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Shallow-Water Waves

Note: this is the accessible print version of this module, which is also available in a multimedia version. To access the multimedia version, visit the module homepage and click Begin.

Outline

1. Introduction

1.1 Overview
1.2 Module Objectives
1.3 Welcome to Coast County State Park

2. Shallow-Water Wave Effects

2.1 How shallow is "shallow"?
2.2 Shoaling
2.2.1 Introduction
2.2.2 Shoaling Magnitude
2.2.3 Depth of Breaking
2.2.4 Exercise
2.3 Refraction
2.3.1 Introduction
2.3.2 Exercise
2.3.3 Refraction and Wave Heights
2.3.4 Exercise
2.4 Shallow-Water Wave Damping
2.4.1 Introduction
2.4.2 Example Exercise
2.5 Reflection
2.6 Combined Exercise

3. Breakers

3.1 Breaker Types
3.1.0 Introduction
3.1.1 Spilling Breakers
3.1.2 Plunging Breakers
3.1.3 Surging and Collapsing Breakers
3.2 Exercise

4. Wave-Current Interactions

4.1 Introduction
4.2 Example
4.3 Exercise

5. Coastal Effects

5.1 Breaker Run-Up
5.2 Wave Set-Up
5.2.1 Introduction
5.2.2 Example Exercise
5.2.3 Significance

6. Summary

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1. Introduction

1.1 Overview

As deep-water waves approach the coastline, they encounter shallower water and begin to interact with the sea floor while evolving into shallow-water waves. These interactions cause the waves to transform as they decelerate and finally break.

This module looks at a variety of shallow-water wave behaviors, including shoaling, refraction, reflection, breaking, attenuation, and coastal run-up and set-up. All are important considerations when forecasting for small craft and other recreational interests in the nearshore environment.

Photo of sailboats.

Two of the main external factors that influence shallow-water wave behaviour are beach orientation (relative to the incoming waves or swell) and nearshore bathymetry, or underwater topography. Without consideration of these factors and their related shallow-water effects, one can greatly underestimate, or overestimate, the wave action at the beach.

To help you understand and visualize these effects, we're going to visit a fictional stretch of coastline, introduced in the next section, which includes a variety of coastal features and bathymetry. As you work through the sections of this module, you'll learn some simple methods for estimating breaker heights and other important parameters. You'll also practice these methods by applying them to our fictional coastline.

This module also introduces a new estimation tool, our "Shallow-Water Wave Calculator," which may also be useful for marine forecast operations. You'll use it to answer exercise questions, and you'll learn some of the simple wave equations it incorporates.

We hope this module will be interesting and beneficial, helping you better understand nearshore wave behaviour and the key interdependencies between waves, sea floor, and coastline.

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1.2 Module Objectives:

By the end of this module, you will have learned:

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1.3 Welcome to Coast County State Park

Welcome to Coast County State Park, a popular weekend destination for pleasure craft sailors and beach goers in your forecast area. This scenic stretch of (fictional) coastline features a lighthouse at the tip of Rocky Point, the diminutive Inlet Cove with its marina and shops, and two beautiful beaches.

Photo of sand castle.

Sandy Beach hosts the Coast County Sand Castle Competition each year, while Surf Beach, as its name suggests, is most popular with the surfing crowd.

Photo of many surfers.

A coastal buoy is situated over deep water, not too far from shore--which comes in handy for assessing coastal hazards and surf conditions.

The variations in coastal features and bathymetry shown on this map allow us to learn about all the important shallow-water effects that modify deep-water waves as they approach the shoreline. We'll return to this setting throughout this learning module to practice the shallow-water wave estimation methods as they are introduced.

Base map of coastal setting

 

2. Shallow-Water Wave Effects

2.1 How shallow is "shallow"

The depth at which a deep-water wave enters "shallow water" (as defined below) depends on the wave's period. Long-period waves have long wavelengths, and their shallow-water zones extend into proportionally deeper water. For the purposes of this module, water depth zones are defined on the basis of the ratio of water depth to wavelength. We have divided these into three zones:

Deep Water:
Water is deeper than one-half of the wavelength
 
Shallow Water:
Water depth is less than or equal to one-twentieth of the wavelength
 
Transition Zone:
Water depth is between the deep- and shallow-water thresholds

Deep, transition, shallow, and shoaling zones.

These divisions are based on simplifications we can make in calculating wave speed, or celerity. In deep water, wave celerity (in knots) is approximately equal to the wave period (T) times three. To get the metric equivalent in meters per second, multiply the wave period by 1.56.

Deep-water Celerity (knots) = T * 3.01

(An easy-to-remember approximation for celerity in knots is 3T.)

Deep-water Celerity (m/s) = T * 1.56

In shallow water, where the depth is equal or less than one-twentieth of the wavelength, the celerity is proportional to the square root of gravity times the water depth.

