Final answer:
For a 216-cm-long string fixed at both ends, the three longest wavelengths for standing waves are approximately 4.32 m, 2.16 m, and 1.44 m, corresponding to the fundamental and first two overtone modes respectively.
Step-by-step explanation:
The way to find the three longest wavelengths for standing waves on a string that is fixed at both ends is by considering the fundamental frequency and the overtone series. Resonance in a string fixed at both ends occurs for wavelengths that fit the condition n * λ = 2L, where n is a positive integer representing the mode number, λ is the wavelength, and L is the length of the string. For a string length of 216 cm (2.16 m), the three longest wavelengths correspond to the fundamental frequency (n=1) and the first two overtones (n=2 and n=3).
The first longest wavelength (n=1, fundamental) is: λ₁ = 2 * L = 2 * 2.16 m = 4.32 m.
The second longest wavelength (n=2, first overtone) would be half of the fundamental: λ₂ = L = 2.16 m.
The third longest wavelength (n=3, second overtone) would be a third of the fundamental: λ₃ = 2 * L / 3 = 4.32 m / 3 ≈ 1.44 m.