Question

The speed of sound in dry air at 20 °C is 343 m/s and the lowest frequency sound wave that the human ear can detect is approximately 20 Hz. (a) What is the wavelength of such a sound wave?

Answer 1

`17.15\; {\rm m}`.

Step-by-step explanation:

The question is asking for the wavelength
`\lambda` of this wave. Wavelength is the distance that the wave travels within each cycle.

The frequency
`f` of a wave is the number of cycles completed in unit time. For example,
`f = 20\; {\rm Hz} = 20\; {\rm s^(-1)}` in this question, meaning that this wave completes
`20` cycles in each second.

The period
`T` of the wave is the time it takes to complete each cycle. Since the wave in this question completes
`20` cycles in
`1\; {\rm s}`, each cycle would have taken
`(1 / 20) \; {\rm s}`. In general, period is the inverse reciprocal of frequency:
`T = (1/f)`.

To find the distance
`\lambda` this wave travels in each cycle, multiply the speed of the wave
`v` by the time
`T` it takes to complete each cycle:

`\begin{aligned} \lambda &= v\, T \\ &= (v)/(f) \\ &= \frac{343\; {\rm m\cdot s^(-1)}}{20\; {\rm s^(-1)}} \\ &= 17.15\; {\rm m} \end{aligned}`.

Thus, the wavelength of this wave would be
`17.15\; {\rm m}`.