Problem 97 - 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. $$I=\frac{1}{2} I_{0}[1+\cos (\pi \sin \theta)]$$

Step-by-Step Solution

Answer

Maximum intensity at \(\theta = 0, \pi, 2\pi\); minimum at \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}\).

Step 1: Understand the Intensity Formula

The given formula for the signal intensity \(I\) is \[ I = \frac{1}{2} I_{0}[1 + \cos(\pi \sin \theta)] \]. This simplifies the problem as a periodic function in \(\theta\) and varies with the wavelength \(\lambda\). Here \(I_{0}\) is the maximum possible intensity.

Step 2: Plot the Intensity Function

To plot the equation \(I = \frac{1}{2} \cdot 5 \cdot (1 + \cos(\pi \sin \theta))\), we will use a range of \(\theta\) values from 0 to \(2\pi\). Plotting the function in polar coordinates will show how the intensity varies with direction.

Step 3: Analyze the Plot for Maximum and Minimum Intensity

Investigate the plotted polar graph to identify where the intensity \(I\) reaches its maximum and minimum values. The cosine term \(\cos(\pi \sin \theta)\) determines these by reaching its minimum (-1) and maximum (1). At these points, calculate \(\theta\) using the condition \(\pi \sin \theta = 0\) for maxima and \(\pi \sin \theta = \pi\) for minima.

Step 4: Determine Angles of Maximum Intensity

The maximum intensity occurs when \(1 + \cos(\pi \sin \theta) = 2\), which simplifies to \(\cos(\pi \sin \theta) = 1\). This happens when \(\pi \sin \theta = 0, 2\pi, 4\pi, \ldots\), meaning \(\sin \theta = 0\). Thus, \(\theta = 0, \pi, 2\pi\) (in the desired range).

Step 5: Determine Angles of Minimum Intensity

The minimum intensity occurs when \(1 + \cos(\pi \sin \theta) = 0\), simplifying to \(\cos(\pi \sin \theta) = -1\). This happens when \(\pi \sin \theta = \pi, 3\pi, 5\pi, \ldots\), meaning \(\sin \theta = 1\) or \(\sin \theta = -1\). Hence, \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}\).

Key Concepts

Understanding the Intensity Formula

The intensity formula provided in the exercise is used to describe how strongly the radio signal is broadcasted in different directions. This formula is given by:\[ I = \frac{1}{2} I_{0}[1 + \cos(\pi \sin \theta)] \] where \( I \) is the intensity of the signal, \( I_{0} \) is the maximum intensity possible, and \( \theta \) is the angle corresponding to the direction of the signal.The factor \( \frac{1}{2} I_{0} \) indicates that the intensity is a fraction of its maximum possible value. Meanwhile, the term \( \cos(\pi \sin \theta) \) adds periodic variation, showing that intensity depends on the angle \( \theta \). This variation is cyclic and repeats due to the trigonometric function of cosine. This means as \( \theta \) changes, the intensity fluctuates between its maximum and minimum values. Understanding how these components interact helps us to predict and analyze signal behavior in a very intuitive way.

Broadcasting Signal Analysis

The concept of a broadcasting signal involves the transmission of radio waves from antennas, which can be analyzed using mathematical formulas. These signals spread out spherically from the source, but their intensity or strength can vary by direction, as shown by the intensity formula. In this specific exercise, the broadcasting signal has two towers set apart by a small distance, contributing to interference effects. Such interference causes the signal to be stronger or weaker in certain directions. This interference pattern is characterized by alternating zones of maximum and minimum intensity, resembling a pattern of light and dark bands, which is a result of the waves either in phase or out of phase. Effective broadcasting depends on strategically placing these towers and understanding how waves interact, to make sure reception is maximized in areas where the signal is desired.

Maximum and Minimum Intensity

Identifying the points of maximum and minimum intensity is crucial for optimizing broadcast signals. In the context of our exercise, this happens by observing the role of the cosine function in the intensity formula.- **Maximum Intensity:** The maximum occurs when the value of \( \cos(\pi \sin \theta) \) is 1. From trigonometry, this happens when \( \sin \theta = 0 \). Thus, this results in the maximum intensity aligning with angles \( \theta = 0, \pi, 2\pi \). - **Minimum Intensity:** Conversely, the minimum occurs when the cosine term equals -1, i.e., \( \cos(\pi \sin \theta) = -1 \). This condition is met when \( \sin \theta = 1 \) or \( \sin \theta = -1 \). Therefore, the minimum intensity aligns with \( \theta = \frac{\pi}{2}, \frac{3\pi}{2} \). Recognizing these points allows engineers to design systems that harness these properties, ensuring effective transmission and coverage.

Understanding Periodic Functions

A periodic function is a function that repeats its values at regular intervals or periods. In our exercise, the intensity formula is periodic, primarily due to the cosine function.The core of periodic functions is the repetition: - They consistently cycle through the same set of values. - For instance, the function \( \cos(\pi \sin \theta) \) will undergo the same rise and fall within each interval of \( 2\pi \) in \( \theta \).Periodic functions are vital in broadcasting as they help predict patterns of maximum and minimum signal intensities. This understanding allows the design of transmission systems that effectively utilize wave behavior for optimal performance. The choice in wavelength and configurations contributes to effective communication, ensuring clear and powerful signal broadcasting over desired areas.