Shallow-water Celerity (m/s) = (gH)1/2

In the transition zone between these two thresholds, celerity is calculated by a more complicated formula.

Transition Zone Celerity (m/s) = [(gL/2π)* tanh(2π h/L)]1/2

Although we have divided depth zones in this way, the transition from deep to shallow water occurs along a continuum, beginning at the point where water depth is equal to one-half of the wavelength. At this depth, frictional drag begins to dissipate wave energy, but the effect is very slight at the deepest depths and becomes progressively more significant as the wave enters shallower water. Also, there is no "magical" transformation that occurs when a wave passes from the transition zone into the shallow-water zone, so these distinctions are somewhat artificial, despite their utility for describing shallow-water wave processes.

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2.2 Shoaling

Shoaling is the process through which wave heights increase as water depth decreases. A fundamental characteristic of waves to remember is the fact that wave period is ALWAYS conserved. If you keep this in mind, the shoaling process is not complicated.

2.2.1. Introduction

As a wave enters the transition zone, the wave bottom begins to drag on the sea floor causing the wave to slow down. We'll attribute this dragging effect to simple friction between the wave and the sea floor, even though the actual process is more complex. The important thing to remember is that waves decelerate as they enter shallow water. Since wave period is always conserved, and since wave speed is the ratio of wavelength to period, any decrease in speed must be accompanied by a decrease in wavelength.

3D visualization of the shoaling process.

As the wave energy becomes concentrated into a narrower and slower-moving band, its kinetic energy is converted into the potential energy of a slower, taller wave. Therefore, shoaling reduces wavelength and speed, while increasing wave height. As the wave moves into progressively shallower water, this process continues until the wave becomes unstable and finally breaks.

Here are some photos of shoaling breakers. Notice, that there is very little visible wave action over the deep-water areas. Closer to the shore, however, you begin to see the wave lines become more prominent, rising up in the shallower water until they finally break. These transitions illustrate the shoaling process.

Surf action on the West Coast of Vancouver Island, near Tofino, B.C.

(Click here for an animated photo sequence showing shoaling.)

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2.2.2. Shoaling Magnitude

The magnitude of shoaling depends on wave period. The longer the period, the larger the shoaling effect will be when a wave reaches shallow water. The shoaling effect is important to account for when forecasting breaker heights, because in some situations, it can increase deep-water wave heights by 50 to 200 percent.

The shoaling effect coefficient for a given deep-water wave can be derived from the deep-water height and period. The shoaling coefficient, k, for a given significant wave height, s, is expressed as the square root of the deep-water group velocity, CgD, divided by the group velocity to which the wave will slow in the breaking zone, CB:

ks = (CgD/CB)1/2

Both of these velocities can be approximated if you know the wave period and deep-water wave height. The deep-water group velocity is one-half the individual wave velocity:

CgD = CD / 2

The group velocity at breaking is proportional to the square root of the gravitational constant, g, times the breaking depth, db:

CB = (g db)1/2

There are a number of ways to estimate breaking depth. The simplest is the so-called "five-thirds rule," which is applicable for "spilling breakers." (You'll learn about spilling breakers in a later section of this module.) This rule simply states that the depth of breaking is approximately five-thirds (or 1.67 times) the deep-water significant wave height, Hs:

db = 5/3 * Hs

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2.2.3. Depth of Breaking

As it turns out, both the shoaling effect and the depth of breaking are dependent on beach slope. If you want greater precision in your estimations of these factors, you should use two reference nomograms developed from empirical studies.

The first gives the shoaling factor for a given deep-water wave steepness:

Breaker heights for various deep-water wave steepnesses and beach slopes.

(Click here for an expandable, interactive version of this nomogram.)

The curves on this chart represent four different beach steepnesses, ranging from a fairly steep 0.1 slope to the much gentler 0.02 slope.

To use this nomogram, first find the deep-water wave steepness, which is the wave height divided by the wavelength. Enter the nomogram from the bottom and trace upward along the line for your wave steepness until you hit one of the beach slope curves. Then, trace to the left along the horizontal axis to find the shoaling factor. Multiply this factor by the significant wave height to obtain the estimated shoaling breaker height.

For example, if you have a 10-foot deep-water wave with a steepness of 0.02, and your beach slope is 0.05 (1:20), the shoaling factor would be 1.4:

Illustration of the example 0.02 steepness and 1.4 shoaling factor.

Multiplying the 1.4 shoaling factor by the deep-water wave height of 10 feet produces an expected shoaling breaker height of 14 feet.

The second nomogram gives the depth of breaking for a given breaker steepness and beach slope:

Breaking depth nomogram

(Click here for an expandable, interactive version of this nomogram.)

The curves on this chart represent a range of different beach steepnesses. To use this nomogram, first find the breaker steepness, which is the breaker height divided by the product of the gravitational constant and the square of the wave period:

Breaker Steepness = HB / g T2

Enter the nomogram from the bottom and trace upward along the line for your breaker steepness until you hit a beach slope curve corresponding to your beach of interest. Then, trace to the left along the horizontal axis to find the breaking depth coefficient. Multiply this coefficient by the expected breaker height to obtain the estimated depth of breaking.

