How do shockwaves in a 1D linear lattice work?

[curious_being]

I am struggling to understand shocks in a one dimensional lattice with a linear spring connecting the masses. Say I have a one dimensional lattice with a linear spring constant, k and lattice spacing a. If the particles in the lattice has mass, m then my speed of sound c is a*sqrt(k/m). That is a small disturbance will propagate with speed c. Now if I move one of the end particles in my lattice at a speed greater than c, I should incite a shockwave? Will the dynamics then be governed by the Hugoniot relations?

I guess my confusion is as to how a shock forms if the system is linear... If it is a linear sound wave, the pressure distribution or the structure of the wave propagates unperturbed. If the interaction between granules is non-linear, I see how the structure of the wave eventually becomes a shock wave as the wavefront gets squished as larger loadings will propagate faster.

Back to the linear system, how does the linear interaction bring about a nonlinear compressibility to generate a shock front if k doesn't change...? I think the density behind the front will increase linearly but I don't know how to relate all the state variables let alone know how to derive an EOS for a linear lattice.

Much help would be greatly appreciated!

 

[boneh3ad]

Maybe I'm missing something, but I've never heard of a shock in a linear system.

 

[jim mcnamara]

Not my area but I bumped into this a couple of days ago -any help?
https://www.amazon.com/dp/1258604019/?tag=pfamazon01-20

 

[curious_being]

Maybe I'm missing something, but I've never heard of a shock in a linear system.

Understandable hence my confusion! What happens when the excitation is greater than the speed of sound then?

 

[sophiecentaur]

Understandable hence my confusion! What happens when the excitation is greater than the speed of sound then?

Perhaps it's to do with the questionable relevance of such a model.

 

[curious_being]

Perhaps it's to do with the questionable relevance of such a model.

Thanks so would the conclusion be:

in a 1d lattice, nothing propagates faster than the maximum phase speed of the dispersion relation. Motion faster than this speed cannot be modeled by such as system without nonlinearity

 

[boneh3ad]

Thanks so would the conclusion be:

in a 1d lattice, nothing propagates faster than the maximum phase speed of the dispersion relation. Motion faster than this speed cannot be modeled by such as system without nonlinearity

That's usually the case. If you have an input that cannot be handled by a model system, then the assumptions underlying that model are generally wrong. In this case, one explanation is that you can't assume a linear relationship.

 

[sophiecentaur]

If you have an input that cannot be handled by a model system

I was thinking of a simple model of a 1D gas with equal mass molecules and elastic collisions, all with random initial ('thermal') velocities. A 'piston' moves at high speed from the left and imparts extra velocity to each molecule it hits and the other molecules eventually carry this extra velocity. There is no mechanism for the built up energy to disperse, as in 3D. I think it would not depend on the actual 'temperature' because you could start off with the balls all stationary.
Just making the point about how non-representative the 1D case would be.

 

[boneh3ad]

I was thinking of a simple model of a 1D gas with equal mass molecules and elastic collisions, all with random initial ('thermal') velocities. A 'piston' moves at high speed from the left and imparts extra velocity to each molecule it hits and the other molecules eventually carry this extra velocity. There is no mechanism for the built up energy to disperse, as in 3D. I think it would not depend on the actual 'temperature' because you could start off with the balls all stationary.
Just making the point about how non-representative the 1D case would be.

Such a piston needn't even be high speed to create a shock in a gas. This is a canonical gas dynamic example. Of course, even the simplified equations are nonlinear.