How to Calculate Group Velocity in Wave Mechanics

The group velocity in wave mechanics is a crucial concept that describes the propagation speed of a wave packet, which is the envelope function of the individual waves within the packet. To calculate the group velocity, we can use the formula vg = dω/dk, where ω is the angular frequency of the wave and k is the wave number. This formula represents the speed at which the wave packet travels, which can be different from the phase velocity, the speed at which the individual waves oscillate.

Understanding the Concept of Group Velocity

The group velocity is a measure of how fast the wave packet propagates, while the phase velocity is the speed at which the individual waves in the packet oscillate. In dispersive media, where different frequencies travel at different speeds, the group velocity and phase velocity can be different.

The group velocity can be thought of as the velocity of the “envelope” of the wave packet, which is the function that describes the overall shape and amplitude of the packet. The individual waves within the packet may be traveling at different speeds, but the group velocity represents the speed at which the entire packet moves.

Calculating Group Velocity for Deep Ocean Waves

Let’s consider a wave packet in a deep ocean with a central wavelength of λ0 = 60 m. The central wavenumber is k0 = 2π/λ0 ≈ 0.1 m−1. The phase speed of deep ocean waves is given by the formula c = (g/k)^(1/2), where g is the acceleration due to gravity.

The frequency of deep ocean waves is ω = (gk)^(1/2). The group velocity is then u = dω/dk = (g/k)^(1/2)/2 = c/2. For the specified central wavenumber, we can calculate the group velocity as:

u ≈ (9.8 m s−2/0.1 m−1)^(1/2)/2 ≈ 5 m s−1

So, the group velocity for the deep ocean wave packet with a central wavelength of 60 m is approximately 5 m/s.

Calculating Group Velocity for a Superposition of Sine Waves

To calculate the group velocity for a superposition of sine waves, we can use the formula:

vg = Δω/Δk = (ω2 - ω1)/(k2 - k1)

where ω1 and ω2 are the angular frequencies, and k1 and k2 are the wave numbers of the individual waves in the packet.

This formula represents the slope of the dispersion relation, which relates the frequency and wave number of the waves in the packet. By calculating the change in angular frequency (Δω) and the change in wave number (Δk) between the individual waves, we can determine the group velocity.

Theorem and Formulas

1. Group Velocity Formula: vg = dω/dk
2. Phase Speed of Deep Ocean Waves: c = (g/k)^(1/2)
3. Frequency of Deep Ocean Waves: ω = (gk)^(1/2)
4. Group Velocity for Deep Ocean Waves: u = dω/dk = (g/k)^(1/2)/2 = c/2
5. Group Velocity for Superposition of Sine Waves: vg = Δω/Δk = (ω2 - ω1)/(k2 - k1)

Examples and Numerical Problems

1. Example 1: Calculate the group velocity for a deep ocean wave packet with a central wavelength of 60 m.
2. Given:
   • Central wavelength, λ0 = 60 m
   • Central wavenumber, k0 = 2π/λ0 ≈ 0.1 m−1
   • Acceleration due to gravity, g = 9.8 m/s^2

3. Solution:

   • Phase speed, c = (g/k0)^(1/2) = (9.8 m/s^2/0.1 m^-1)^(1/2) = 9.9 m/s
   • Group velocity, u = c/2 = 9.9 m/s/2 = 4.95 m/s

4. Numerical Problem 1: A wave packet in a dispersive medium has a central frequency of 1000 Hz and a central wavenumber of 100 rad/m. Calculate the group velocity of the wave packet.

5. Given:
   • Central frequency, f0 = 1000 Hz
   • Central wavenumber, k0 = 100 rad/m

6. Solution:

   • Central angular frequency, ω0 = 2πf0 = 2π × 1000 = 6283.185 rad/s
   • Group velocity, vg = dω/dk = (dω/df)/(dk/df) = (dω/df)/1/c = ω0/k0 = 6283.185 rad/s / 100 rad/m = 62.83 m/s

7. Numerical Problem 2: A wave packet consists of two sine waves with frequencies of 100 Hz and 102 Hz, and wavenumbers of 10 rad/m and 10.2 rad/m, respectively. Calculate the group velocity of the wave packet.

8. Given:
   • Frequency 1, f1 = 100 Hz
   • Frequency 2, f2 = 102 Hz
   • Wavenumber 1, k1 = 10 rad/m
   • Wavenumber 2, k2 = 10.2 rad/m
9. Solution:
   • Angular frequency 1, ω1 = 2πf1 = 2π × 100 = 628.319 rad/s
   • Angular frequency 2, ω2 = 2πf2 = 2π × 102 = 640.885 rad/s
   • Group velocity, vg = Δω/Δk = (ω2 - ω1)/(k2 - k1) = (640.885 - 628.319)/(10.2 - 10) = 12.566 rad/s / 0.2 rad/m = 62.83 m/s

Figures and Data Points

Figure 1: Dispersion relation showing the relationship between group velocity and phase velocity.

Table 1: Comparison of group velocity and phase velocity for different types of waves.

Wave Type	Group Velocity	Phase Velocity
Deep Ocean Waves	c/2	c
Electromagnetic Waves in Vacuum	c	c
Electromagnetic Waves in Dispersive Media	< c	> c
Plasma Waves	< c	> c

Conclusion

In summary, to calculate the group velocity in wave mechanics, we can use the formula vg = dω/dk, which represents the speed of the wave packet. This formula can be applied to different types of waves, such as deep ocean waves or superposition of sine waves, to obtain the group velocity and understand how the wave packet propagates.

The group velocity is an important concept in wave mechanics, as it describes the speed at which the overall wave packet moves, which can be different from the phase velocity of the individual waves. By understanding and calculating the group velocity, we can gain insights into the behavior of wave packets in various physical systems.

References:

1. Group Velocity.pdf – https://people.eecs.ku.edu/~demarest/470/Group%20Velocity.pdf
2. Group Velocity – Physics LibreTexts – https://phys.libretexts.org/Bookshelves/University_Physics/Radically_Modern_Introductory_Physics_Text_I_%28Raymond%29/01:_Waves_in_One_Dimension/1.09:_Group_Velocity
3. Group Velocity: Dispersion & Phase Comparison – StudySmarter – https://www.studysmarter.co.uk/explanations/physics/waves-physics/group-velocity/