# The relation between shockwave thickness and shockwave strength

What is the relation between shockwave thickness and shockwave strength? I mean with increasing altitude and increase shockwave thickness, shock become stronger or weaker?

Defining strength of a shock wave: Strength of a shock wave is defined as the ratio of increase in static pressure across the shock to the inlet static pressure. $$\text{Strength of shock} = \frac{p_y - p_x}{p_x}$$

Here we see that with decreasing upstream pressure (high altitudes), shock becomes stronger but how it can be possible? In high altitudes, density is too low and mean free path between particles is large. As a result shock must be weaker, not stronger! I'm confused!

In high altitudes, density is too low and mean free path between particles is large. As a result shock must be weaker, not stronger!

First, see the following answer regarding the speed of sound versus altitude and how sound waves propagate at high altitudes:
https://physics.stackexchange.com/a/266046/59023

Generally as one goes to higher and higher altitudes, it becomes more and more difficult for sound waves to propagate without significant dissipation/loss due to acoustic attenuation. Further, only certain wavelengths will survive as the mean free path continues to increase.

Here we see that with decreasing upstream pressure (high altitudes), shock becomes stronger but how it can be possible?

This presumes that the downstream pressure remains the same at all altitudes, which it will not. The pressure increase will be more limited in more tenuous gases due to the much lower densities. One can still get large normalized changes in pressure (i.e., your shock strength calculation), but note the absolute values of the upstream and downstream pressures will dramatically decrease with increasing altitude.

What is the relation between shockwave thickness and shockwave strength?

This depends largely on the mechanism dissipating energy (e.g., see https://physics.stackexchange.com/a/139436/59023 and https://physics.stackexchange.com/a/210097/59023), not necessarily altitude (unless you get into space where the gas becomes ionized, i.e., a plasma). At altitudes where binary particle collisions and viscosity mediate the energy dissipation necessary to maintain a stable discontinuity like a shock wave, the thickness of the shock will mostly be controlled by the mean free path of the particles. If the mean free path increases, then the shock thickness will increase accordingly. If the gas becomes so tenuous as to be effectively collisionless, then other mechanisms would be necessary to dissipate the excess kinetic energy across a shock. Note that if the gas is super tenuous and somehow still neutral, it can very difficult to actually initiate a shock without extreme speeds because the effective speed of sound can diverge.

Note that it is the accumulation of particles which cannot move fast enough to get around an obstacle that leads to the density pulse steepening into a shock. If there are so few particles present that they can easily get around the obstacle before any significant accumulation takes effect, then a shock could not form.

I mean with increasing altitude and increase shockwave thickness, shock become stronger or weaker?

Again, this depends on the Mach number and other factors of the obstacle. Above some speed, the obstacle will generate a shock just fine because it's moving so fast even in a tenuous media it still causes a significant pile-up of particles to generate a shock.

If the obstacle moves at a constant, true air speed (TAS) -- speed of obstacle relative to rest frame of media through which obstacle moves -- V at all altitudes, then yes, the downstream absolute pressure will decrease at higher altitudes. The true Mach number (i.e., Mach number calculated from TAS) can increase just by moving at a constant TAS at a higher altitude.