Question

One cold and windy winter day. Zach notices a humming sound coming from his chimney when the chimney is open at the top and closed at the bottom. He opens the chimney at the bottom and notices that the sound changes. He goes over to the piano to try to match the note that the chimney is producing with the bottom open. He finds that the "C" three octaves below middle "C" matches the chimney's fundamental frequency. Zach knows that the frequency of middle "C" is \(261.6 \mathrm{Hz}\), and each lower octave is \(\frac{1}{2}\) of the frequency of the octave above. From this information, Zach finds the height of the chimney and the fundamental frequency of the note that was produced when the chimney was closed at the bottom. Assuming that the speed of sound in the cold air is \(330 \mathrm{m} / \mathrm{s}\), reproduce Zach's calculations to find (a) the height of the chimney and (b) the fundamental frequency of the chimney when it is closed at the bottom.

Short answer

The height of the chimney is approximately 5.04 meters, and the fundamental frequency with the chimney closed at the bottom is about 16.35 Hz.

Step-by-step solution

Determine the frequency of the note matched

Identify the frequency of the note Zach matched with the chimney's sound. Middle "C" is 261.6 Hz, and each lower octave halves this frequency. The note three octaves below middle "C" can be calculated as: \[ f = \frac{261.6}{2^3} = \frac{261.6}{8} = 32.7 \text{ Hz} \] Thus, the frequency of the note Zach matched is 32.7 Hz.

Relate frequency to the speed of sound and the height of the chimney

We know that the chimney behaves like an open-ended air column with fundamental frequency given by \( f = \frac{v}{2L} \), where \( v \) is the speed of sound (330 m/s), and \( L \) is the height of the chimney. Rearrange to solve for \( L \): \[ L = \frac{v}{2f} = \frac{330}{2 \times 32.7} = \frac{330}{65.4} \approx 5.04 \text{ meters} \] So, the height of the chimney is approximately 5.04 meters.

Calculate the fundamental frequency with the chimney closed at the bottom

When the chimney is closed at the bottom, it acts as a closed-end air column. The fundamental frequency is given by \( f = \frac{v}{4L} \). Substitute \( L \) from the previous calculation: \[ f = \frac{330}{4 \times 5.04} \approx \frac{330}{20.16} \approx 16.35 \text{ Hz} \] Thus, the fundamental frequency of the chimney when it is closed at the bottom is approximately 16.35 Hz.

Key concepts

Chimney Resonance

Chimney resonance is a fascinating occurrence where sound waves resonate within a chimney. When you hear a humming or whistling sound from a chimney on a breezy day, it's caused by air vibrating in the chimney column. This can happen in chimneys because they act as natural resonators due to their tube-like shape. Different lengths and openings influence how these sound waves bounce inside the chimney.

To understand this, think of the chimney like a musical instrument. When air moves through, it creates sound waves that bounce back and forth. The specific frequencies that resonate depend on the chimney's height and whether the openings are open or closed. These frequencies are what we call the chimney's natural resonances, and each has a "fundamental frequency"—the lowest frequency that can sustain a stable sound wave in that space.

Speed of Sound

The speed of sound is crucial for understanding resonance in chimneys and other air columns. It tells us how fast sound waves travel through the air. Typically, the speed of sound in dry air at room temperature is about 343 meters per second. However, this speed can change with temperature and other conditions like humidity or air pressure.

In this exercise, it's noted that the speed of sound in the cold winter air is 330 meters per second. The speed plays a key role in calculating resonant frequencies because the formulae used involve dividing this speed by a constant related to the chimney's dimensions. Essentially, the speed of sound dictates how quickly sound waves can reflect within the space, thus affecting the resonant frequency.

Open and Closed Air Columns

Air columns can be "open" or "closed," and this status affects their resonant frequencies. An open air column has both ends open, allowing air to move freely through it. In contrast, a closed air column has one end sealed, which changes how sound waves reflect and resonate.

For chimneys, when they are open at the top and closed at the bottom, they act like a closed air column. This scenario produces a specific set of resonant frequencies distinct from when they are entirely open. With a closed end, only odd harmonics are allowed, and the fundamental frequency is lower because only a quarter of the wavelength fits into the column at a time.

• Open air column: Resembles a flute or organ pipe open at both ends.
• Closed air column: Similar to a bottle, open at one end and closed at the other.

Understanding the difference helps in predicting and calculating the resonant frequencies correctly.

Frequency Calculation

Calculating frequencies involves using formulas based on whether the column is open-ended or closed. These formulas enable us to determine the fundamental frequency, which is crucial for understanding how sound behaves in different structures.

For open columns:- The formula used is \( f = \frac{v}{2L} \) where \( f \) is frequency, \( v \) is the speed of sound, and \( L \) is the length of the column.

For closed columns:- The formula changes to \( f = \frac{v}{4L} \) because only odd-numbered harmonics are possible.

In the example with the chimney, with a known speed of sound and using Zach's measurement, we calculated the height of the chimney and the fundamental frequency when it's closed at the bottom. With sound resonant phenomena, mastering these calculations allows one to predict how sound will interact with various lengths and ends of columns.