In a simple vibrating string, what happens to the frequency as string length increases?

In a simple vibrating string, what happens to the frequency as string length increases?

• A. The frequency increases
• B. The frequency decreases
• C. The frequency stays the same
• D. The frequency doubles

Answer: (B)

In a simple string fixed at both ends, the fundamental mode has a wavelength that is twice the length of the string, so λ = 2L. The wave speed v is determined by the tension and the string’s linear density (v = √(T/μ)). Frequency is the speed divided by wavelength, f = v/λ, which for the fundamental becomes f = v/(2L). Therefore, as the length L increases while tension and density stay the same, the frequency decreases. This matches the intuition that a longer string vibrates more slowly, producing a lower pitch.

In a simple string fixed at both ends, the fundamental mode has a wavelength that is twice the length of the string, so λ = 2L. The wave speed v is determined by the tension and the string’s linear density (v = √(T/μ)). Frequency is the speed divided by wavelength, f = v/λ, which for the fundamental becomes f = v/(2L). Therefore, as the length L increases while tension and density stay the same, the frequency decreases. This matches the intuition that a longer string vibrates more slowly, producing a lower pitch.