Question

The clarinet is well‑modeled as a cylindrical pipe that is open at one end and closed at the other. Find the wavelength and frequency of the third normal mode of vibration of a clarinet's air column with effective length of 0.393 m. Take 344 m/s for the speed of sound inside the instrument.

Answer 1

The wavelength of the third normal mode of vibration for a clarinet's air column with an effective length of 0.393 m is 1.572 m, and the corresponding frequency is 218.7 Hz.

Step-by-step explanation:

The third normal mode of vibration for a clarinet's air column, with an effective length of 0.393 m and at a speed of sound of 344 m/s, corresponds to the third harmonic. In a closed-open cylindrical pipe, the wavelength of the third harmonic is equal to four times the length of the pipe. Thus, the wavelength of the third harmonic is 4 * 0.393 m = 1.572 m.

Using the formula v = f * λ, where v is the speed of sound and λ is the wavelength, we can rearrange the formula to solve for the frequency (f): f = v / λ. Plugging in the values, the frequency of the third normal mode is 344 m/s / 1.572 m = 218.7 Hz.