Question

What is the equation that relates the wavelength and frequency of a light wave? (Define all symbols used.)

Short answer

The equation is \( c = \lambda f \) where \( c \) is the speed of light (\(3 \times 10^8 \, m/s\)), \( \lambda \) is the wavelength in meters, and \( f \) is the frequency in hertz.

Step-by-step solution

Understanding the Wave Equation

The equation relating the wavelength \( \lambda \) and frequency \( f \) of a light wave is given by \( c = \lambda f \) where \( c \) is the speed of light in a vacuum and is approximately \( 3 \times 10^8 \, m/s \). The wavelength \( \lambda \) is the distance between successive peaks of the wave and is measured in meters (m). The frequency \( f \) is the number of wave cycles that pass a given point per second and is measured in hertz (Hz).

Key concepts

Speed of Light

The speed of light is a fundamental constant in physics, symbolized by the letter 'c'. It represents the velocity at which light waves propagate through a vacuum and is approximately equal to 299,792,458 meters per second, although for many calculations it is approximated to be around 300,000 kilometers per second or, more conveniently, as \(3 \times 10^8 \, m/s\).

This incredible speed means that light can travel around the Earth approximately 7.5 times in one second. The constancy of the speed of light underpins many of the principles of modern physics, including the theory of relativity. It is essential to note that while light can travel through various media such as air, water, or glass, its speed will be slower than in a vacuum.

The speed of light is not just essential for understanding wave phenomena but also a major component in equations across physics and astronomy, including determining distances in space, known as light-years, and in technologies such as lasers and fiber optic communications.

Wavelength and Frequency Relationship

The wavelength \(\lambda\) and frequency \(f\) of a light wave are inversely proportional to each other and are related by the wave equation \(c = \lambda f\). In simple terms, this equation tells us that the speed of light \(c\) is the product of the wave's wavelength and its frequency.

Wavelength is the distance between two consecutive peaks (or troughs) of the wave and is typically measured in meters. Frequency, on the other hand, is the number of complete wave cycles that pass a given point in one second, with the unit of measurement being hertz (Hz).

As the frequency increases, the wavelength decreases, and vice versa, since the speed of light remains constant in a vacuum. This relationship helps us understand the spectrum of light, from radio waves with long wavelengths and low frequencies to gamma rays with short wavelengths and high frequencies.

To visualize this, imagine plucking a string with tight tension (higher frequency) that generates shorter wave patterns compared to a looser string (lower frequency) that has longer wave patterns.

Wave Properties

Understanding wave properties is crucial to comprehend various physical phenomena. The basic properties of waves include amplitude, wavelength, frequency, and speed. Amplitude refers to the height of the wave's crest or the depth of its trough, which correlates with the wave's energy.

Wavelength \(\lambda\), as previously explained, is the distance between consecutive crests, while frequency \(f\) is the number of times the wave oscillates in one second. Waves also exhibit behaviors such as reflection, when they bounce off surfaces; refraction, when they change direction entering a new medium; and interference, when different waves coincide to produce a new wave pattern.

These properties not only apply to light but also to other types of waves, such as sound and water waves. In each case, the medium through which the wave is traveling can affect these properties and the behavior of the wave itself. For example, light waves can bend when moving from air into water, illustrating refraction, while overlapping waves from two different sources can create a pattern of constructive and destructive interference, leading to areas of increased or decreased amplitude.