Calculating Wavelengths: What Are the Wavelengths of Audible Sounds?

To find wavelength from frequency, we can use $v_{\text{w}}=\mathrm{f\lambda }$.

1. Identify knowns. The value for $v_{\text{w}}$, is given by
   $$v_{\text{w}}=\left(\text{331}\text{m/s}\right)\sqrt{\frac{T}{\text{273}\text{K}}}.$$

   17.5
2. Convert the temperature into kelvin and then enter the temperature into the equation
   $$v_{\text{w}}=\left(\text{331}\text{m/s}\right)\sqrt{\frac{\text{303 K}}{\text{273}\text{K}}}=\text{348}\text{.}7\text{m/s}.$$

   17.6
3. Solve the relationship between speed and wavelength for $\lambda$:
   $$\lambda =\frac{v_{\mathrm{w}}}{f}.$$

   17.7
4. Enter the speed and the minimum frequency to give the maximum wavelength:
   $${\lambda }_{\text{max}}=\frac{\text{348}\text{.}7\text{m/s}}{\text{20 Hz}}=\text{17}\text{m}.$$

   17.8
5. Enter the speed and the maximum frequency to give the minimum wavelength:
   $${\lambda }_{\text{min}}=\frac{\text{348}\text{.}7\text{m/s}}{\text{20}\text{,}\text{000 Hz}}=0\text{.}\text{017}\text{m}=1\text{.}\text{7 cm}.$$

   17.9

Because the product of $f$ multiplied by $\lambda$ equals a constant, the smaller $f$ is, the larger $\lambda$ must be, and vice versa.