An old mining tunnel disappears into a hillside. You would like to know how long the tunnel is, but it's too dangerous to go inside. Recalling your recent physics class, you decide to try setting up standing-wave resonances inside the tunnel. Using your subsonic amplifier and loudspeaker, you find resonances at 4.5 Hz and 6.3 Hz, and at no frequencies between these. It's rather chilly inside the tunnel, so you estimate the sound speed to be 335 m/s . Based on your measurements, how far is it to the end of the tunnel?

Verified step by step guidance

1. Step 1: Understand the concept of standing-wave resonances. Standing waves occur when sound waves reflect back and forth in a confined space, creating nodes and antinodes. The resonant frequencies correspond to the natural frequencies of the tunnel, which depend on its length and the speed of sound.
2. Step 2: Recognize that the resonances at 4.5 Hz and 6.3 Hz represent consecutive harmonics. The difference between these frequencies gives the fundamental frequency of the tunnel. Calculate the fundamental frequency using the formula: \( f_1 = f_2 - f_1 \), where \( f_2 \) and \( f_1 \) are consecutive resonant frequencies.
3. Step 3: Use the relationship between the fundamental frequency, the speed of sound, and the length of the tunnel. The formula is \( f_1 = \frac{v}{2L} \), where \( v \) is the speed of sound and \( L \) is the length of the tunnel. Rearrange the formula to solve for \( L \): \( L = \frac{v}{2f_1} \).
4. Step 4: Substitute the given values into the formula. The speed of sound \( v \) is 335 m/s, and the fundamental frequency \( f_1 \) was calculated in Step 2. Perform the substitution to find \( L \).
5. Step 5: Interpret the result. The calculated \( L \) represents the distance to the end of the tunnel. Ensure the units are consistent and verify the calculation steps for accuracy.

Key Concepts

Standing Waves

Standing waves occur when two waves of the same frequency and amplitude traveling in opposite directions interfere with each other, creating nodes and antinodes. In a confined space like a tunnel, standing waves can form at specific frequencies, known as resonant frequencies, which depend on the length of the tunnel and the speed of sound in the medium.

Resonant Frequencies

Resonant frequencies are specific frequencies at which a system naturally oscillates with greater amplitude. In the context of the tunnel, the observed frequencies (4.5 Hz and 6.3 Hz) indicate the fundamental and harmonic modes of vibration, which can be used to determine the length of the tunnel based on the relationship between frequency, wavelength, and the speed of sound.

Speed of Sound

The speed of sound is the rate at which sound waves propagate through a medium, influenced by factors such as temperature and density. In this scenario, the estimated speed of sound is 335 m/s, which is crucial for calculating the wavelength associated with the resonant frequencies and ultimately determining the length of the tunnel using the formula: length = (n * wavelength) / 2, where n is the mode number.