Shoaling is extremely important to consider in a wave forecast because it can significantly increase breaker heights. The shoaling factor is also important because it can be further amplified by other nearshore and coastal effects.

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2.2.4. Exercise

To aid in estimation of shoaling breaker heights, we have developed a Shallow-Water Wave Calculator, which you can use both to answer the exercise questions in this module and on the job when preparing coastal hazards and surf zone forecasts. Click the Shallow-Water Wave Calculator link to open this tool and use it to complete the following exercise.

The deep water buoy reports 15-second waves from the northwest, with a significant height of 10 feet. What estimates can you make?

1. What is the approximate deep water velocity? (Use the "Deep Water" tab on the Shallow-Water Wave Calculator.) (Choose the best answer.)

a) 14 knots
b) 4-20 knots
c) 22.5 knots
d) 45 knots

2. At what depth will these waves enter the transition zone?
(Choose the best answer.)

a) 10 feet
b) 10 fathoms
c) 96 fathoms
d) 176 meters

3. At what depth will these waves enter the shallow-water zone?
(Choose the best answer.)

a) 10 feet
b) 10 fathoms
c) 96 fathoms
d) 176 meters

4. What approximate height will these waves reach as shoaling breakers on the west side of Rocky Point, where the bathymetry slope of 0.065 is steeper than average? (HINT: Use the "Shoaling" tab on the Shallow-Water Wave Calculator and refer to the Coast County State Park map to see the bathymetry contours along the west side of Rocky Point.)
(Choose the best answer.)

a) 10-12 feet
b) 13-14 feet
c) 15-16 feet
d) 17-18 feet

5. What would be the depth of breaking for these waves on a beach with average slope (1:20)? (HINT: Use the "Shoaling" tab on the Shallow-Water Wave Calculator to find the breaker steepness, then open the Breaking Depth Nomogram and use the breaker steepness to find the breaking depth coefficient. Multiply this coefficient by the estimated shoaling breaker heights.)
(Choose the best answer.)

a) 14-17 feet
b) 18-21 feet
c) 22-25 feet

6. The regional wave model predicts the following swell periods and heights over the next two days:

Forecast Time Height Period
12-hr
5 feet
18 seconds
24-hr
8 feet
16 seconds
36-hr
10 feet
11 seconds
48-hr
8 feet
8 seconds

At which forecast time should you expect the greatest breaker heights to occur on a a steep beach (1:10 slope)? (HINT: Use the Shallow-Water Wave Calculator to compare shoaling breaker heights for the different forecast swell sets.)
(Choose the best answer.)

a) 12-hr
b) 24-hr
c) 36-hr
d) 48-hr

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ANSWERS

1. What is the approximate deep water velocity? (Use the "Deep Water" tab on the Shallow-Water Wave Calculator.)

Answer: 22.5 knots. This is given on the Deep Water tab of the Shallow-Water Wave Calculator.

2. At what depth will these waves enter the transition zone?

Answer: 96 fathoms. This is equal to 576 feet, or one-half of the wavelength.

3. At what depth will these waves enter the shallow-water zone?

Answer: 10 fathoms. This is equal to 60 feet, or one-twentieth of the wavelength.

4. What approximate height will these waves reach as shoaling breakers on the west side of Rocky Point, where the bathymetry slope of 0.065 is steeper than average?

Answer: 17-18 feet. This is given on the Shoaling tab of the Shallow-Water Wave Calculator. Shoaling breaker heights for a 10-foot, 15-second wave range from 15 to 18 feet, with the higher breakers occuring on steeper beaches.

5. What would be the depth of breaking for these waves on a beach with average slope (1:20)?

Answer: 14-17 feet. The shoaling breaker heights for average to steep beaches are 17-18 feet. The associated breaker steepnesses are 0.00235 to 0.00249. Looking these values up on the breaking depth nomogram obtains breaking depth factors ranging from about 0.78 to 0.94. When multiplied by the 17-18 foot shoaling breaker heights, these correspond to breaking depths of 14 to 17 feet.


6. The regional wave model predicts the following swell periods and heights over the next two days:

Forecast Time Height Period
12-hr
5 feet
18 seconds
24-hr
8 feet
16 seconds
36-hr
10 feet
11 seconds
48-hr
8 feet
8 seconds

At which forecast time should you expect the greatest breaker heights to occur on a a steep beach (1:10 slope)?

Answer: 24-hr. As the following table shows, the greatest breaker heights could be expected to occur on steep beaches at least 12 hours before the highest swell packet arrives. This is because the shoaling effect is greater for longer-period waves, so an 8-foot, 16-second wave has a higher breaker than a 10-foot, 11-second wave.

