[Physics] Equivalent spring-constant for infinite square grid of springs

newtonian-mechanicsspring

Consider an infinite square grid, where each side of a square is a spring following Hooke's law, with spring constant $k$.

What is the relation between the force and displacement between two points? If they are proportional, what is the equivalent spring constant between the origin and the point $(x,y)$ (integers) ?

Edit 1:
I also want to know this:
Suppose you make the springs so small that this can be treated as a continuous sheet, at what speed will a wave propagate? Assuming a wave starting as an initial displacement perpendicular to the sheet.

Given some initial state, is there an equation for the time-evolution of the continuous sheet?

Edit 2:
Suppose there is a mass at every node, and its $(x,y)$-coordinates is fixed, it only vibrates out of the plane. Consider that we take the continuous limit, such that we get a 2D membrane of mass density $\mu$.

  1. Is the membrane isotropic?
  2. Suppose we use another tiling (like hexagonal) before taking the continuous limit, will this sheet behaves the same way?
  3. If not, but they are both isotropic, how does one characterize their difference, can they be made to behave the same way by changing the spring constant $k$?
  4. What is the equation of motion for the square sheet with spring constant $k$?
  5. What is the equation of motion for the square sheet if the springs obey a generalized Force law, $F=kx^n$, where $n$ is a variable.
  6. What is the equation of motion for a 3D cubic grid?

I am particularly interested in answers to 1., 2. and 3.
I dont expect anyone to answer all these and will also accept an answer which does not explain anything but simply provides a good reference.

Best Answer

I'll answer only the third one (for now at least); the movement with limit to small vertical oscillations will be governed by the drum equation:

$\ddot{s}(x,y)=c^2 \nabla^2 s(x,y)$

where $s(x,y)$ is a vertical displacement in point $(x,y)$ and $c$ is the weave speed; using dimensional analysis I would say that $c\sim\sqrt{\frac{k}{\sigma}}$, where $\sigma$ is the mass density. Of course everything is getting much more complex with larger amplitudes.

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