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.

Verified step by step guidance

1. 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.
2. Calculate the wavelength of the sound wave with the lowest frequency (30 s\(^{-1}\)) using the formula \( \lambda_{\text{low}} = \frac{344 \text{ m/s}}{30 \text{ s}^{-1}} \).
3. Calculate the wavelength of the sound wave with the highest frequency (1.5 \times 10^4 s\(^{-1}\)) using the formula \( \lambda_{\text{high}} = \frac{344 \text{ m/s}}{1.5 \times 10^4 \text{ s}^{-1}} \).
4. Determine the difference in wavelength between the two sounds by subtracting the wavelength of the highest frequency from the wavelength of the lowest frequency: \( \Delta \lambda = \lambda_{\text{low}} - \lambda_{\text{high}} \).
5. Express the final result for the difference in wavelength in meters.

Key Concepts

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.