Forecast Time Height Period Shoaling Breaker Heights
(steep beach)
12-hr
5 feet
18 seconds
11 ft (3.3 m)
24-hr
8 feet
16 seconds
16 ft (4.9 m)
36-hr
10 feet
11 seconds
15 ft (4.7 m)
48-hr
8 feet
8 seconds
11 ft (3.4 m)

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2.3. Refraction

2.3.1. Introduction

Refraction is the bending of wave energy due to a complex process that we will broadly term "bottom friction."

Refraction along a straight beach.

In the above illustration, the wave is not approaching the coastline straight on, so the section of the wave that is in shallower water closest to the shore is experiencing more frictional drag and deceleration than the portion in deeper water. The speed differences cause the wave to bend toward shallower water, so that the wave conforms to the shoreline. The amount of refraction depends on the incoming wave's angle of incidence; a wave will not refract where its angle of incidence is perpendicular to the nearshore bathymetry.

The following picture shows a wave conforming to a curved beach. Because of the bathymetry, the wave is slowing and turning toward shallower water, attempting to approach the beach straight on from all sides.

Photo of refraction in a bay.

The following picture illustrates the refraction effect on a smaller scale. Notice how the incoming wave energy bends around the large rock outcropping and propagates along the narrow channel at the base of the rock.

Photo showing wave energy bending around a rock.

Refraction has the largest effect on long-period waves because wavelength, L, is directly proportional to the square of wave period, T:

L(meters) = 1.56 T2 ; L(feet) = 5.12 T2

The longer the wavelength, the deeper its "wave depth" becomes, because wave depth is equal to one-half of the deep-water wavelength:

Wave Depth = LD / 2

When wave periods are long, refraction begins in deeper water; when periods are short, bending begins much closer to shore and the refraction effects are diminished. Consequently, short-period wave orientations are not altered much when moving from deep to shallow water.

In short, refraction is minimal with short-period waves and steep sea floors or abrupt transitions from deep to shallow water. Maximum refraction occurs with long-period waves and mild sea floor slopes.

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2.3.2. Exercise

An offshore buoy reports North swell with a 7-second period. What predictions can you make?

1. What is the significant wavelength? (HINT: Use the "Deep Water" tab on the Shallow-Water Wave Calculator.)
(Choose the best answer.)

a) 13 feet
b) 76 feet
c) 251 feet

2. What is the transition zone starting depth?
(Choose the best answer.)

a) 13 feet
b) 21 fathoms
c) 38 meters

3. At what depth will this swell begin to refract? (HINT: Recall that the transition zone begins at a depth equal to one-half the deep-water wavelength.)
(Choose the best answer.)

a) 13 feet
b) 21 fathoms
c) 251 feet

4. About how far from shore will waves begin to refract along the west side of Rocky Point? (HINT: Use the bathymetry contours on the Coast County State Park map to determine how far from shore the waves will enter the transition zone.)
(Choose the best answer.)

a) Within 200 yards (183 m)
b) Approx. 200 yards (183 m)
c) Farther than 200 yards (183 m)

5. For which of the following wave periods would refraction effects be the greatest?
(Choose the best answer.)

a) 3-second
b) 7-second
c) 15-second

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ANSWERS

An offshore buoy reports North swell with a 7-second period. What predictions can you make?

1. What is the significant wavelength?

Answer: 251 feet.

2. What is the transition zone starting depth?

Answer: 21 fathoms.

3. At what depth will this swell begin to refract?

Answer: 21 fathoms.

4. About how far from shore will waves begin to refract along the west side of Rocky Point?

Answer: Farther than 200 yards.

5. For which of the following wave periods would refraction effects be the greatest?

Answer: 15-second.

The wave period can tell you a lot, including the wavelength, velocity, and depths at which the wave will enter the transition and shallow-water zones.

A seven-second wave begins to interact with the bottom at a depth of approximately 40 meters or 21 fathoms. This is over 200 yards from shore along the west side of Rocky Point. In general, refraction effects are greater for long-period swell than for short-period waves.

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2.3.3. Refraction and Wave Heights

Along shallow, straight, and uniform stretches of shoreline, refraction always reduces breaker heights as wave energy is lost to bottom friction. However, refraction effects are minimal for short wave periods and/or when the wave's direction of propagation is nearly perpendicular to the beach orientation.

Refraction can significantly alter breaker heights in areas with sharp variations in coastline orientation, especially points, headlands, bays, and inlets. As waves bend toward shallow water, refraction concentrates and focuses the wave energy at the tips of points and headlands. The converse is true in bays and inlets, where refraction spreads the wave energy, reducing wave heights.

Refraction demonstrated at the tip of Rocky Point

(Click here to view the animated version of this illustration.)

In the above example, the section of deep-water swell that eventually strikes the headland is approximately 80 meters (about 260 feet) wide. Because of the convex shape of the headland, refraction causes the deep-water portions of the waves to bend toward shallow water, focusing the waves into a smaller area. The length of shoreline over which the focused waves break is 40 meters (about 130 feet) wide, roughly half its original width. Hence, energy density (and wave height) will be doubled.

