Question

A well with vertical sides and water at the bottom resonates at \(7.20 \mathrm{~Hz}\) and at no lower frequency. The air in the well has a density of \(1.21 \mathrm{~kg} / \mathrm{m}^{3}\) and a bulk modulus of \(1.41 \times 10^{5} \mathrm{~Pa}\). How deep is the well?

Short answer

The final calculation from the last step provides the depth of the well, given the resonance frequency and the properties of the air in the well.

Step-by-step solution

Identifying the relevant formula

The formula to recall to solve this problem is \(v = f \lambda\), where \(v\) is the velocity of the wave, \(f\) is the frequency, and \(\lambda\) is the wavelength. In this situation, the wave is the sound wave in the air column of the well.

Calculating the speed of sound in the air

Before using the wave equation, we need to find the speed of sound in the air using the formula \(v = \sqrt{\frac{B}{\rho}}\), where \(B\) is the bulk modulus of the air and \(\rho\) is the density of the air. Substituting the given values \(B = 1.41 \times 10^{5} Pa\) and \(\rho = 1.21 kg/m^{3}\), the speed of sound in the air can be calculated.

Determining the wavelength

Since the well is open at one end and closed at the other end, the air column forms a tube with one closed end. The resonating wave is thus the fundamental or first harmonic, which is a wave with a node at the bottom and an antinode at the top, implying the length of the well is the same as the length of the wave, i.e., the depth of the well equals one-fourth of the wavelength. Thus, to obtain the wavelength, one could manipulate the wave speed equation to \(\lambda = \frac{v}{f}\). With the calculated speed and the given frequency \(f = 7.20 Hz\), we can determine the wavelength.

Calculating the depth of the well

Recall from Step 3 that the depth of the well, \(d\), equals a quarter of the wavelength. So, the depth can be calculated by taking the calculated wavelength and dividing it by 4.

Key concepts

Wave Mechanics

Understanding wave mechanics is essential to grasp how sound travels through different media, like the air in a well. Sound waves are a type of longitudinal wave, where the particles of the medium move parallel to the direction of wave propagation. In our exercise, as sound waves travel through the air column in the well, they reflect off the boundaries, creating standing waves with nodes and antinodes.

When these waves resonate at a certain frequency, specific patterns called harmonics are formed. In the case of the well in the exercise, the air column resonates at its fundamental frequency, creating a standing wave with one node at the water surface and an antinode at the top. This behavior is dictated by the wave mechanics principle that the length of the resonating air column correlates to specific wave harmonics.

Frequency of Sound Waves

Frequency of sound waves refers to the number of vibrations or cycles that occur per second, measured in Hertz (Hz). The frequency of sound is a critical factor in determining the pitch of the sound. In our exercise, the well vibrates at a frequency of 7.20 Hz, which means there are 7.20 cycles of the sound wave in one second. This is the fundamental frequency of the well's air column — the lowest frequency at which the air column can resonate.

When a sound wave resonates at the fundamental frequency, the maximum energy is transferred, and the sound is the loudest. The frequency is inversely proportional to the wavelength — as the frequency increases, the wavelength decreases. This relationship is an essential aspect of wave mechanics and helps in solving problems related to sound resonance in air columns.

Speed of Sound

The speed of sound is the rate at which sound waves propagate through a medium. It's influenced by the medium’s properties, such as density (\rho) and the bulk modulus (B), which is a measure of the medium's resistance to compression. In our exercise, using the provided density and bulk modulus of the air, the speed of sound is calculated with the formula
\[v = \sqrt{\frac{B}{\rho}}.\]
In this case, it's important to remember that the speed of sound is more than just a numerical value; it determines how quickly sound can travel from its source to an observer and dictates how the sound resonates in a confined space like an air column.

Harmonics in Physics

Harmonics in physics are integral to understanding sound resonance. These are the distinct standing wave patterns that form within a medium due to constructive and destructive interference at different frequencies. The first harmonic, or the fundamental frequency, is the simplest pattern and is characterized by the fact that it has the longest wavelength and the lowest frequency that can resonate in a given system.

In the context of the well from the textbook exercise, the fundamental harmonic is considered because the well resonates at no lower frequency. This first harmonic determines the length of the well since at this frequency, the length of the well is one-fourth of the wavelength of the sound wave. By analyzing harmonics, one can decode the complex phenomena of sound and its interactions with physical environments.