Michael Courtney
Member
Physics of the Ballistic Pressure Wave
One might wonder why a given 135 grain .40 caliber JHP bullet at 1350 FPS creates a larger ballistic pressure wave many other JHP bullets in the .40 S&W cartridge.
The origin of the pressure wave is Newton’s third law. The bullet slows down in tissue due the force the tissue applies to the bullet. By Newton’s third law, the bullet exerts an equal and opposite force on the tissue. When a force is applied to a fluid or a visco-elastic material such as tissue or ballistic gelatin, a pressure wave radiates outward in all directions from the location where the force is applied.
The instantaneous magnitude of the force, F, between the bullet and the tissue is given by
F = dE/dx,
Where E = ½ m V*V is the instantaneous kinetic energy of the bullet, and x is the penetration distance. dE/dx is the first derivative of the energy with respect to the penetration depth. In other words, it is the instantaneous rate of kinetic energy loss per inch of penetration depth. Losing 100 ft-lbs of kinetic energy in 0.02 feet of penetration would create a force of 5,000 lbs because 100 ft-lbs/0.02 ft = 5,000 lbs.
It is important to note that this force (equal to the rate of energy loss) changes continuously and depends on both the loss of velocity and the loss of mass (unless the mass is constant). By the chain rule of calculus,
F = dE/dx = ½ V*V dm/dx + m V dV/dx,
Where dm/dx and dV/dx are the instantaneous rate of mass and velocity loss with respect to penetration depth.
Applying this formula directly requires detailed knowledge of the instantaneous mass and velocity changes of a bullet at every point along the wound channel. The instantaneous force can be accurately determined by shooting the same bullet through varying thicknesses of ballistic gelatin. In other words, one might shoot through a 0.05 ft thick block of gelatin to determine the loss of energy in the first 0.05 ft of penetration. Then one might shoot through a 0.1 ft thick block of gelatin to determine the loss of energy in a 0.1 ft thick block of gelatin. Then one might shoot through a 0.15 ft thick block of gelatin to determine the loss of energy in a 0.15 ft thick block of gelatin. Repeating this process in small increments, and applying standard techniques for estimating derivatives from measured values at closely spaced points would yield an accurate measurement of the instantaneous force at every penetration depth.
There are some simple and reasonable estimates that can be made more easily.
In cases where the mass is constant, the average force Fave between the tissue and bullet is simply the initial kinetic energy E divided by the penetration depth d.
Fave = E/d.
However, in cases where the bullet loses mass along the wound channel, the average force is increased by between 1 and 2 times the fraction of mass lost. If the bullet loses 20% of its mass distributed evenly along the wound channel, it creates an average force
Fave = 1.2 E/d.
Many bullets which lose mass, lose more mass early in the wound channel rather than late. This can enhance the average force even more. If the average depth of lost mass is one third of the penetration depth (d/3), the enhancement of the average force is twice the lost mass fraction. In other words, a bullet which loses 20% of its mass at an average depth of one third the penetration depth will create an average force
Fave = 1.4 E/d.
The peak of any variable force is larger than the average value. The peak to average ratio usually occurs during or right after expansion, and most bullets have peak to average force ratios between 3 and 8. Bullets which do not expand, penetrate deeply, and lose energy gradually have a peak to average ratio close to 3. Bullets which expand rapidly, lose a lot of energy early, erode down to a smaller diameter and then penetrate deeply have a peak to average ratio close to 8. Nosler Partitions with their soft lead front section which expands rapidly and erodes away quickly leaving the base containing roughly 60% of the original mass at little more than the unexpanded diameter is an example of large peak to average force ratio. Most JHP handgun bullets have a peak to average ratio close to 5, so this can provide a reasonably accurate estimate of the peak force in many cases.
If we go with the more conservative estimate that the average distance of penetration for lost mass is the middle of the total penetration (d/2) and we call the faction of lost mass f, then we estimate the peak pressure as
Fpeak = (1 + f) 5 E/d,
This allows us to see quite simply why the 135 grain bullet has a larger peak force than other .40 caliber S&W loads. The table below shows values of mass, energy, penetration, fraction of lost mass, and the peak force estimate for several .40 S&W loads
Load m (gr) V (FPS) d (in) E (ft-lbs) f F (lbs)
DT135JHP(N) 135 1350 11.9 547 0.36 3752
Rem165GS 165 1150 12 485 0 2427
Fed155HS 155 1140 13.3 448 0 2021
Fed180HS 180 950 15 361 0 1445
This force acts as a point of origin for the pressure wave which radiates outward in all directions. The pressure falls off with distance because the area it covers increases. However, we can compare the peak pressure generated by different loads if we standardize the point of interest to be the surface of a sphere with diameter of 1” centered at the point of origin of the force. This gives peak pressure wave levels of the different loads:
Load P(1") PSI
DT135JHP(N) 1195
Rem165GS 773
Fed155HS 643
Fed180HS 460
In summary, the .40 S&W Double Tap loading of the Nosler 135 grain JHP has a larger pressure wave than many other JHP bullets in that cartridge for three reasons:
1. It has greater kinetic energy.
2. It penetrates less.
3. It fragments and loses more of its mass.
One can apply this identical analysis to other JHP loads in this and other cartridges. Doing so reveals that there are a few other JHP loads which generate comparable peak pressure levels. Among them are the 125 grain Federal and Remington JHP loads in .357 Magnum that are known for their ability to incapacitate quickly.
