Problem 20 - Chemistry: An Atoms-Focused Approach

Question

Submarine Communications The Russian and American navies developed extremely low-frequency communications networks to send messages to submerged submarines. The frequency of the carrier wave of the Russian network was \(82 \mathrm{Hz},\) while the Americans used \(76 \mathrm{Hz}\) a. What was the ratio of the wavelengths of the Russian network to the American network? b. To calculate the actual underwater wavelength of the transmissions in either network, what additional information would you need?

Step-by-Step Solution

Answer

Answer: The ratio of the wavelengths of the Russian and American networks can be found using the formula: Wavelength Ratio = λ1 / λ2. To calculate the actual underwater wavelength, we need the speed of radio waves underwater, as it will be different than the speed of light in a vacuum (or air).

Step 1: Write Down the Known and Unknown Values

We are given the frequency of the two networks: - Russian network: f1 = 82 Hz - American network: f2 = 76 Hz We want to find the ratio of wavelengths: - Russian network: λ1 - American network: λ2

Step 2: Apply Formula for Wavelength

To obtain the speed of light, we need to multiply the frequency and the wavelength. speed of light (c) = frequency × wavelength To isolate the wavelength, we divide both sides by the frequency: wavelength = speed of light / frequency

Step 3: Calculate the Wavelengths for Both Networks

Use the given frequencies for the Russian and American networks and the speed of light (c = 3 x 10^8 m/s) to calculate the corresponding wavelengths. λ1 = c / f1 = (3 × 10^8 m/s) / (82 Hz) λ2 = c / f2 = (3 × 10^8 m/s) / (76 Hz)

Step 4: Calculate the Ratio of Wavelengths

Divide λ1 by λ2 to find the wavelength ratio. Wavelength Ratio = λ1 / λ2

Step 5: Answer Part b of the Exercise

In order to calculate the actual underwater wavelength, we need the speed of the radio waves underwater, as it will be different than the speed of light in a vacuum (or air). So, we need the speed of radio waves underwater in addition to the given information to calculate the actual underwater wavelength.

Key Concepts

Frequency

Frequency is an important concept in understanding how waves, such as those used in communications, behave. It refers to the number of times a wave oscillates or cycles per second. The unit of frequency is Hertz (Hz).

• In the context of radio communications, different frequencies can help divide signals into separate channels.
• For submarine communications, the frequencies are quite low, namely 82 Hz for the Russian network and 76 Hz for the American network.

Frequency directly affects the wavelength and the capacity of transmissions. The lower the frequency, the longer the wavelength, which is beneficial for penetrating water, a much denser medium than air.

Speed of Light

The speed of light is a fundamental constant in physics, denoted by the symbol \(c\), and is approximately \(3 \times 10^8\) meters per second in a vacuum. This constant speed forms the backbone of calculating wavelengths in various environments. In the context of communication systems, this speed applies to electromagnetic waves in a vacuum. The relationship between speed, frequency, and wavelength is expressed through the equation:\[ \text{wavelength} = \frac{\text{speed of light}}{\text{frequency}} \]This equation highlights the inverse relationship between frequency and wavelength: as one increases, the other decreases. However, when the wave propagates through different mediums, such as underwater, its speed changes, affecting the wavelength.

Ultra-Low Frequency Communications

Ultra-low frequency (ULF) communications represent a method where very low frequencies, typically in the range of 3 Hz to 300 Hz, are used for transmitting signals. This is particularly advantageous for certain applications because these frequencies have longer wavelengths and can penetrate dense media like water and land.

• ULF is suitable for long-distance communications.
• It is mainly used where other technologies struggle, such as communicating with submarines.

Despite their advantages, these frequencies carry less data due to their low bandwidth, making them less suitable for applications requiring high data rates, such as video streaming or large file transmission.

Submarine Communications

Submarine communications are crucial for naval operations, allowing communication with vessels below the ocean's surface. Due to water's density, conventional radio frequencies cannot penetrate significant depths. Using low-frequency waves, such as those mentioned in the original exercise, provides a solution:

• The low frequency means longer wavelengths, which can travel further through water.
• This method facilitates reliable communication even in challenging environments.

Adding to the challenge, the speed of radio waves underwater is not the same as in air, necessitating specific adjustments in calculations to determine the exact underwater wavelengths, ensuring message integrity and clarity.