For symmetrical headlands surrounded by somewhat evenly sloped bathymetry, the refraction effect on breaker heights can be approximated by the following formula:

Cr = 1 + 0.5 * [sin(a) * ((L/2) / 50)]

In this formula, Cr is the refraction coefficient; L/2 is the wave depth; and a, as the following graphic illustrates, is the angle over which an incoming wave that is parallel to the surrounding shoreline will bend to conform to the shape of the headland. In this case "a" would be 24 degrees.

Angle of symmetrical refraction

Note that this formula can only be applied reliably for angles of refraction up to about 40 degrees. For sharper angles, it is not as useful because extremely small variations in bathymetry and underwater features can have large effects on narrow headland areas.

When you need to forecast breaker heights for such locations, there are two possible approaches:

  1. Use a high-resolution, nearshore wave refraction model, or
  2. Develop an empirical data set, correlating buoy observations with observed breaker heights for a range of orientations and wave periods.
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2.3.4. Exercise

To aid in estimation of the refraction effect for points and headlands, we have included a "Refraction Effect" tab in our Shallow-Water Wave Calculator, which you can use both to answer the exercise questions in this module and on the job when preparing coastal hazards and surf zone forecasts. Open it now and use it to complete the following exercise.

The deep-water buoy near Coast County State Park reports 15-second NNW swell at a significant height of 10 feet. What predictions can you make? (Choose the best answers for each of the following questions then click Done.)

1. Assuming that the angle of convergence for Surf Beach is 20 degrees and the beach slope is steep (1:10), what height would refraction cause breakers to grow to at the point? (HINT: Choose the best answer for Hs waves, not H1/10 or larger.)
(Choose the best answer.)

a) 15 feet
b) 16 feet
c) 17 feet
d) 18 feet
e) 24 feet
f) 29 feet

2. The regional wave model predicts the following NNW swell with the following periods and heights over the next two days:

Forecast Time Height Period
12-hr
6 feet
20 seconds
24-hr
9 feet
14 seconds
36-hr
10 feet
12 seconds
48-hr
8 feet
8 seconds

At which forecast time should you expect the greatest refracted breaker heights to occur on at Surf Beach (1:10 slope)?
(Choose the best answer.)

a) 12-hr
b) 24-hr
c) 36-hr
d) 48-hour

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ANSWERS

The deep-water buoy near Coast County State Park reports 15-second NNW swell at a significant height of 10 feet. What predictions can you make?

1. Assuming that the angle of convergence for Surf Beach is 20 degrees and the beach slope is steep (1:10), what height would refraction cause breakers to grow to at the point?

Answer: 29 feet. The best answer is 29 feet (8.8 m), which is significantly larger than the 15 feet. shoaling breaker heights that could be expected along a straight stretch of coastline. But these estimations are given only for significant wave heights; when you consider Hmax waves, the refracted shoaling breakers could reach heights over 57 feet (17.6 m)! This shows the dramatic influences refraction can have on wave heights at the points of headlands and other convex coastline features.

The converse effect will occur in Sandy Beach, where the concave geometry of this bay will cause refraction effects to significantly reduce shoaling breaker heights in the center part of this beach.

2. The regional wave model predicts the following NNW swell with the following periods and heights over the next two days:

Forecast Time Height Period
12-hr
6 feet
20 seconds
24-hr
9 feet
14 seconds
36-hr
10 feet
12 seconds
48-hr
8 feet
8 seconds

At which forecast time should you expect the greatest refracted breaker heights to occur on at Surf Beach (1:10 slope)?

Answer: The 12-hour forecast. As the following table shows, the refraction effect will cause the greatest breaker heights to occur at least 24 hours before the highest swell packet arrives. This is because the 20-degree angle of convergence at the point at Surf Beach allows the refraction effect to magnify the impact of shoaling when there is northwest swell.

Forecast Time Height Period Shoaling Breaker Heights
(Hs)
Refracted Shoaling Breaker Heights
(Surf Beach)
12-hr
6 feet
20 seconds
13 ft (4 m)
27 ft (8.3 m)
24-hr
9 feet
14 seconds
16 ft (4.9 m)
25 ft (7.5 m)
36-hr
10 feet
12 seconds
16 ft (4.8 m)
22 ft (6.7 m)
48-hr
8 feet
8 seconds
11 ft (2.4 m)
13 ft (4 m)

There are three key points to remember about refraction:

1. Except in convex sections of coastline where wave energy is concentrated, refraction always reduces breaker heights because wave energy is lost to frictional drag along the bottom.