Michael Courtney
One might wonder why a given 135 grain .40 caliber JHP bullet at 1350 FPS creates a larger ballistic pressure wave many other JHP bullets in the .40 S&W cartridge.
The origin of the pressure wave is Newton’s third law. The bullet slows down in tissue due the force the tissue applies to the bullet. By Newton’s third law, the bullet exerts an equal and opposite force on the tissue. When a force is applied to a fluid or a visco-elastic material such as tissue or ballistic gelatin, a pressure wave radiates outward in all directions from the location where the force is applied.
The instantaneous magnitude of the force, F, between the bullet and the tissue is given by
F = dE/dx,
Where E = ½ m V*V is the instantaneous kinetic energy of the bullet, and x is the penetration distance. dE/dx is the first derivative of the energy with respect to the penetration depth. In other words, it is the instantaneous rate of kinetic energy loss per inch of penetration depth. Losing 100 ft-lbs of kinetic energy in 0.02 feet of penetration would create a force of 5,000 lbs because 100 ft-lbs/0.02 ft = 5,000 lbs.
It is important to note that this force (equal to the rate of energy loss) changes continuously and depends on both the loss of velocity and the loss of mass (unless the mass is constant). By the chain rule of calculus,
F = dE/dx = ½ V*V dm/dx + m V dV/dx,
Where dm/dx and dV/dx are the instantaneous rate of mass and velocity loss with respect to penetration depth.
Applying this formula directly requires detailed knowledge of the instantaneous mass and velocity changes of a bullet at every point along the wound channel. The instantaneous force can be accurately determined by shooting the same bullet through varying thicknesses of ballistic gelatin. In other words, one might shoot through a 0.05 ft thick block of gelatin to determine the loss of energy in the first 0.05 ft of penetration. Then one might shoot through a 0.1 ft thick block of gelatin to determine the loss of energy in a 0.1 ft thick block of gelatin. Then one might shoot through a 0.15 ft thick block of gelatin to determine the loss of energy in a 0.15 ft thick block of gelatin. Repeating this process in small increments, and applying standard techniques for estimating derivatives from measured values at closely spaced points would yield an accurate measurement of the instantaneous force at every penetration depth.
There are some simple and reasonable estimates that can be made more easily.
In cases where the mass is constant, the average force Fave between the tissue and bullet is simply the initial kinetic energy E divided by the penetration depth d.
Fave = E/d.
However, in cases where the bullet loses mass along the wound channel, the average force is increased by between 1 and 2 times the fraction of mass lost. If the bullet loses 20% of its mass distributed evenly along the wound channel, it creates an average force
Fave = 1.2 E/d.
Many bullets which lose mass, lose more mass early in the wound channel rather than late. This can enhance the average force even more. If the average depth of lost mass is one third of the penetration depth (d/3), the enhancement of the average force is twice the lost mass fraction. In other words, a bullet which loses 20% of its mass at an average depth of one third the penetration depth will create an average force
Fave = 1.4 E/d.
The peak of any variable force is larger than the average value. The peak to average ratio usually occurs during or right after expansion, and most bullets have peak to average force ratios between 3 and 8. Bullets which do not expand, penetrate deeply, and lose energy gradually have a peak to average ratio close to 3. Bullets which expand rapidly, lose a lot of energy early, erode down to a smaller diameter and then penetrate deeply have a peak to average ratio close to 8. Nosler Partitions with their soft lead front section which expands rapidly and erodes away quickly leaving the base containing roughly 60% of the original mass at little more than the unexpanded diameter is an example of large peak to average force ratio. Most JHP handgun bullets have a peak to average ratio close to 5, so this can provide a reasonably accurate estimate of the peak force in many cases.
If we go with the more conservative estimate that the average distance of penetration for lost mass is the middle of the total penetration (d/2) and we call the faction of lost mass f, then we estimate the peak pressure as
Fpeak = (1 + f) 5 E/d,
This allows us to see quite simply why the 135 grain bullet has a larger peak force than other .40 caliber S&W loads. The table below shows values of mass, energy, penetration, fraction of lost mass, and the peak force estimate for several .40 S&W loads
Load m (gr) V (FPS) d (in) E (ft-lbs) f F (lbs)
DT135JHP(N) 135 1350 11.9 547 0.36 3752
Rem165GS 165 1150 12 485 0 2427
Fed155HS 155 1140 13.3 448 0 2021
Fed180HS 180 950 15 361 0 1445
This force acts as a point of origin for the pressure wave which radiates outward in all directions. The pressure falls off with distance because the area it covers increases. However, we can compare the peak pressure generated by different loads if we standardize the point of interest to be the surface of a sphere with diameter of 1” centered at the point of origin of the force. This gives peak pressure wave levels of the different loads:
Load P(1") PSI
DT135JHP(N) 1195
Rem165GS 773
Fed155HS 643
Fed180HS 460
In summary, the .40 S&W Double Tap loading of the Nosler 135 grain JHP has a larger pressure wave than many other JHP bullets in that cartridge for three reasons:
1. It has greater kinetic energy.
2. It penetrates less.
3. It fragments and loses more of its mass.
One can apply this identical analysis to other JHP loads in this and other cartridges. Doing so reveals that there are a few other JHP loads which generate comparable peak pressure levels. Among them are the 125 grain Federal and Remington JHP loads in .357 Magnum that are known for their ability to incapacitate quickly.
Michael Courtney