Question

Write an equation for a sinusoidal radio wave of amplitude 10 and frequency 600 kilohertz. Hint: The velocity of a radio wave is the velocity of light, \(c=3 \cdot 10^{8} \mathrm{m} / \mathrm{sec}.\)

Short answer

\[ y(t) = 10 \sin(1,200,000\pi t) \]

Step-by-step solution

Understand the Amplitude

The amplitude of the sinusoidal wave is given as 10 units. This means the maximum value of the wave is 10.

Identify the Frequency

The frequency of the wave is provided as 600 kilohertz. Convert this to hertz: \( 600 \text{ kHz} = 600 \times 10^3 \text{ Hz} = 600,000 \text{ Hz} \)

Calculate the Angular Frequency

The angular frequency \( \omega \) can be calculated using the formula: \( \omega = 2\pi f \)Substitute the frequency: \( \omega = 2\pi \times 600,000 \)\( \omega = 1,200,000\pi \text{ rad/sec} \)

Construct the General Form of the Equation

The general form of a sinusoidal wave equation is:\[ y(t) = A \sin(\omega t + \phi) \]where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase shift. Assuming no phase shift (\( \phi = 0 \)): \[ y(t) = 10 \sin(1,200,000\pi t) \]

Key concepts

Amplitude

The term 'amplitude' refers to the maximum value or height of a wave from its equilibrium position. In other words, it measures how far the wave moves from the middle to its highest point. In this exercise, the amplitude is given as 10 units. This means the wave will rise to 10 units above and fall to 10 units below its central axis.
Amplitude is a crucial factor because it indicates the energy and intensity of the wave. Larger amplitudes correspond to more energetic waves. For example, in the context of a radio wave, a higher amplitude can mean a stronger signal.

Frequency

Frequency is the number of cycles a wave completes per unit of time. It is usually measured in hertz (Hz), where one hertz equals one cycle per second. In the exercise, the frequency is given as 600 kilohertz (kHz). To put this into standard units, we convert it:

• 600 kHz = 600,000 Hz or 600 × 10³ Hz

Understanding frequency is vital in applications like radio broadcasting, where different stations transmit at different frequencies to avoid interference. The frequency determines the pitch of sound waves and the color of light waves.

Angular Frequency

Angular frequency, denoted by the symbol \( \omega \), represents how fast something is oscillating in a circular path. It is related to the frequency but is measured in radians per second. The formula to calculate angular frequency is: \( \omega = 2\pi f \) Here, \( f \) is the frequency in hertz.
For a wave with a frequency of 600 kHz:

• \( \omega = 2\pi \times 600,000 \)
• \( \omega = 1,200,000\pi \text{ rad/sec} \)

Understanding angular frequency helps in analyzing wave properties more comprehensively, especially for waves traveling in medium or oscillating systems, such as radio waves studied in the given exercise.