Question

Project ELF, described in Sec. 4.4.1, was an extremely low frequency, \(76 \mathrm{~Hz},\) radio system set by the military to communicate with submarines. It had facilities near Clam Lake, Wisconsin and Republic, Michigan, 148 miles apart [52]. Because these facilities were located a fraction of a wavelength apart, antennas at these locations acted as part of a single array. The length of all antenna elements was 84 miles [52]. Assume it took 18 minutes to transmit a three letters message using 8 bit ASCII, and assume signals travel close to the speed of light in free space. (a) Calculate the ratio of the distance between the transmitting facilities to the wavelength. (b) Calculate the ratio of the length of all antenna elements to the wavelength. (c) What was the speed of communication in bits per second? (d) How many wavelengths long were each bit?

Short answer

(a) Ratio is approximately 0.0603, (b) Ratio is approximately 0.0342, (c) Speed is approximately 0.0222 bits/second, (d) Each bit is about 177,822,426.13 meters.

Step-by-step solution

Calculate the wavelength

The wavelength \( \lambda \) can be calculated using the formula \( \lambda = \frac{c}{f} \), where \( c = 3 \times 10^8 \text{ m/s} \) is the speed of light and \( f = 76 \text{ Hz} \).\[\lambda = \frac{3 \times 10^8}{76} \approx 3,947,368.42 \text{ meters}\]

Calculate the distance ratio for the facilities

The distance between the facilities is 148 miles. Converting miles to meters, we have:\[148 \text{ miles} \times 1609.34 \text{ meters/mile} = 238,183.32 \text{ meters}\]The ratio of the distance between the transmitting facilities to the wavelength is:\[\text{Ratio} = \frac{238,183.32 \text{ meters}}{3,947,368.42 \text{ meters}} \approx 0.0603\]

Calculate the antenna element length ratio

The length of all antenna elements is 84 miles. Converting miles to meters, we have:\[84 \text{ miles} \times 1609.34 \text{ meters/mile} = 135,184.56 \text{ meters}\]The ratio of the length of all antenna elements to the wavelength is:\[\text{Ratio} = \frac{135,184.56 \text{ meters}}{3,947,368.42 \text{ meters}} \approx 0.0342\]

Calculate the speed of communication in bits per second

A three-letter message in 8-bit ASCII is \( 3 \times 8 = 24 \text{ bits} \). Since it takes 18 minutes to send this message:\[18 \text{ minutes} \times 60 \text{ seconds/minute} = 1080 \text{ seconds}\]The speed of communication is:\[\text{Speed} = \frac{24 \text{ bits}}{1080 \text{ seconds}} \approx 0.0222 \text{ bits per second} \]

Calculate the wavelength per bit

With the communication speed of 0.0222 bits per second and the wavelength of approximately 3,947,368.42 meters, each bit corresponds to:\[\text{Wavelength per bit} = \frac{3,947,368.42 \text{ meters/bit}}{0.0222 \text{ bits per second}} \approx 177,822,426.13 \text{ meters per bit} \]

Key concepts

Frequency Calculation

To understand the operation of extremely low frequency (ELF) radio systems like Project ELF, it's crucial to grasp the concept of frequency calculation. Radio waves, much like all electromagnetic waves, have specific frequencies and wavelengths. The frequency (\( f \)) represents how many cycles of the wave occur per second, measured in hertz (Hz). For ELF systems, this frequency is particularly low.
For Project ELF, the frequency was given as \( 76 \text{ Hz} \). The wavelength (\( \lambda \)) of a wave, which is the distance over which the wave's shape repeats, can be determined using the formula:\[ \lambda = \frac{c}{f} \]where \( c \) is the speed of light (\( 3 \times 10^8 \text{ m/s} \)). Calculating this for Project ELF gives us a wavelength of approximately 3,947,368.42 meters. This means that a single wave cycle is close to 4,000 kilometers long! Such lengths are typical for ELF, enabling communication over vast distances necessary for submarine communication, as they can penetrate underwater depths better than higher frequencies.

Speed of Communication

The speed at which information is transmitted over an ELF system is another fascinating aspect. It's calculated in bits per second (bps), showing how many bits of data can be sent every second. For instance, in Project ELF, communicating a simple three-letter message using 8-bit ASCII took a significant amount of time.
To find the transmission speed, let's consider that each letter is encoded into 8 bits, totaling 24 bits for three letters. The transmission duration was 18 minutes, or 1,080 seconds. Therefore, the speed of communication can be calculated as:\[ \text{Speed} = \frac{24 \text{ bits}}{1080 \text{ seconds}} \approx 0.0222 \text{ bits per second} \]While this speed seems incredibly slow by today's standards, it's essential to note that ELF was primarily designed for reliability over extreme distances, not speed. These signals could still effectively transmit short, critical messages where no other method was viable.

Antenna Design

Antenna design for ELF communication is a unique challenge due to the vast wavelengths involved. In Project ELF, antennas were distributed over miles to function effectively as part of a single array. This arrangement was crucial because typical, smaller antennas are impractical at such low frequencies.
The length of the combined antenna system in Project ELF was 84 miles, equating to 135,184.56 meters. Given the calculated wavelength of about 3,947,368.42 meters, the length of the antennas was only a fraction of the wavelength:\[ \frac{135,184.56 \text{ meters}}{3,947,368.42 \text{ meters}} \approx 0.0342 \]This means the antennas were much shorter than the wavelength. Antennas of such lengths create a radiative pattern where parts of the electromagnetic wave reinforce other parts, successfully boosting the signal. The design and strategic positioning allowed ELF systems to maintain long-range communication capabilities, crucial for underwater and underground communication with submarines.