General PDE Dispersion Relation Calculator

Calculator inputs

Example data table

The default example represents the one dimensional wave equation: u_tt - 4u_xx = 0. It should return ω = ±2k.

Formula used

For a constant coefficient linear PDE, this calculator applies the trial wave:

u = A exp(i(kx + ly - ωt))

Each derivative becomes an algebraic factor. For example, u_t = -iωu, u_tt = -ω²u, u_xx = -k²u, and u_xxxx = k⁴u.

The computed dispersion equation is:

-c_ttω² - i c_tω + S(k,l) = 0

Here, S(k,l) is the spatial symbol built from the x, y, mixed, and fourth order coefficients. Phase speed is Re(ω)/sqrt(k²+l²). Group speeds are estimated with centered finite differences.

How to use this calculator

1. Write your PDE in constant coefficient linear form.
2. Enter each coefficient beside the matching derivative term.
3. Enter the target wave numbers k and l.
4. Set a k range for the graph.
5. Press the calculation button.
6. Review both frequency roots, phase speed, damping, and group trends.
7. Use CSV or PDF export for reports and records.

Understanding dispersion relations

What the relation means

A dispersion relation links frequency with wave number. It shows how a wave travels inside a model. In many linear PDEs, waves of different wavelengths move at different speeds. This calculator converts derivative terms into algebraic factors, then solves for angular frequency.

Why coefficients matter

Each coefficient changes the spatial symbol. A second spatial derivative often adds a squared wave number term. A fourth derivative adds a stronger bending or smoothing term. First and third derivatives can shift the imaginary part of the symbol. Time derivatives control whether the frequency equation is linear or quadratic.

Reading the roots

The real part of omega describes oscillation. The imaginary part describes growth or decay when the selected wave convention is used. A positive imaginary part means the amplitude grows in time. A negative value means the mode decays. Near zero values are usually neutral, but numerical tolerance should be considered before making a final judgment.

Phase and group speed

Phase speed measures how a single crest moves. Group speed estimates how a packet of nearby waves moves. The calculator reports x and y group trends using small centered changes. These values are most reliable when the root branch is smooth and no branch crossing occurs.

Practical interpretation

A flat curve means frequency changes slowly with wave number. A steep curve means the packet speed may be high. Curves with strong imaginary parts suggest damping, instability, or numerical stiffness. Compare the graph with the result table. The table gives exact values at one selected wave number. The plot shows behavior across a wider range.

Good modeling practice

Use nondimensional coefficients when possible. Check signs before trusting a result. Compare simple cases with known equations. Increase graph points for smoother curves. Avoid interpreting unstable roots as coding errors. They may show a real instability in the model. For complex systems, test one term at a time, then add more terms gradually.

FAQs