Question

In the case of plane-wave propagation in the x direction within a uniform medium, the homogeneous momentum equation (3.9) for shear waves can be expressed as ∂²u/∂t² = β² ∂²u/∂x² where u is the displacement. Write a computer program that uses finite differences to solve this equation for a bar 100 km in length, assuming β = 4 km/s. Use dx = 1 km for the length spacing and dt= 0 s for the time spacing. Assume a source–time function at u(50 km) of the form

Answer 1

To solve the given equation for shear waves using finite differences, we can discretize the equation and solve it iteratively for different spatial and temporal points. By substituting the approximations for the second derivatives and rearranging the equation, we can compute the displacement at each point in the bar. It is important to note the initial condition and the number of temporal points required for the solution.

Step-by-step explanation:

To solve the homogeneous momentum equation ∂²u/∂t² = β² ∂²u/∂x² using finite differences, we can discretize the equation into a set of equations at different spatial and temporal points. In this case, we have a bar 100 km in length, with β = 4 km/s. We can use a time spacing of dt = 0 s (since it's not specified) and a length spacing of dx = 1 km.

Let's start by discretizing the equation at each spatial and temporal point. We can represent the displacement u as a matrix, where each element corresponds to a specific spatial and temporal point. We can denote the elements of the matrix as u(i, j), where i represents the spatial point (i = 1, 2, ..., 100) and j represents the temporal point (j = 1, 2, ...).

Using finite differences, we can approximate the second derivatives as: ∂²u/∂t² ≈ (u(i, j+1) - 2u(i, j) + u(i, j-1))/dt² and ∂²u/∂x² ≈ (u(i+1, j) - 2u(i, j) + u(i-1, j))/dx².

Substituting these approximations into the equation, we get the finite difference equation as: (u(i, j+1) - 2u(i, j) + u(i, j-1))/dt² = β²(u(i+1, j) - 2u(i, j) + u(i-1, j))/dx².

Now, we can rearrange this equation to solve for u(i, j+1): u(i, j+1) = (β²dt²/dx²)(u(i+1, j) - 2u(i, j) + u(i-1, j)) + 2u(i, j) - u(i, j-1).

We can use this equation to iteratively solve for the displacement u at each spatial and temporal point. We start with an initial condition at j = 1, where u(i, 1) represents the displacement at the initial time. Then, we can use the finite difference equation to compute the displacement at j = 2, j = 3, and so on, until we reach the desired number of temporal points.