The speed of sound in air is 344 m/s at room temperature. The lowest frequency of a large organ pipe is 30 s^–1 and the highest frequency of a piccolo is 1.5×10⁴ s^–1. Find the difference in wavelength between these two sounds.

검증된 단계별 안내

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

Calculate the wavelength of the sound wave with the lowest frequency (30 s\(^{-1}\)) using the formula \( \lambda_{\text{low}} = \frac{344 \text{ m/s}}{30 \text{ s}^{-1}} \).

Calculate the wavelength of the sound wave with the highest frequency (1.5 \(\times\) 10^4 s\(^{-1}\)) using the formula \( \lambda_{\text{high}} = \frac{344 \text{ m/s}}{1.5 \times 10^4 \text{ s}^{-1}} \).

Determine the difference in wavelength between the two sounds by subtracting the wavelength of the highest frequency from the wavelength of the lowest frequency: \( \Delta \lambda = \lambda_{\text{low}} - \lambda_{\text{high}} \).

Express the final result for the difference in wavelength in meters.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Wave Speed

Wave speed is the distance traveled by a wave per unit time. In the context of sound, it is determined by the medium through which it travels, such as air. The speed of sound in air at room temperature is approximately 344 m/s, which is essential for calculating the wavelength of sound waves using the formula: wavelength = speed/frequency.

Frequency

Frequency is the number of cycles of a wave that pass a point in one second, measured in hertz (Hz). In this question, the lowest frequency of the organ pipe is 30 Hz, and the highest frequency of the piccolo is 15,000 Hz. Frequency is inversely related to wavelength; as frequency increases, wavelength decreases, which is crucial for understanding the difference in wavelengths of the two sounds.

Wavelength

Wavelength is the distance between successive crests (or troughs) of a wave. It can be calculated using the formula: wavelength = speed/frequency. For the organ pipe and piccolo, calculating their respective wavelengths will allow us to find the difference in wavelength, which is the main goal of the question. Understanding how to manipulate this formula is key to solving the problem.

The speed of sound in air is 344 m/s at room temperature. The lowest frequency of a large organ pipe is 30 s–1 and the highest frequency of a piccolo is 1.5×104 s–1. Find the difference in wavelength between these two sounds.