Wave Refraction Calculator

Calculator input

Input mode

Use depth-based mode for coastal wave mechanics. Use direct mode for known celerity interfaces.

Incident ray angle θ₁ (deg from contour normal)

0° means normal approach. Larger values indicate more oblique incidence.

Initial wave height H₁ (m)

Optional if you want a refraction-only wave height estimate.

Wave period T (s)

Required in depth-based mode. Optional in direct mode for wavelength output.

Gravity g (m/s²)

Use 9.80665 for standard Earth gravity.

Starting depth h₁ (m)

Usually the deeper reference location.

Target depth h₂ (m)

The calculation predicts the new angle at this depth.

In this mode, the page solves the linear wave dispersion relation, then applies Snell's law using computed celerities.

Starting celerity C₁ (m/s)

Known wave speed in the first region.

Target celerity C₂ (m/s)

Known wave speed in the second region.

Use direct mode for abrupt interfaces, laboratory studies, or when celerities are already known from external analysis.

Formula used

1) Linear wave dispersion relation

ω² = gk tanh(kh)

For depth-based mode, the page solves the wave number k iteratively. Here, ω = 2π/T, h is water depth, and T is wave period.

2) Wavelength and celerity

L = 2π / k

C = L / T

Once the dispersion relation is solved, wavelength and phase speed follow directly from the wave number and period.

3) Snell’s law for wave rays

sin(θ₁) / C₁ = sin(θ₂) / C₂

This relationship keeps the wave ray parameter constant as the wave moves into a region with different celerity.

4) Refraction coefficient

K_r = √[cos(θ₁) / cos(θ₂)]

H₂ = H₁ × K_r

The calculator reports a refraction-only wave-height change. It does not automatically apply shoaling or breaking limits.

5) Group velocity

n = 0.5 [1 + (2kh / sinh(2kh))]

C_g = nC

Group velocity is included in depth-based mode because it is useful for broader coastal transformation studies.

How to use this calculator

Select Depth-based dispersion when you know period and depths.
Select Direct celerity transition when wave speeds are already known.
Enter the incident ray angle measured from the contour normal.
Add initial wave height if you want the refraction-only height estimate.
Click Submit to display the result above the form.
Review the Plotly chart to inspect how angle and celerity evolve through the transition.
Use the CSV or PDF buttons to export the current calculation.

Example data table

Mode	Period (s)	h₁ (m)	h₂ (m)	θ₁ (deg)	H₁ (m)	C₁ (m/s)	C₂ (m/s)	θ₂ (deg)	K_r	H₂ (m)
Depth-based	12.00	20.00	5.00	30.00	2.00	12.6940	6.8389	15.6272	0.9483	1.8966

This example shows a wave ray rotating toward the contour normal as it enters shallower water and slows down.

FAQs

1) What does this calculator estimate?

It estimates the refracted wave-ray angle, celerity change, wavelength, refraction coefficient, and a refraction-only wave-height adjustment for coastal engineering analysis.

2) Why does the wave angle usually decrease in shallow water?

As depth decreases, wave celerity commonly decreases too. Snell’s law then rotates the wave ray toward the contour normal, reducing the approach angle.

3) What is the difference between depth-based and direct mode?

Depth-based mode solves celerity from period and depth using the dispersion relation. Direct mode skips that step and uses celerities you already know.

4) Does the reported wave height include shoaling?

No. The displayed H₂ value applies the refraction coefficient only. Shoaling, breaking, diffraction, and current effects must be added separately if needed.

5) How is the angle measured here?

The input angle is measured from the bottom-contour normal. A value of 0° means the wave ray approaches straight toward shore-normal alignment.

6) When should I use direct celerity mode?

Use it for abrupt interfaces, scaled physical models, imported simulation outputs, or any case where phase speeds are known more reliably than depths.

7) Can this tool be used for harbor or shoreline studies?

Yes. It is useful for first-pass screening in harbor alignment, shoreline exposure checks, coastal structures, and conceptual wave-transformation studies.

8) What limitations should I remember?

This tool assumes linear wave theory and a refraction-only transformation. It does not model diffraction, reflection, nonlinear breaking, currents, or complex bathymetry grids.