For the harmonic wavelengths of a string fixed at both ends, lambda_n = 2L/n. Which statement is true as n increases?

• A. Lambda decreases as n increases.
• B. Lambda increases as n increases.
• C. Lambda stays the same for all n.
• D. Lambda becomes negative.

Answer: (A)

Higher harmonic numbers produce shorter wavelengths on a string fixed at both ends. Since the allowed wavelengths are lambda_n = 2L / n, increasing n makes lambda_n smaller. Physically, more nodes along the string mean a shorter distance between adjacent nodes, so a shorter half-wavelength fits in the same length. Because the wave speed on the string is set by tension and linear density, a shorter wavelength corresponds to a higher frequency, f_n = v / lambda_n, which increases with n. The wavelength cannot be negative, and it does not stay the same or grow with increasing n, so it must decrease as n grows.