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, \(c=3\). \(10^{8} \mathrm{~m} / \mathrm{sec}\).

Short answer

The sinusoidal wave equation is \(y(t) = 10 \, \text{sin}(2\pi \, 600 \, 10^3 \, t)\).

Step-by-step solution

Understanding the Given Values

Identify the amplitude, frequency, and speed of the radio wave. Given: amplitude = 10, frequency = 600 kilohertz (600 × 10^3 Hz), and speed of light, c = 3 × 10^8 m/s.

Formula for a Sinusoidal Wave

The general equation for a sinusoidal wave is given by \[ y(t) = A \, \text{sin}(2\pi ft + \theta) \] where A is the amplitude, f is the frequency, and \(\theta\) is the phase constant (which can be taken as zero if not specified).

Substitute the Amplitude and Frequency

Substitute the given values of amplitude (A) and frequency (f) into the sinusoidal wave equation: \[ y(t) = 10 \, \text{sin}(2\pi \, 600 \, 10^3 \, t) \].

Key concepts

amplitude

Amplitude is the measure of the maximum displacement of a wave from its mean or equilibrium position. It shows how 'tall' or 'strong' the wave is. For example, in the given exercise, the amplitude is 10. This means the wave reaches 10 units above and below its average position.
Amplitude is crucial because it affects the wave's energy.
In sound waves, higher amplitude means louder sound. In light waves, higher amplitude means brighter light.

frequency

Frequency refers to how often the wave passes a certain point in one second. It's measured in Hertz (Hz), which equals one cycle per second.
In the exercise, the given frequency is 600 kilohertz, or 600,000 Hz. This means the wave cycles 600,000 times every second!
Frequency affects the pitch of sound and the color of light. Higher frequency means higher pitch in sound, and in light, it shifts from red to blue. To sum up, frequency tells us about the wave's 'pace'.

speed of light

The speed of light in a vacuum is a fundamental constant of nature. It travels at approximately 3 \times 10^8 meters per second, or 300,000 kilometers per second.
In the given problem, speed of light is crucial because radio waves travel at this speed.
This value helps to connect time and distance in wave equations. For instance, if we know the frequency, we can use the speed of light to find the wavelength and vice versa, using the formula: \[ \text{speed} = \text{frequency} \times \text{wavelength} \]

sinusoidal wave

A sinusoidal wave refers to a wave that can be mathematically described using the sine function. It represents the smoothest and simplest type of periodic wave.
This type of wave has key parameters: amplitude (A), frequency (f), and phase (\theta).
In the general sinusoidal wave equation: \[ y(t) = A \, \text{sin}(2\pi ft + \theta) \], phase (\theta) often represents shifts along the time axis, but it can be zero if not mentioned. For our exercise: \[ y(t) = 10 \, \text{sin}(2\pi \, 600 \, 10^3 \, t) \]

radio wave

Radio waves are a type of electromagnetic wave, same as light, but with much longer wavelengths and lower frequencies.
These waves are used in many applications like broadcasting, GPS, and cell phones.
In the given exercise, we are dealing with a radio wave of 600 kilohertz. Using its frequency and knowing that it travels at the speed of light, we can describe its behavior using the sinusoidal wave equation.
Thus, given the amplitude and frequency, we wrote the formula for a specific radio wave. This describes its oscillations across time, making it useful for various communication technologies.