Question

A standing wave with a fundamental mode wavelength of 60 cm forms in an air column open at both ends. How long is the column for the fundamental mode?

Answer 1

The condition for the resonance frequency for open at both ends tube is the following:

$$f = \dfrac{nv}{2L}$$

where $n$ is the resonance mode (for fundamental mode $n = 1$), $v = 343 m/s$ is the speed of sound in air and $L$ is the length of the air column.

Expressing $L$, obtain:

$$L = \dfrac{nv}{2f}$$

The frequency of a standing wave can be calculated as following:

$$f = v/\lambda$$

where $\lambda = 0.6 m$ is the wavelength.

Thus, obtain:

$$L = \dfrac{nv}{2f} = \dfrac{nv}{2v/\lambda} = \dfrac{n\lambda}{2} = \dfrac{1\cdot 0.6}{2} = 0.3m$$

Answer. The lenght of the tube is 0.3m.