Final answer:
The problem deals with adjusting the length of a stopped pipe to match its fundamental frequency with the second overtone of a taut wire, causing resonance. It involves understanding the wave relationships for string harmonics and sound propagation in air to achieve resonance. The speed of sound and string properties like tension and mass per unit length are crucial to finding the resonant conditions.
Step-by-step explanation:
Understanding Resonance in Pipes and Strings
When a stopped pipe is adjusted to its fundamental frequency, it can induce resonance in a nearby wire or string. The condition for resonance is that the frequency of the pipe matches one of the natural frequencies (harmonics) of the string. For a string under tension, the frequency of its harmonics can be found using the wave relationship f = √(T/μ) / (2L), where T is the tension, μ is the mass per unit length of the string, and L is the length of the string.
The speed of sound is also essential in determining the resonant condition of the pipe. In air, the speed of sound is approximately 344 m/s, which affects the wavelength and frequency of the resonating sound wave. A tube closed at one end has a fundamental frequency (n = 1 mode) where the wavelength is four times the length of the tube, and harmonics at frequencies where the wavelength fits into the length of the tube with additional nodes and antinodes.
Therefore, to achieve the second overtone in the string (which corresponds to the third harmonic for a vibrating string, since the second overtone is the third mode of vibration), the pipe's fundamental frequency must match the frequency of the string's second overtone. By using wave relationships, we can calculate this correct adjustment for the pipe length and subsequently determine the string's resonant behavior.