2. The effect of refraction is always greater for long-period than for short-period waves.

3. Refraction may also change the timing of when maximum breaking waves are observed compared to buoy observations.

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2.4. Shallow-Water Wave Damping

Whenever waves move into shallow water, they lose energy through frictional drag along the bottom. The amount of energy loss depends on two factors, bottom roughness and the distance over which a wave propagates through shallow water of a given depth. In some regions and locations, wide zones of shallow water near the coast can significantly attenuate wave energy. Examples are found in many parts along the coast of the Gulf of Mexico, and in the shallow waters offshore from Long Beach, California, shown in these bathymetry charts.

Gulf of Mexico bathymetry

Long Beach, CA bathymetry

 

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2.4.1. Introduction

Assuming an average bottom roughness, it is possible to estimate the amount of attenuation that will occur for a given wave traveling over an extended stretch of shallow water.

Frictional decay can be roughly estimated from this Frictional Decay Coefficient formula, which is based on bathymetry slope and wave period:

Frictional Decay Coefficient = (bathymetry slope / 0.15)1/2 * (6 / T)1/2

Multiply this coefficient by the expected local shoaling and refracted breaker heights to find the resulting approximate attenuated heights to expect after the wave travels over the stretch of water (in the shallow-water zone) defined by the bathymetry slope. Note that this formula provides only crude estimates based on generic slope values; in many situations, better results could likely be obtained by comparing statistical records of deep-water waves to observed breaker heights for your coastal area of interest.

Attenuation effects, based on this approximation, have been incorporated into our Shallow-Water Wave Calculator, but attenuation can also occur through wave-wind interactions. For example, strong offshore winds can rapidly dissipate incoming waves moving through shallow waters. Wind-wave interactions are not accounted for by the Shallow-Water Wave Calculator

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2.4.2. Example Exercise

Consider an example of 10-second, 15-foot waves traveling through an extended shallow-water zone. (Use the "Attenuation" tab on the Shallow-Water Wave Calculator and refer to the Coast County State Park map to find the best answers for each of the following questions.)

1. What is a rough estimate of the attenuated significant wave height for waves that travel through a shallow-water zone with a bathymetry slope of 0.05 (1:20)?
(Choose the best answer.)

a) ~7 feet (2.1 m)
b) ~9 feet (2.7 m)
c) ~11 feet (3.4 m)
d) ~13 feet (3.9 m)
e) ~15 feet (4.6 m)

2. What if, instead of a 0.05 (1:20) slope, the waves travel through a shallow-water zone with a much milder 0.005 (1:200) slope? What is a rough estimate of the attenuated wave heights?
(Choose the best answer.)

a) ~7 feet (2.1 m)
b) ~9 feet (2.7 m)
c) ~11 feet (3.4 m)
d) ~13 feet (3.9 m)
e) ~15 feet (4.6 m)

3. How about small, short-period waves? What would be a rough estimate of attenuated wave heights for 4-foot, 4-second waves traveling through a shallow-water zone with the same 0.005 (1:200) bathymetry?
(Choose the best answer.)

a) ~1 feet (0.3 m)
b) ~2 feet (0.6 m)
c) ~3 feet (0.9 m)
d) ~4 feet (1.2 m)

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ANSWERS

Consider an example of 10-second, 15-foot waves traveling through an extended shallow-water zone.

1. What is a rough estimate of the attenuated significant wave height for waves that travel through a shallow-water zone with a bathymetry slope of 0.05 (1:20)?

Answer: ~15 feet (4.6 m). Since attenuation effects are small for relatively steep bathymetry slopes, negligible attenuation could be expected for this wavelength and slope -- IF bottom roughness is minimal.

2. What if, instead of a 0.05 (1:20) slope, the waves travel through a shallow-water zone with a much milder 0.005 (1:200) slope? What is a rough estimate of the attenuated wave heights?

Answer: ~7 feet (2.1 m). With this milder slope, the attenuation is much greater, reducing wave heights by over 50%.

3. How about small, short-period waves? What would be a rough estimate of attenuated wave heights for 4-foot, 4-second waves traveling through a shallow-water zone with the same 0.005 (1:200) bathymetry?

Answer: ~3 feet (0.9 m). The reduction by attenuation is much less for short-period waves, reducing 4-second waves by only 25% compared to the 50% reduction for the longer-period wave example above.

The main point to remember about attenuation is that it can significantly reduce wave heights in areas with mild, gentle, shallow-water bathymetry. The effects are greater for the longer-period waves and milder bathymetry slopes. However, the estimates provided by the Shallow-Water Wave Calculator are rough approximations because attenuation is so dependent on bottom roughness and local variations in bathymetry. For the best results, look to statistical records for a given coastal area, which compare deep-water swell of various wavelenghts, heights, and directions to observed breaker heights at the beaches.

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2.5 Reflection

Reflection occurs when a wave hits an obstacle, bounces off, and continues in a new direction. Waves may reflect from seawalls, jetties, natural cliffs, and even large ship hulls. At certain angles of incidence, the reflected waves can be superimposed on other incoming waves resulting in significantly increased wave heights. These reflected waves may be large and pose hazards for people and small boats.

