Normal Modes Visualization

Mode n = 0 (Uniform Translation)

y(x,t) = A₀ cos(ω₀t)

Frequency: ω₀ = 0 (no oscillation in this idealized case)

Physical meaning: The entire rope moves up and down as one unit. This mode only exists because the ends can slide freely!

Key feature: No nodes, constant displacement across the rope.

Mode n = 1 (Fundamental)

y(x,t) = A₁ cos(πx/L) cos(ω₁t)

Frequency: ω₁ = π√(T/μ)/L

Physical meaning: One half wavelength fits in the rope length.

Key feature: One antinode at center, horizontal at both ends.

Mode n = 2 (Second Harmonic)

y(x,t) = A₂ cos(2πx/L) cos(ω₂t)

Frequency: ω₂ = 2π√(T/μ)/L = 2ω₁

Physical meaning: One complete wavelength fits in the rope.

Key feature: One node at center, two antinodes, horizontal at ends.

🔑 Key Differences from Fixed-End Boundary Conditions

Horizontal at boundaries: The rope enters the rods horizontally (zero slope), shown by the dashed cyan lines.
Cosine spatial dependence: Only cos(nπx/L) terms appear, no sine terms.
n = 0 mode exists: Unlike fixed ends, the rope can translate uniformly up and down.
Maximum displacement at ends: For odd n modes, the ends have maximum displacement (antinodes), opposite to fixed-end conditions.
Energy consideration: No work is done at the boundaries since the force is perpendicular to the motion.