A student was investigating the difference in wavelength of a bass guitar and a whistle. The student noticed that the frequency of the whistle was high, and the bass guitar had a low frequency. Given the students' information and the model below, what is the difference in wavelength for each instrument and explain how frequency affects the wavelength?

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Answer 1

To understand the difference in wavelength between the bass guitar and the whistle, we first need to recall the relationship between frequency, wavelength, and the speed of sound. The formula that describes this relationship is:

\[ v = f \times \lambda \]

where:

• \( v \) is the speed of sound (approximately 343 meters per second in air at room temperature),
• \( f \) is the frequency (in Hertz),
• \( \lambda \) is the wavelength (in meters).

1. Bass Guitar: Typically, a bass guitar produces low frequencies, usually in the range of 40 Hz to 400 Hz. Let's assume a frequency of 100 Hz for our calculations.

   \[ \lambda_{\text{bass}} = \frac{v}{f} = \frac{343 , \text{m/s}}{100 , \text{Hz}} = 3.43 , \text{m} \]

2. Whistle: A whistle generally produces high frequencies, which can range from 1 kHz (1000 Hz) and higher. Let’s assume a frequency of 2000 Hz for our calculations.

   \[ \lambda_{\text{whistle}} = \frac{v}{f} = \frac{343 , \text{m/s}}{2000 , \text{Hz}} = 0.1715 , \text{m} \]

3. Difference in Wavelength:

   • Wavelength of the bass guitar: \( 3.43 , \text{m} \)
   • Wavelength of the whistle: \( 0.1715 , \text{m} \)

   The difference in wavelength can be calculated as follows:

   \[ \Delta \lambda = \lambda_{\text{bass}} - \lambda_{\text{whistle}} = 3.43 , \text{m} - 0.1715 , \text{m} \approx 3.2585 , \text{m} \]

4. Explanation of How Frequency Affects Wavelength:

   The wavelength of a sound wave is inversely proportional to its frequency. This means that as the frequency increases, the wavelength decreases. Conversely, lower frequencies result in longer wavelengths.

   • For the bass guitar, the low frequency (100 Hz) results in a long wavelength (3.43 m), allowing it to produce deep sounds.
   • For the whistle, the high frequency (2000 Hz) leads to a much shorter wavelength (0.1715 m), producing a sharp, high-pitched sound.

In summary, the student observed that the bass guitar has a much longer wavelength compared to the whistle because it has a lower frequency. The difference in their wavelengths is approximately \( 3.2585 , \text{m} \), illustrating how frequency and wavelength are interconnected in sound production.