Question

A soprano sings the note \(C 6(1046 \mathrm{~Hz})\) across the mouth of a soda bottle. To excite a fundamental frequency in the soda bottle equal to this note, describe how far the top of the liquid must be below the top of the bottle.

Short answer

Answer: The liquid's surface should be placed 8.20 cm below the top of the bottle to excite a fundamental frequency equal to the sung note C6 (1046 Hz).

Step-by-step solution

Calculate the wavelength of the sound wave

To find the wavelength of the sound wave, we need to know the speed of sound in air. The speed of sound (\(v\)) is approximately \(343 \frac{m}{s}\) at room temperature. Given the frequency (\(f\)) of the sound wave, we can use the formula \(v = f\lambda\) to calculate the wavelength (\(\lambda\)) of the sound wave. Rearranging the formula for wavelength, \[\lambda = \frac{v}{f}\] We are given the frequency \(f = 1046 \mathrm{~Hz}\), and the speed of sound \(v = 343 \frac{m}{s}\). So, \[\lambda = \frac{343 \frac{\mathrm{m}}{\mathrm{s}}}{1046 \mathrm{~Hz}}\]

Calculate the wavelength of the sound wave

Now we substitute the given values of \(f\) and \(v\) in the formula, \[\lambda = \frac{343 \frac{\mathrm{m}}{\mathrm{s}}}{1046 \mathrm{~Hz}} \approx 0.3279 \mathrm{~m}\] This is the wavelength of the sound wave produced by the singing soprano.

Calculate the length of the air column in the bottle

To find the length of the air column inside the bottle that would resonate at this given frequency, we need to consider the fact that, for a closed-end air column, the length corresponds to one-fourth the wavelength. For an air column with one closed end (such as a bottle with liquid inside), the length (\(L\)) is given by: \[L = \frac{1}{4} \lambda\] Here, the length \(L\) is the distance from the top of the liquid to the top of the bottle. Now substitute the calculated wavelength value: \[L = \frac{1}{4} \cdot 0.3279 \mathrm{~m} \approx 0.0820 \mathrm{~m}\]

Convert the length to centimeters

Finally, we convert the calculated length of the air column in meters to centimeters for a more practical representation of the distance: \[L = 0.0820 \mathrm{~m} \cdot \frac{100 \mathrm{~cm}}{1 \mathrm{~m}} = 8.20 \mathrm{~cm}\] Thus, the liquid's surface should be \(8.20 \mathrm{~cm}\) below the top of the bottle in order to excite a fundamental frequency equal to the sung note \(C 6(1046 \mathrm{~Hz})\).

Key concepts

Resonance

Resonance occurs when an object or system vibrates at a particular frequency with larger amplitude. This happens when the frequency of the externally applied vibrations matches the natural frequency of the system. In the context of acoustics, resonance is crucial to understanding how certain frequencies get amplified. For example, if a singer hits a note, the air inside a bottle can resonate and create a loud, clear sound if the frequency of the note matches the bottle's fundamental frequency. This is what happens in the described problem when a soprano sings a note across a soda bottle. The air column inside the bottle resonates at the frequency of the sung note if the lengths and conditions are just right, enhancing the sound as it reflects and builds upon itself within the bottle's confines.

Wavelength

Wavelength is the distance between successive crests of a wave, especially in sound or light. In acoustics, understanding wavelength is key to determining how waves behave in different environments. A wavelength (\(\lambda\)) is tied to both the speed of sound and the frequency of the wave through the equation:

• \[ v = f\lambda \]

Where \(v\) is the speed of sound and \(f\) is the frequency. For the note \(C 6(1046\;\mathrm{Hz})\) sung across the soda bottle, the wavelength is calculated to be approximately \(0.3279\;\mathrm{m}\). This wavelength determines how the wave fits within the bottle and is essential for achieving resonance. By understanding the wavelength, you can predict the conditions needed for resonance in closed or open air columns.

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system naturally vibrates. In the case of the soda bottle, this is the frequency at which the air column vibrates when a minimal amount of energy is introduced. It is determined by the physical dimensions of the bottle and the characteristics of the air inside. For a bottle partially filled with liquid, the air column acts like a tube closed at one end, which means its fundamental frequency corresponds to a quarter of a wavelength fitting within the tube's length. This resonant frequency amplifies the sound at this pitch, which is precisely what the soprano is aiming to match with her singing.

Speed of Sound

The speed of sound is a measure of how quickly sound waves travel through a medium. It is affected by the medium itself (air, water, etc.) and also its temperature. For this exercise, sound travels through air at a speed of approximately \(343\;\frac{m}{s}\) at room temperature. Knowing the speed of sound is vital for calculating the wavelength of a sound wave and thus for determining the wavelengths and frequencies that will resonate in a given space. These concepts are interconnected and are essential to understanding how sound behaves in various conditions, including the resonance in a soda bottle when a specific note is sung across its mouth. The precise calculation of the sound speed assists in accurately predicting and achieving resonance.