A source emits sound of wavelengths 2.54 m and 2.72 m in air. How many beats per second will be heard? (Assume T = 20°C.)

검증된 단계별 안내

Determine the formula for the frequency of a sound wave: \( f = \frac{v}{\lambda} \), where \( f \) is the frequency, \( v \) is the speed of sound in air, and \( \lambda \) is the wavelength.

At \( T = 20^{\circ}C \), the speed of sound in air is approximately \( v = 343 \, \text{m/s} \). Use this value to calculate the frequencies corresponding to the two given wavelengths: \( \lambda_1 = 2.54 \) m and \( \lambda_2 = 2.72 \) m.

Substitute \( \lambda_1 \) into the formula \( f_1 = \frac{v}{\lambda_1} \) to calculate the first frequency \( f_1 \). Similarly, substitute \( \lambda_2 \) into the formula \( f_2 = \frac{v}{\lambda_2} \) to calculate the second frequency \( f_2 \).

The beat frequency is the absolute difference between the two frequencies: \( f_{\text{beat}} = |f_1 - f_2| \).

Substitute the calculated values of \( f_1 \) and \( f_2 \) into the formula for \( f_{\text{beat}} \) to determine the number of beats per second.

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주요 개념

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Wavelength and Frequency Relationship

The relationship between wavelength and frequency is described by the equation v = fλ, where v is the speed of sound in air, f is the frequency, and λ is the wavelength. As the wavelength increases, the frequency decreases, and vice versa. This concept is crucial for determining the frequencies of the sound waves emitted by the source.

Beats Frequency

Beats occur when two sound waves of slightly different frequencies interfere with each other. The beat frequency is calculated as the absolute difference between the two frequencies: f_beat = |f1 - f2|. This concept is essential for solving the problem, as it allows us to find how many beats per second will be heard from the two emitted wavelengths.

Speed of Sound in Air

The speed of sound in air at 20°C is approximately 343 meters per second. This value is important for calculating the frequencies of the sound waves from their respective wavelengths. Knowing the speed of sound allows us to apply the wavelength-frequency relationship to find the frequencies needed to determine the beat frequency.