Boundary condition: Zero slope (horizontal) at both ends
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.
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.
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.