Step 1: Planes waves.

For a homogeneous, isotropic, elastic medium, we find that the three-dimensional equation of motion can be decomposed into P-wave and S-wave components. Plane waves provide general solutions of the wave equation in which the displacement only varies in the direction of wave propagation. We denote the direction of wave propagation by the slowness vector s. The particle motion is denoted by the polarization vector A.

1. What is the relation between s and A in case of a P-wave. Give an example of a plane wave displacement field which represents a P-wave. Compute the divergence and rotation of this displacement field.

2. Now provide the answers to the questions in 1. in case of an S-wave.

If the medium properties change across an interface we find effects of reflection and refraction. The next part of this lab is to show Snell's law using plane wave solutions.

3. Assume that the displacement above the interface (in medium 1) is given by u₁(x, t) = A₁ cos (t-s₁ . x ).
Analogously, the displacement below the interface (in medium 2) is given by u₂(x, t) = A₂ cos (t-s₂ . x ).
Derive Snell's law using these expressions together with the condition of continuity of displacement at the interface.

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