Group Velocity in Wave Mechanics §

Introduction §

Group velocity is a fundamental concept in wave mechanics, particularly in the study of wave propagation in various media. It describes the speed at which the overall shape of a wave’s amplitudes—known as the envelope or the group of waves—propagates through space.

Definition and Formula §

The group velocity ($v_g$) is defined as the derivative of the wave’s angular frequency ($\omega$) with respect to its wave number ($k$): $v_g=\frac{d\omega }{dk}$

This relationship is crucial in understanding how different wave components travel together in a medium.

Physical Significance §

• Energy Transport: Group velocity is often associated with the transport of energy in a wave.
• Dispersion: In dispersive media, where the phase velocity varies with frequency, the group velocity can differ from the phase velocity.

Historical Context §

The concept of group velocity was first introduced by Lord Rayleigh in the 19th century while studying the propagation of waves in different media. This was a significant advancement in wave theory, contributing to the understanding of phenomena like ocean waves, sound waves, and later, electromagnetic waves.

Application in Wave Mechanics §

• Optics: In fiber optics, group velocity is crucial for understanding signal propagation and dispersion.
• Quantum Mechanics: The concept is used in the study of matter waves, where the group velocity can be associated with the velocity of particles.

Examples §

1. Light Waves in Glass: In a dispersive medium like glass, different colors (frequencies) of light travel at different speeds, affecting the group velocity.
2. Sound Waves: The group velocity of sound waves can vary in different gases, affecting how sound is propagated.

Conclusion §

Group velocity is a key concept in wave mechanics, offering insights into how wave packets move in various media. It’s essential for understanding wave behaviour in fields like optics, acoustics, and quantum mechanics.

Test Questions §

1. STARTI [Basic] Question: What is the formula for group velocity in wave mechanics? Back: The formula for group velocity is $v_g=\frac{d\omega }{dk}$, where $v_g$ is the group velocity, $\omega$ is the angular frequency, and $k$ is the wave number. ENDI
2. STARTI [Basic] Question: How does group velocity differ in a dispersive medium? Back: In a dispersive medium, the phase velocity varies with frequency, leading to a difference between the phase and group velocities. ENDI
3. STARTI [Basic] Question: What is the significance of group velocity in fibre optics? Back: In fibre optics, group velocity is crucial for understanding how light signals propagate and how dispersion affects signal transmission. ENDI