Which statement is true about f_n = n f_1 for a string's harmonic series?

Which statement is true about f_n = n f_1 for a string's harmonic series?

• A. f_n is inversely proportional to n.
• B. f_n equals f_1 for all n.
• C. f_n equals n times the fundamental frequency f_1.
• D. f_n is independent of n.

Answer: (C)

Frequencies of harmonics on a string fixed at both ends scale linearly with the mode number. The allowed wavelengths are λn = 2L/n, and with a constant wave speed v the frequency is f_n = v/λn = v/(2L/n) = n(v/2L) = n f1, where f1 = v/(2L) is the fundamental frequency. So the nth harmonic is simply n times the fundamental. This isn’t inversely proportional to n, nor is it independent of n or equal to f1 for all n.

Frequencies of harmonics on a string fixed at both ends scale linearly with the mode number. The allowed wavelengths are λn = 2L/n, and with a constant wave speed v the frequency is f_n = v/λn = v/(2L/n) = n(v/2L) = n f1, where f1 = v/(2L) is the fundamental frequency. So the nth harmonic is simply n times the fundamental. This isn’t inversely proportional to n, nor is it independent of n or equal to f1 for all n.