Joules). 
Even though Wang et al. document remote neural damage for low levels of energy transfer (roughly 100 ftlbf), these levels of neural damage are probably too small to contribute to rapid incapacitation.  Courtney and Courtney believe that remote neural effects only begin to make significant contributions to rapid incapacitation for ballistic pressure wave levels above 500 PSI (corresponds to transferring roughly 300  (Expression error: Missing operand for *. ) in 12 inches of penetration) and become easily observable above 1000 PSI (corresponds to transferring roughly 600  (Expression error: Missing operand for *. ) in 1 foot of penetration).  Incapacitating effects in this range of energy transfer are consistent with observations of remote spinal injuries , observations of suppressed EEGs and apnea in pigs.   and with observations of incapacitating effects of ballistic pressure waves without a wound channel. 

# Other Scientific Findings
The scientific literature contains significant other findings regarding injury mechanisms of ballistic pressure waves.  Ming et al. found that ballistic pressure waves can break bones.  Tikka et al. reports abdominal pressure changes produced in pigs hit in one thigh.    Akimov et al. report on injuries to the nerve trunk from gunshot wounds to the extremities.  

# Recommendations
The FBI recommends that loads intended for self-defense and law enforcement applications meet a minimum penetration requirement of 12 in ballistic gelatin .   Maximizing the ballistic pressure wave effects requires transferring maximum energy in a penetration distance that meets this requirement.  In addition, bullets that fragment and meet minimum penetration requirements generate higher pressure waves than bullets which do not fragment.