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Chapter 12: Problem 59

At what frequency \(f\) does a sound wave in air have a wavelength of $15 \mathrm{cm},$ about half the diameter of the human head? Some methods of localization work well only for frequencies below \(f\), while others work well only above \(f\). (See Conceptual Questions 4 and 5 .)

Short Answer

Expert verified
Answer: The frequency of the sound wave is approximately 2287 Hz.

Step by step solution

01

Convert wavelength to meters

Since the given wavelength is 15 cm, we need to convert it to meters: \(\lambda = 15 \mathrm{cm} = 0.15 \,\mathrm{m}\). This will allow us to use the wavelength in the formula with the correct units.
02

Find the velocity of sound in air

The velocity of sound in air, at room temperature (20°C), is approximately 343 m/s. Keep in mind that the velocity of sound depends on the temperature.
03

Use the formula to find the frequency

We know the velocity of sound in air, \(v=343 \, \mathrm{m/s}\), and the wavelength, \(\lambda = 0.15 \, \mathrm{m}\). We will plug in these values into the formula \(v = f\lambda\) and solve for \(f\): $$ f = \frac{v}{\lambda} = \frac{343 \, \mathrm{m/s}}{0.15 \, \mathrm{m}} \approx 2287 \, \mathrm{Hz} $$
04

Interpret the result

The sound wave in air has a frequency of approximately 2287 Hz when its wavelength is 15 cm (0.15 m). Some methods of localization work well only for frequencies below 2287 Hz, while others work well only above 2287 Hz.

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