Visualization of two waves colliding.

(Click here to view the animated version of this illustration.)

The amount of wave energy reflected by an obstacle depends on the breaking state of the wave as well as the slope and hardness of the obstacle. For non-breaking waves in deep water, nearly 100% of the energy can be reflected by a hard, vertical surface like a sea wall. Such reflected waves, when superimposed on other incoming waves, could double in height.

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2.6. Combined Exercise

The deep water buoy reports 20-second swell from the northwest at a significant height of 16 feet. What predictions can you make?

1. What is approximate deep-water group velocity?
(Choose the best answer.)

a) 16 knots (~8 m/s)
b) 30.5 knots (~16 m/s)
c) 61 knots (~31 m/s)

2. What is the significant wavelength?
(Choose the best answer.)

a) 16 feet (4.9 m)
b) 32 feet (9.8 m)
c) 624 feet (190 m)
d) 2047 feet (624 m)

3. At what depth will this swell begin to refract?
(Choose the best answer.)

a) 32 feet (9.8 m)
b) 17 fathoms (31 m)
c) 171 fathoms (312 m)
d) 2047 feet (624 m)

4. What is the shallow-water starting depth for these waves?
(Choose the best answer.)

a) 32 feet (9.8 m)
b) 17 fathoms (31 m)
c) 171 fathoms (312 m)
d) 2047 feet (624 m)

5. What approximate height will these waves reach as shoaling breakers on beaches where there are no refraction effects?
(Choose the best answer.)

a) 15-19 feet (4.6-5.8 m)
b) 20-24 feet (6.1-7.3 m)
c) 25-30 feet (7.5-9.1 m)

6. What would be the approximate depth of breaking on a moderately steep beach, like the west shore of Rocky Point (1:15 slope)? (HINT: Use the Breaking Depth Nomogram to find the answer.)
(Choose the best answer.)

a) ~20 feet (6.1 m)
b) ~25 feet (7.6 m)
c) ~30 feet (9.1 m)
d) ~35 feet (10.7 m)

7. What would be the resulting height of shoaling, refracted breakers at the point of Surf Beach? (HINT: Assume an angle of refraction of 20 degrees and a steep (1:10) beach slope.)
(Choose the best answer.)

a) 25 feet (7.6 m)
b) 51 feet (15.5 m)
c) 55 feet (16.7 m)
d) 58 feet (17.7 m)
e) 62feet (18.9 m)

8. What would happen if this swell travels over the shallow-water area north and east of Coast County State Park, where the bathymetry slope is 0.02? What would be the resulting attenuated wave heights? (HINT: Use the "Attenuation" tab on the Shallow-Water Wave Calculator to find the answer.)
(Choose the best answer.)

a) ~10 feet (3.1 m)
b) ~12 feet (3.7 m)
c) ~14 feet (4.3 m)
d) ~16 feet (4.9 m)

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ANSWERS

The deep water buoy reports 20-second swell from the northwest at a significant height of 16 feet. What predictions can you make?

1. What is approximate deep-water group velocity?

Answer: 30.5 knots (~16 m/s). The deep-water group velocity is one half of the single-wave velocity.

2. What is the significant wavelength?

Answer: 2047 feet (624 m). A quick mental calculation for wavelength in feet is to multiply T2 times 5; the quick metric approximation is T2 times 1.5.

3. At what depth will this swell begin to refract?

Answer: 171 fathoms (312 m). Refraction begins at a depth equal to one-half the wavelength, which is the transition zone starting depth. However, refraction effects are very slight at this depth and don't become significant until the wave enters shallower water.

4. What is the shallow-water starting depth for these waves?

Answer: 17 fathoms (31 m). The shallow water starting depth is equal to one twentieth of the wavelength.

5. What approximate height will these waves reach as shoaling breakers on beaches where there are no refraction effects?

Answer: 25-30 feet (7.6-9.1 m). Shoaling factors for these large, long-period waves range from 1.54 to 1.86. The resulting breaker heights are significant and dangerous, but actual breaker heights will depend on the location. In some areas refraction and/or attenuation effects will significantly reduce breaker heights. In other locations, refraction effects can magnify the shoaling effect.

6. What would be the approximate depth of breaking on a moderately steep beach, like the west shore of Rocky Point (1:15 slope)?

Answer: 25 feet (7.6 m). Breaker steepnesses on a 1:15 slope beach is about 0.00226. This corresponds to a breaking depth factor of 0.86 on the nomogram. When multiplied by the shoaling breaker height of about 29 feet (the shoaling breaker height on a beach with 1:15 slope), these factors predict breaking depths of ~25 feet, or 6.1 m.

7. What would be the resulting height of shoaling, refracted breakers at the point of Surf Beach?

Answer: 62 feet (18.9 m). Incredible as it may seem, shoaling breakers for a 16-feet 20-second wave could reach over 60-feet in height when magnified by the refraction effect at the point of a 20-degree promontary of land. Such waves would be extremely dangerous and damaging. After such an episode, "Surf Beach" may be eroded beyond recognition.

