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 $\mu$.
- Is the membrane isotropic?
- Suppose we use another tiling (like hexagonal) before taking the continuous limit, will this sheet behaves the same way?
- 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 $k$?
- What is the equation of motion for the square sheet with spring constant $k$?
- 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.
- 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.