Calculating Wavelength of a Tuning Fork in a Resonant Tube

Introduction to Tuning Forks and Resonance

A tuning fork is a device commonly used to produce a precise frequency of sound. When struck, it resonates with a specific pitch, which can be observed in the airwaves it produces. These sound waves can then be observed in a medium such as an air tube, where certain lengths allow for resonant standing waves to form. This phenomenon is known as resonance.

Understanding Resonance in Air Tubes

Resonance in a closed or open-air tube is dependent on the ends of the tube. In a closed pipe, the first resonance occurs at a length of (frac{1}{4}) of the wavelength, while an open pipe resonates at a length that is (frac{1}{2}) of the wavelength. The formula for the wavelength ((lambda)) is given by:

(lambda frac{v}{f})

where (v) is the speed of sound in the medium (typically air), and (f) is the frequency of the sound wave produced by the tuning fork. For air at standard temperature and pressure, the speed of sound ((v)) is approximately 343 meters per second.

Calculating Wavelength from Given Resonance Lengths

In the problem at hand, two resonant lengths are provided: 40 cm and 85 cm. Let's calculate the wavelength. The difference in these resonant lengths is:

85 cm - 40 cm 45 cm

This difference represents the length of one full wavelength ((lambda)). Hence, we have:

(lambda 45 , text{cm})

Adjusting for End Correction

To get a more accurate measurement, one must consider end correction, which is the distance from the theoretical end of the tube to the actual end where standing waves form. This correction is particularly important for open tubes, as air can leak around the open end, affecting the actual length of the standing wave formed.

Final Calculation

Given that the difference in the lengths is 45 cm, and assuming no significant end correction, the wavelength of the tuning fork can be calculated as follows:

(lambda 45 , text{cm})

With the speed of sound in air ((v 343 , text{m/s})), if the frequency of the tuning fork is (f), then: (f frac{343}{0.45} approx 762.22 , text{Hz})

This calculation uses the relationship between speed, wavelength, and frequency: (v lambda f).

Conclusion

The wavelength of the tuning fork can be calculated based on the resonant lengths observed in the air tube. The difference in the lengths at which resonance occurs (40 cm and 85 cm) gives a direct measurement of the wavelength. Taking into account end correction and the speed of sound in air, a more accurate frequency can be determined.