Solved Problem on Acoustics

The audible sounds by the human ear result from vibratory movements whose frequencies are between 16 Hz and 20000 Hz. Knowing that at 20 °C, the speed of propagation of sound in water is 1450 m/s and in air 344 m/s. Calculate:
a) The wavelengths corresponding to the maximum and minimum frequencies in air and water;
b) The ratio between the wavelengths in water and air.

Problem data:

Minimum audible frequency: f_min = 16 Hz;
Maximum audible frequency: f_max = 20000 Hz;
Speed of sound in water 20°C: v_w = 1450 m/s;
Speed of sound in air at 20 °C: v_a = 344 m/s;
Ambient temperature: T = 20 °C.

Solution

a) We use the expression for the speed of sound as a function of frequency and wavelength

\[ \begin{gather} \bbox[#99CCFF,10px] {v=\lambda f} \end{gather} \]

for the wavelength, we write

\[ \begin{gather} \lambda =\frac{v}{f} \end{gather} \]

For the air:

Maximum wavelength (when the frequency is minimum)

\[ \begin{gather} \lambda_{a max}=\frac{344}{16} \end{gather} \]

\[ \begin{gather} \bbox[#FFCCCC,10px] {\lambda_{a max}=21.5\;\text{m}} \end{gather} \]

Minimum wavelength (when the frequency is maximum)

\[ \begin{gather} \lambda_{a min}=\frac{344}{20000} \end{gather} \]

\[ \begin{gather} \bbox[#FFCCCC,10px] {\lambda_{a min}=0.0172\;\text{m}} \end{gather} \]

For the water:

Maximum wavelength (when the frequency is minimum)

\[ \begin{gather} \lambda_{w max}=\frac{1450}{16} \end{gather} \]

\[ \begin{gather} \bbox[#FFCCCC,10px] {\lambda_{w max}=90.625\;\text{m}} \end{gather} \]

Minimum wavelength (when the frequency is maximum)

\[ \begin{gather} \lambda_{w min}=\frac{1450}{20000} \end{gather} \]

\[ \begin{gather} \bbox[#FFCCCC,10px] {\lambda_{w min}=0.0725\;\text{m}} \end{gather} \]

b) Using the maximum wavelengths of water and air the ratio will be

\[ \begin{gather} \frac{\lambda_{w max}}{\lambda_{a max}}=\frac{90.625}{21.5} \end{gather} \]

\[ \begin{gather} \bbox[#FFCCCC,10px] {\frac{\lambda_{a max}}{\lambda_{a max}}=4.2} \end{gather} \]

Note: If the minimum wavelengths were used, the result would be the same.

Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .