Dispersion relation
Nontrivial plane waves require three frequency branches.
ω(ω²−f²−gH|k|²)=0
Wave Modes of the Shallow Water Equations
How rotation and gravity shape fluid motion
The rotating shallow-water equations contain a balanced mode and two propagating inertia-gravity waves.
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The essential idea: rotation controls the low-wavenumber frequency, while gravity sets the high-wavenumber wave speed.
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Wave Modes of the Shallow Water Equations
Presented by Charlotte Moser · Coming soon
What to notice
- One geostrophic mode has ω=0.
- Two inertia-gravity modes have opposite frequencies.
- The gravity-wave speed is c=√(gH).
The mathematical structure
Three modes
The steady balanced branch coexists with two oscillatory branches.
ω₀=0, ω±=±√(f²+gH|k|²)
Read the shallow-water dispersion relation
Change rotation, depth, and wavenumber. Track the geostrophic and inertia-gravity branches from the slides.
Shallow-water dynamics separate into one balanced geostrophic mode and two propagating inertia-gravity waves.