1. Apr 12, 2014

### Cantique

1. The problem statement, all variables and given/known data

Derive the strong shock condition given the Rankine-Hugoniot conditions:

$\frac{ρ_{2}}{ρ_{1}}$ = $\frac{u_{1}}{u_{2}}$ = $\frac{γ+1}{γ-1}$

where u1, ρ1 are velocity and density upstream of the shock and u2 ρ2 are velocity and density downstream of the shock.

Two identical gas clouds collide with a relative velocity of 2V, where V is very much larger than the sound speed in the clouds. A strong, planar, adiabatic shock wave propagates into the clouds away from the surface of contact. Explain why, in the zero-momentum frame, the shocked gas must be stationary, while the unshocked gas continues to travel at its initial speed V.

How fast does the shock travel in the zero-momentum frame?

Without detailed calculation discuss what happens when the shock reaches the side of the cloud distant from the surface of contact.

2. Relevant equations

Rankine Hugoniot equations

3. The attempt at a solution

I have completed the first part and second part asking for the explanation. I am stuck on the problem of how fast the shock travels in the zero-momentum frame. In my opinion it should do one of two things:

Stay stationary at the centre of the two clouds - The reason being that the fact that the downstream speed is zero in the zero-momentum frame means that all the gas collides only at the centre where it then remains. However, this has the problem of creating an infinitely large pressure at the centre which would eventually result in the sound speed being higher than V (the speed of the unshocked gas) so the shocked gas would start to spread out under the thermal pressure and so the position of the shock would move out to each edge which leads to my second case:

The shock moves at speed V towards the edge. Following on from the reasoning of the other case.

I think the second case is more likely than the first, but don't believe it to be correct. Am I thinking about this problem incorrectly?
The text passage contains a physics problem that involves the derivation of a strong shock condition from the Rankine-Hugoniot conditions and a series of questions related to the behavior of gas clouds in a shock wave scenario. The passage includes a specific question about the speed of the shock in the zero-momentum frame, and the author provides their thoughts on possible answers.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
1. Apr 12, 2014

### Cantique

1. The problem statement, all variables and given/known data

Derive the strong shock condition given the Rankine-Hugoniot conditions:

$\frac{ρ_{2}}{ρ_{1}}$ = $\frac{u_{1}}{u_{2}}$ = $\frac{γ+1}{γ-1}$

where u1, ρ1 are velocity and density upstream of the shock and u2 ρ2 are velocity and density downstream of the shock.

Two identical gas clouds collide with a relative velocity of 2V, where V is very much larger than the sound speed in the clouds. A strong, planar, adiabatic shock wave propagates into the clouds away from the surface of contact. Explain why, in the zero-momentum frame, the shocked gas must be stationary, while the unshocked gas continues to travel at its initial speed V.

How fast does the shock travel in the zero-momentum frame?

Without detailed calculation discuss what happens when the shock reaches the side of the cloud distant from the surface of contact.

2. Relevant equations

Rankine Hugoniot equations

3. The attempt at a solution

I have completed the first part and second part asking for the explanation. I am stuck on the problem of how fast the shock travels in the zero-momentum frame. In my opinion it should do one of two things:

Stay stationary at the centre of the two clouds - The reason being that the fact that the downstream speed is zero in the zero-momentum frame means that all the gas collides only at the centre where it then remains. However, this has the problem of creating an infinitely large pressure at the centre which would eventually result in the sound speed being higher than V (the speed of the unshocked gas) so the shocked gas would start to spread out under the thermal pressure and so the position of the shock would move out to each edge which leads to my second case:

The shock moves at speed V towards the edge. Following on from the reasoning of the other case.

I think the second case is more likely than the first, but don't believe it to be correct. Am I thinking about this problem incorrectly?
user
The text passage contains a physics problem that involves the derivation of a strong shock condition from the Rankine-Hugoniot conditions and a series of questions related to the behavior of gas clouds in a shock wave scenario. The passage includes a specific question about the speed of the shock in the zero-momentum frame, and the author provides their thoughts on possible answers.

TYPE: SINGLE_QUESTION_WITH_ANSWER