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The wave equation describes how electromagnetic waves, such as light, move through space. It mathematically expresses something that varies with both position and time. The general form is written as:

• For an electric field: \( E(x, t) = E_0 \cos(kx - \omega t) \)

Here, \( E_0 \) is the amplitude of the wave, \( k \) is the wave number, \( \omega \) is the angular frequency, \( x \) is the position, and \( t \) is the time.
A wave traveling in the positive x-direction means it has a negative \( \omega t \) term.
Think of this equation as a pattern that repeats over time and space, telling how strong the electric field is at any point.

Angular frequency is a measure of how quickly the wave cycles through its motion. It is denoted by \( \omega \) and is measured in radians per second. The formula for calculating angular frequency is:

• \( \omega = \frac{2\pi c}{\lambda} \)

Here, \( c \) is the speed of light (approximately \( 3 \times 10^8 \) m/s), and \( \lambda \) is the wavelength of the wave.
Angular frequency relates to how fast the peaks and troughs of the wave pass a given point. The larger the \( \omega \), the more rapidly the wave oscillates. For electromagnetic waves, this speed is extremely fast, making them appear continuous to our senses.

Wave number is another key characteristic of a wave. It tells how many wave cycles are present in a unit distance. Represented by \( k \), it’s calculated using:

• \( k = \frac{2\pi}{\lambda} \)

Where \( \lambda \) is the wavelength. The wave number unit is \( \mathrm{m^{-1}} \) (inverse meters).
Wave number reflects the spatial frequency. That is, how tight or loose the wave is over distance. A higher wave number means more cycles are packed into a smaller space, implying a shorter wavelength and thus, higher energy.

The electric field component of an electromagnetic wave represents the force that would act on a charged particle placed in the field. It’s a vector quantity with both magnitude and direction.
In our context, the electric field has an amplitude \( E_0 \), based on the initial strength given to the wave; in this problem, it's given as \( 100 \, \mathrm{N/C} \).
It expresses how strong the wave’s electric component is at any point in space and time. This strength decreases and increases as the wave propagates, cycling between -\( E_0 \) and \( E_0 \).
The variability captured by the wave equation shows how this field interacts with other charges and fields around it.

Wavelength, denoted as \( \lambda \), is the distance between successive crests or troughs of a wave. Think of it as the 'length' of just one cycle of the wave. For electromagnetic waves, it's often measured in meters.
Wavelength and frequency are inversely related: longer wavelengths mean lower frequencies and vice versa. In simpler terms, if a wave has a long wavelength, it oscillates less frequently.

• Relation: \( c = \lambda f \)

Understanding this helps explain how different colors of light and parts of the electromagnetic spectrum have different properties. Wavelength has a direct impact on energy; shorter wavelengths are associated with more energetic waves, like gamma rays, while longer wavelengths are typical of radio waves, which carry less energy.