Problem 28 - Algebra and Trigonometry

Question

FM Radio Signals The carrier wave for an FM radio signal is modeled by the function $$ y=a \sin \left(2 \pi\left(9.15 \times 10^{7}\right) t\right) $$ where \(t\) is measured in seconds. Find the period and frequency of the carrier wave.

Step-by-Step Solution

Answer

The frequency is \( 9.15 \times 10^7 \) Hz, and the period is approximately \( 1.093 \times 10^{-8} \) seconds.

Step 1: Identify the Wave Equation

The wave function given is \( y = a \sin (2\pi(9.15 \times 10^7)t) \). This is in the form of \( y = a \sin(2\pi f t) \), where \( f \) is the frequency of the wave.

Step 2: Determine the Frequency

From the equation \( y = a \sin(2\pi(9.15 \times 10^7)t) \), we can directly see that the frequency \( f \) is \( 9.15 \times 10^7 \) Hz (hertz).

Step 3: Calculate the Period of the Wave

The period \( T \) of a wave is the reciprocal of the frequency, so \( T = \frac{1}{f} \). Substitute \( f = 9.15 \times 10^7 \) Hz to find the period: \[ T = \frac{1}{9.15 \times 10^7} \approx 1.093 \times 10^{-8} \text{ seconds}. \]

Key Concepts

Understanding Frequency

Frequency is a critical concept when studying waves and signals, such as FM radio signals. In simple terms, frequency tells us how often a wave repeats itself over a set period of time. It's like counting how often a wave hits a particular point within one second.
For FM radio signals, frequency is usually measured in hertz (Hz), where one hertz is equal to one cycle (or repetition) per second. In our exercise, the frequency of the carrier wave is given as \(9.15 \times 10^7\) Hz. This high frequency means that the wave completes \(9.15 \times 10^7\) cycles in just one second!
When dealing with wave equations, frequency is often visible in the equation itself. In the form \(y = a \sin(2\pi f t)\), the variable \(f\) represents the frequency. Knowing this helps you understand how quickly the signal oscillates, which is fundamental in radio technology.

Exploring the Period of a Wave

The period of a wave is closely related to its frequency. While frequency tells us how many waves pass a point in a second, the period tells us how long it takes for one full wave to pass a point. In other words, it's the time taken for one complete cycle of the wave.
This is calculated easily by finding the reciprocal of the frequency:

• If the frequency \(f\) is known, the period \(T\) is \(T = \frac{1}{f}\).

In our example, where the frequency is \(9.15 \times 10^7\) Hz, the period is accordingly \(1.093 \times 10^{-8}\) seconds. This indicates that the wave's cycle is extremely fast, characteristic of electromagnetic signals like those used in FM radio broadcasting.
Understanding the concept of the period is essential to mastering the intricacies of wave behavior, particularly in technologies like communication systems, where timing is everything.

The Wave Equation in Practice

In physics and engineering, the wave equation forms the backbone for analyzing wave behaviors, including FM radio signals. The wave equation \(y = a \sin(2\pi f t)\) illustrates how a wave function behaves over time.
In this equation, \(a\) represents the amplitude, or the peak height of the wave. Amplitude is a separate concept that shows the wave's strength, but here we're focusing on \(f\), the frequency, which critically influences how often the wave cycles.

• The expression \(2\pi f t\) inside the sine function is crucial. It's directly linked to the phase of the wave, representing the swift movement of the wave as \(t\) (time) progresses.

In practical applications like radio transmission, the wave equation allows engineers to model how radio waves propagate and interact with various elements.
So, when you tune into your favorite FM station, you're essentially engaging with these intricate but fascinating wave dynamics, governed by the wave equation.