Wave Dispersion Relation Calculator

Dispersion Relation: \\( \omega = \sqrt{g k \tanh(k h)} \\)

Phase Velocity: \\( c = \frac{\omega}{k} = \sqrt{\frac{g}{k} \tanh(k h)} \\)

Where:

• \\(\omega\\): Angular frequency (rad/s)
• \\(k\\): Wave number (1/m)
• \\(h\\): Water depth (m)
• \\(g\\): Gravitational acceleration (9.81 m/s²)
• \\(c\\): Phase velocity (m/s)

Deep water (\\(k h > \pi\\)): \\( \tanh(k h) \approx 1 \\), so \\( \omega \approx \sqrt{g k} \\).
Shallow water (\\(k h < 0.1\\)): \\( \tanh(k h) \approx k h \\), so \\( \omega \approx k \sqrt{g h} \\).

Example 1: Deep Water Wave
\\(h = 100 \, \text{m}\\), \\(k = 0.1 \, \text{1/m}\\), \\(g = 9.81 \, \text{m/s}^2\\)
Angular Frequency: \\( \omega = \sqrt{9.81 \times 0.1 \times \tanh(0.1 \times 100)} \approx 0.990 \, \text{rad/s} \\)
Phase Velocity: \\( c = \frac{0.990}{0.1} \approx 9.90 \, \text{m/s} \\)

Example 2: Shallow Water Wave
\\(h = 1 \, \text{m}\\), \\(k = 0.05 \, \text{1/m}\\), \\(g = 9.81 \, \text{m/s}^2\\)
Angular Frequency: \\( \omega = \sqrt{9.81 \times 0.05 \times \tanh(0.05 \times 1)} \approx 0.698 \, \text{rad/s} \\)
Phase Velocity: \\( c = \frac{0.698}{0.05} \approx 13.96 \, \text{m/s} \\)

Example 3: Intermediate Depth
\\(h = 10 \, \text{m}\\), \\(k = 0.2 \, \text{1/m}\\), \\(g = 9.81 \, \text{m/s}^2\\)
Angular Frequency: \\( \omega = \sqrt{9.81 \times 0.2 \times \tanh(0.2 \times 10)} \approx 1.350 \, \text{rad/s} \\)
Phase Velocity: \\( c = \frac{1.350}{0.2} \approx 6.75 \, \text{m/s} \\)

Example 4: Very Deep Water
\\(h = 1000 \, \text{m}\\), \\(k = 0.01 \, \text{1/m}\\), \\(g = 9.81 \, \text{m/s}^2\\)
Angular Frequency: \\( \omega = \sqrt{9.81 \times 0.01 \times \tanh(0.01 \times 1000)} \approx 0.313 \, \text{rad/s} \\)
Phase Velocity: \\( c = \frac{0.313}{0.01} \approx 31.30 \, \text{m/s} \\)