Visualize the reflection and refraction of elastic waves in a 1D heterogeneous elastic medium, building upon the program constructed in the lab meeting.

The simulation models a heterogeneous elastic medium consisting of 100 coupled oscillators with different material properties in two distinct regions.

1. Medium Structure: 100 oscillators total

   • First 50 oscillators: Mass m1, connected by springs with constant k1
   • Next 50 oscillators: Mass m2, connected by springs with constant k2

2. Boundary Conditions:

   • Right-hand end: Fixed boundary (eta_d = 0)
   • Left-hand end: Driven by sinusoidal pulse input

3. Input Signal:

   • Single "semi-wave" pulse of sinusoidal type
   • eta_s grows sinusoidally from t = 0 to maximum amplitude
   • Then it decreases to zero and remains zero for the rest of the simulation
   • Represents one half-period of harmonic oscillation

4. Wave Propagation Visualization:

   • Pulse propagation toward the medium interface
   • Formation and propagation of reflected pulse
   • Formation and propagation of refracted (transmitted) pulse

5. Analysis Requirements:

   • Understand the relationship between wave velocity and material properties (m and k)
   • Observe wave behavior at the material interface
   • Analyze reflection and transmission coefficients

• Create a 100-oscillator array with heterogeneous properties
• Define the material interface at oscillator 50
• Set appropriate initial conditions (all oscillators at rest)

• Left boundary: Implement sinusoidal pulse driver
• Right boundary: Fix displacement (eta_d = 0)

• Generate a single semi-wave sinusoidal pulse
• Control pulse duration and amplitude
• Ensure smooth pulse injection

• Implement equations of motion for coupled oscillators
• Account for different masses and spring constants
• Use an appropriate numerical integration method
• Handle interface conditions properly

• Real-time animation of oscillator displacements
• Track pulse propagation, reflection, and refraction
• Plot displacement vs. position over time
• Include material property indicators

• Calculate wave velocities in each medium
• Determine reflection and transmission coefficients
• Verify theoretical relationships: v = √(k/m)

The wave velocity in each medium is given by:

v₁ = √(k₁/m₁)  (first medium)
v₂ = √(k₂/m₂)  (second medium)

• Reflection: Portion of the wave bounces back into the first medium
• Refraction: Portion of the wave transmits into a second medium
• Impedance matching: Determines reflection/transmission ratios

• Reflection coefficient: R = (Z₂ - Z₁)/(Z₂ + Z₁)
• Transmission coefficient: T = 2Z₂/(Z₂ + Z₁)
• Where Z = √(mk) is the mechanical impedance

• Initial pulse propagation in the first medium
• Partial reflection at the interface (pulse traveling leftward)
• Partial transmission into a second medium (possibly different velocity)
• Multiple reflections from the fixed right boundary

• Measure wave velocities in both media
• Compare with theoretical predictions
• Analyze amplitude ratios of reflected/transmitted waves

• Choose an appropriate time step for stability
• Handle discontinuity at the material interface
• Ensure energy conservation (within numerical limits)

• Color coding for different media regions
• Real-time displacement plotting
• Optional: Energy density visualization
• Interface position marking

• Choose m₁, m₂, k₁, k₂ for clear wave behavior demonstration
• Pulse frequency should be appropriate for the system response
• Simulation duration should capture multiple reflections

• Clear visualization of incident, reflected, and transmitted pulses
• Quantitative agreement with theoretical wave velocities
• Demonstration of wave impedance effects at the interface
• Understanding of the relationship between material properties and wave behavior