Question

Find the resonance frequency of the ear canal. Treat it as a half-open pipe of diameter \(8.0 \mathrm{~mm}\) and length \(25 \mathrm{~mm}\). Assume that the temperature inside the ear canal is body temperature \(\left(37^{\circ} \mathrm{C}\right)\).

Short answer

Answer: The resonance frequency of the ear canal at body temperature (37°C) is 3490.2 Hz.

Step-by-step solution

Convert temperature to Kelvin

First, we need to convert the given temperature from Celsius to Kelvin. To do this, we add 273.15 to the Celsius temperature: $$ T_K = T_C + 273.15 $$ Substitute the given temperature. $$ T_K = 37^{\circ} \mathrm{C} + 273.15 = 310.15 \mathrm{~K} $$ The temperature inside the ear canal is 310.15 K.

Calculate the speed of sound inside the ear canal

Next, we will calculate the speed of sound (v) inside the ear canal using the ideal gas law and the following formula: $$ v = \sqrt{\frac{\gamma RT}{M}} $$ Here, \(\gamma\) is the adiabatic index for air (1.4), R is the specific gas constant for air (287 J/kg·K), T is the temperature in Kelvin, and M is the molar mass of air (0.029 kg/mol). Substitute the known values. $$ v = \sqrt{\frac{1.4 \times 287 \mathrm{~J/kg \cdot K} \times 310.15 \mathrm{~K}}{0.029 \mathrm{~kg/mol}}} = 349.02 \mathrm{~m/s} $$ The speed of sound in the ear canal is 349.02 m/s.

Calculate the fundamental frequency of the half-opened pipe

Now, we will calculate the fundamental frequency of the half-opened pipe using its length (L) and the speed of sound (v) in the following formula: $$ f_1 = \frac{v}{4L} $$ Given the length of the ear canal is 25 mm, we need to convert this to meters: $$ L_m = 25 \mathrm{~mm} \times \frac{1 \mathrm{~m}}{1000 \mathrm{~mm}} = 0.025 \mathrm{~m} $$ Substitute the values for v and L_m: $$ f_1 = \frac{349.02 \mathrm{~m/s}}{4 \times 0.025 \mathrm{~m}} = 3490.2 \mathrm{~Hz} $$ The fundamental frequency of the half-opened pipe is 3490.2 Hz.

Present the final result for the resonance frequency

The resonance frequency of the ear canal, which is treated as a half-opened pipe, is 3490.2 Hz at body temperature (37°C).