# How do shockwaves in a 1D linear lattice work?

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• curious_being
In summary: 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.
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!

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

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
curious_being said:
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
sophiecentaur said:
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

curious_being said:
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.

curious_being
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.

sophiecentaur said:
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.

sophiecentaur

## 1. How are shockwaves created in a 1D linear lattice?

Shockwaves in a 1D linear lattice are created when there is a sudden change in the properties of the lattice, such as a change in density or pressure. This creates a disturbance that propagates through the lattice as a shockwave.

## 2. What factors affect the speed of shockwaves in a 1D linear lattice?

The speed of shockwaves in a 1D linear lattice is affected by the properties of the lattice, such as the density and stiffness, as well as the initial conditions that create the shockwave. The speed of the shockwave also depends on the magnitude of the change in properties that caused it.

## 3. How do shockwaves in a 1D linear lattice interact with each other?

When two shockwaves in a 1D linear lattice meet, they can either pass through each other without affecting each other, or they can merge and create a larger shockwave. This depends on the relative speeds and amplitudes of the two shockwaves.

## 4. Can shockwaves in a 1D linear lattice be controlled or manipulated?

Yes, shockwaves in a 1D linear lattice can be controlled and manipulated by changing the properties of the lattice, such as the density and stiffness, or by introducing external forces. This can be useful in applications such as shockwave therapy in medicine.

## 5. What are the applications of shockwaves in a 1D linear lattice?

Shockwaves in a 1D linear lattice have various applications in different fields, such as shockwave therapy in medicine, shockwave lithotripsy for breaking up kidney stones, and shockwave testing in materials science. They are also used in research to study the behavior of waves in one-dimensional systems.

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