Problem 98 - Precalculus: Functions and Graphs

Question

Suppose that a radio station has two broadcasting towers located along a north-south line and that the towers are separated by a distance of \(\frac{1}{2} \lambda,\) where \(\lambda\) is the wavelength of the station's broadcasting signal. Then the intensity \(I\) of the signal in the direction \(\theta\) can be expressed by the given equation, where \(I_{0}\) is the maximum intensity of the signal. (a) Plot \(I\) using polar coordinates with \(I_{0}=5\) for \(\boldsymbol{\theta} \in \mathbf{[ 0 , 2 \pi ]}\) (b) Determine the directions in which the radio signal has maximum and minimum intensity. $$I=\frac{1}{2} I_{0}[1+\cos (\pi \sin 2 \theta)]$$

Step-by-Step Solution

Answer

Max intensity at \( \theta = 0, \pi/2, \pi, 3\pi/2 \). Min intensity at \( \theta = \pi/4, 3\pi/4, 5\pi/4, 7\pi/4 \).

Step 1: Understand the Intensity Function

The intensity function is given as \( I = \frac{1}{2} I_{0}[1+\cos (\pi \sin 2 \theta)] \). Here, \( I_{0} \) is the maximum intensity, set to 5. The equation is a function of \( \theta \), given in polar coordinates.

Step 2: Define Parameters for the Plot

For the polar plot, you need intensity \( I \) and angle \( \theta \). \( I_{0} \) is given as 5, and \( \theta \) varies from \( 0 \) to \( 2\pi \).

Step 3: Calculate Intensity for Different Angles

Substitute \( \theta \) values ranging from \( 0 \) to \( 2\pi \) into the equation \( I = \frac{1}{2} \times 5 \times [1+\cos (\pi \sin 2 \theta)] \) and calculate \( I \) for each. This results in intensity values that depend on \( \theta \).

Step 4: Plot the Function in Polar Coordinates

Use a polar plot to represent the relationship between \( \theta \) and \( I \). Each angle \( \theta \) has an associated intensity \( I \), and the plot will visualize this equation with radius \( r = I \) for each \( \theta \).

Step 5: Analyze the Plot for Maximum and Minimum Intensity

In the polar plot, observe the directions (angles \( \theta \)) where \( I \) reaches its highest and lowest values. Maximum intensity occurs where the cosine term equals 1, and minimum intensity occurs where it reaches -1.

Step 6: Calculate Maximum and Minimum Intensity Directions

\( \cos(\pi \sin 2\theta) = 1 \) at \( \pi \sin 2\theta = 0, 2\pi, 4\pi, \dots \), giving \( \theta = n\frac{\pi}{4} \) when \( n \) is even, and \( \cos(\pi \sin 2\theta) = -1 \) at odd \( n \). Thus, maximum intensities occur at \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) and minima at \( \theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \).

Key Concepts

Radio Signal Intensity

The intensity of a radio signal refers to the strength or power of the signal transmitted by a radio wave. Think of it as the 'loudness' of the signal as it travels from the broadcasting towers. In our context, the intensity is represented by the function \( I = \frac{1}{2}I_{0}[1+\cos (\pi \sin 2\theta)] \). Here, \( I_{0} \) signifies the peak or maximum possible intensity of the signal, and it's set to a value of 5 in our exercise.

Polar coordinates are used here to describe the radio signal's intensity as a function of the angle \( \theta \). By understanding these angles, we can see how the signal's strength changes as we move around the broadcasting tower. This plays an important role in determining which directions the signal is strongest and weakest.

Wavelength

The wavelength \( \lambda \) is a crucial aspect of any wave-related discussion, including radio signals. It determines the distance over which the wave's shape repeats. In the context of the original exercise, the wavelength affects the positioning of the radio towers, which are separated by \( \frac{1}{2} \lambda \).

Wavelength is integral in defining the spatial characteristics of the radio wave. Shorter wavelengths could result in a different distribution of signal intensity across different directions, while longer wavelengths could lead to more spread-out intensity peaks and troughs.

• A shorter wavelength means more frequent oscillations of the wave.
• A longer wavelength implies fewer oscillations over the same distance.

This understanding helps in predicting the pattern and reach of radio waves, especially important when considering how signals might interfere or align under different conditions.

Cosine Function

The cosine function is a critical component in the intensity equation of the radio signal. It influences how the intensity changes with different angles \( \theta \). Specifically, it's represented in the form \( \cos(\pi \sin 2\theta) \).

The cosine function oscillates between -1 and 1, which directly affects the intensity value. When \( \cos(\pi \sin 2\theta) = 1 \), the intensity is at its maximum, while at \( \cos(\pi \sin 2\theta) = -1 \), the intensity is at its minimum.

• When maximum, intensity peaks are sharp and clear.
• When minimum, the signal is weakest.

These oscillations form a crucial part of the plot in polar coordinates, depicting the varying signal strength in different directions. Understanding the behavior of the cosine function is key to predicting these intensity changes.

Maximum and Minimum Intensity

Understanding the maximum and minimum intensities of the radio signal helps determine the effectiveness of broadcasting in certain directions. The intensity equation sheds light on these peaks and troughs by examining where the cosine term \( \cos(\pi \sin 2\theta) \) becomes 1 or -1.

Maximum intensity occurs when \( \cos(\pi \sin 2\theta) = 1 \), meaning the signal is at its strongest. This happens at specific angles such as \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \). In contrast, minimum intensity, where the signal loses strength, occurs when \( \cos(\pi \sin 2\theta) = -1 \) at angles like \( \theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \).

Understanding these intensities helps in optimizing the broadcasting of radio signals and could be used to target specific areas effectively while avoiding areas of signal drop-offs. By recognizing these directions, broadcasters can adjust strategies to maintain consistency in coverage.