Wave Refraction Angle Calculator
Calculate wave refraction angle with our free science calculator. Uses standard scientific formulas with unit conversions and explanations.
Reviewed by Daniel Agrici, Founder & Lead Developer
Formula
sin(alpha2)/sin(alpha1) = C2/C1 (Snell Law)
Where alpha1 is the incident wave angle, alpha2 is the refracted wave angle, C1 is wave celerity at the initial depth, and C2 is wave celerity at the final depth. The refraction coefficient Kr = sqrt(cos(alpha1)/cos(alpha2)).
Worked Examples
Example 1: Wave Approaching Beach at Angle
Problem:A 10-second period wave with 2 m height approaches a beach at 30 degrees from deep water (100 m) to a nearshore depth of 5 m. Calculate the refracted angle and wave height.
Solution:Deep water wavelength L0 = gT^2/(2*pi) = 9.81*100/6.2832 = 156.1 m\nDeep water celerity C0 = 15.61 m/s\nShallow celerity C2 = sqrt(9.81*5) = 7.00 m/s\nSnell law: sin(alpha2) = (C2/C0)*sin(30) = (7.00/15.61)*0.5 = 0.2242\nalpha2 = arcsin(0.2242) = 12.96 degrees\nKr = sqrt(cos(30)/cos(12.96)) = sqrt(0.866/0.974) = 0.943\nKs = sqrt(Cg0/Cg2) = sqrt(7.81/7.00) = 1.056\nH2 = 2 * 0.943 * 1.056 = 1.99 m
Result:Refracted Angle: 12.96 deg | Kr: 0.943 | Ks: 1.056 | H2: 1.99 m
Example 2: Oblique Wave Approaching Reef
Problem:Waves with a 45-degree approach angle and 8-second period travel from 50 m depth over a reef at 3 m depth. Find the refracted angle.
Solution:L0 = 9.81*64/6.2832 = 99.9 m\nC0 = 99.9/8 = 12.49 m/s\nC_shallow = sqrt(9.81*3) = 5.42 m/s\nsin(alpha2) = (5.42/12.49)*sin(45) = 0.434*0.707 = 0.307\nalpha2 = arcsin(0.307) = 17.87 degrees\nTurning angle = 45 - 17.87 = 27.13 degrees
Result:Refracted Angle: 17.87 deg | Wave turned 27.13 deg toward shore normal
Frequently Asked Questions
What is wave refraction and why do waves bend toward shore?
Wave refraction is the bending of wave crests as they propagate from deep water into shallow water, caused by the variation of wave speed with water depth. In shallow water, wave celerity equals the square root of gravity times depth, so portions of a wave crest in shallower water travel slower than portions in deeper water. This speed difference causes the wave crest to pivot, bending toward the shallower region. The result is that waves approaching a straight shoreline at an angle will progressively turn to become more parallel to the beach contours. This process is analogous to the refraction of light passing between media of different densities and follows the same mathematical framework as Snell law of optics. Wave refraction is fundamental to understanding wave patterns along complex coastlines.
How is Snell law applied to wave refraction?
Snell law for water waves states that the ratio of the sine of the wave angle to the wave celerity remains constant along a wave ray: sin(alpha1)/C1 = sin(alpha2)/C2. This is mathematically identical to Snell law for light refraction in optics. To apply it, you need the wave approach angle at the initial depth and the wave celerity at both the initial and final depths. Since shallow water celerity depends only on depth through C = sqrt(g*d), you can calculate how the wave angle changes between any two depth contours. The law predicts that waves always bend toward regions of slower propagation speed, meaning toward shallower water. When waves approach perfectly perpendicular to the depth contours, no refraction occurs because the entire wave crest experiences the same speed change simultaneously.
What is the refraction coefficient and how does it affect wave height?
The refraction coefficient Kr quantifies how wave height changes due to the convergence or divergence of wave rays during refraction. It is calculated as the square root of the ratio of the spacing between adjacent wave rays at the initial position to the spacing at the final position. For straight parallel contours, Kr equals the square root of the cosine of the initial angle divided by the cosine of the refracted angle. When wave rays converge (such as at headlands), Kr exceeds 1.0 and wave height increases. When wave rays diverge (such as in bays), Kr is less than 1.0 and wave height decreases. The total change in wave height from deep to shallow water involves both the refraction coefficient and the shoaling coefficient, with the combined effect determining the actual wave height at any location.
How does wave refraction affect coastal erosion patterns?
Wave refraction concentrates wave energy on headlands and disperses it in bays, creating characteristic erosion and deposition patterns along irregular coastlines. At headlands, wave rays converge as waves wrap around the protruding landform, increasing wave height and energy density, which leads to accelerated erosion. In bays, wave rays diverge as the wider area is filled, reducing wave height and energy density, which promotes sediment deposition. Over geological time scales, this differential energy distribution tends to straighten coastlines by eroding headlands and filling bays. Understanding these refraction patterns is essential for coastal management decisions including where to build structures, where beach nourishment is most effective, and where natural erosion should be allowed to proceed.
References
Reviewed by Daniel Agrici, Founder & Lead Developer · Editorial policy