8. What would happen if this swell travels over the shallow-water area north and east of Coast County State Park, where the bathymetry slope is 0.02? What would be the resulting attenuated wave heights?

Answer: ~10 feet (3.1 m). These large waves would be attenuated significantly after travelling over an extended shallow-water zone like the one to northeast of Surf Beach.

This extreme example illustrates the range of variations that may be expected for large, long-period waves entering shallow water and striking a non-uniform coastline.

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3. Breakers

Photo of a breaking wave.

Why do waves break?

As a wave enters shallower water, its wavelength shortens and its height and steepness increase due to interaction with the bottom. The lower part of the wave slows through frictional drag along the bottom, while the top of the wave continues to propagate at a higher rate of speed. The top of the wave eventually outruns the slower moving base and it begins to spill over, breaking down the front of the wave.

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3.1 Breaker Types

3.1.0 Introduction

The manner in which a wave breaks depends on the wave's depth with respect to the bathymetry. Understanding this dependency will help you identify locations where hazardous breakers and significant beach erosion could be expected for a given wave period and tidal phase.

There are four basic types of breaking waves: spilling, plunging, surging, and collapsing. Since breaker types are dependent on the depth of the water, tidal variations must be taken into consideration when anticipating breaker type.

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3.1.1 Spilling Breakers

Spilling breakers occur in areas with gradual, gently sloped bathymetry. They are characterized by their gentle slope and the way the breaking water spills directly down along the forward slope of the wave. The depth at which a spilling wave will begin to break is approximately 1.67H, where H is the height at breaking.

Schematic illustration of a spilling breaker.

(Click here to view the animated version of this illustration.)

 

Notice in the following picture that the wave is not very steep and once a certain height is reached, the water tumbles down the front of the wave.

Photo of a spilling breaker.

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3.1.2 Plunging Breakers

Plunging breakers are the type surfers like.

Photo of a surfer hitting a curl.

They occur when the wave encounters an abrupt transition from deep to shallow water. The base of the wave decelerates rapidly, while the top of the wave continues moving at a higher speed. With this large speed differential, the top of the wave pitches out in front, forming a curl or tube.

Schematic illustration of a plunging breaker.

(Click here to view the animated version of this illustration.)

Photo of a plunging breaker.

Plunging breakers can be more dangerous for beachgoers because the breaking action is much more sudden and concentrated. This type of breaker can also cause much more beach erosion than other types because it breaks with such concentrated force in shallow water, scouring sand away. Notice in this picture, the brown coloration within the plunging wave and the murkiness of the water in the foreground. These are signs of sand transport and beach erosion.

Photo of a murky plunging breaker.

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3.1.3 Surging and Collapsing Breakers

Surging breakers are relatively uncommon for many operational marine forecasters. However, there are areas, such as the Pacific Northwest, where surging breakers are deadly because they can knock the unsuspecting beachgoer into deep water.

Surging breakers occur along stretches of coastline where there is no shoaling zone due to extremely rapid drop-offs. If this is the case, the deep water waves never break before reaching land. Instead, the water just pushes or surges up onto the beach

Schematic of surging wave

(Click here to view the animated version of this illustration.)

Since surging breakers do not break off shore, they do not lose energy and therefore may run up the beach with force and distance.

Photo of a surging breaker.

Collapsing breakers occur with very small, short-period waves, which shoal and collapse right onto the beach. These breakers are operationally insignificant because they are so small and pose negligible hazard.

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3.2 Exercise

Which breaker types would you expect to occur most commonly at which shore areas in Coast County State Park?

1. Sandy Beach
(Choose the best answer.)

a) Plunging Breakers
b) Spilling Breakers
c) Surging Breakers

2. Surf Beach
(Choose the best answer.)

a) Plunging Breakers
b) Spilling Breakers
c) Surging Breakers

3. The northeast side of Rocky Point
(Choose the best answer.)

a) Plunging Breakers
b) Spilling Breakers
c) Surging Breakers

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ANSWERS

Which breaker types would you expect to occur most commonly at which shore areas in Coast County State Park?

1. Sandy Beach

Answer: Spilling Breakers. The concave shape and mild bathymetry of Sandy Beach make spilling breakers the most likely of these three breaker types to occur on that beach.

2. Surf Beach

Answer: Plunging Breakers. Plunging breakers may be more likely at Surf Beach, especially with northwest swell, because the change in water depth is so abrupt.

3. The northeast side of Rocky Point

Answer: Surging Breakers. The dropoff along the northeast shore of Rocky Point, near the lighthouse, is extremely sudden. This would favor surging breakers in most situations.
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4. Wave-Current Interactions

4.1 Introduction

Waves may be affected by sea currents, depending on their orientation with respect to the direction of wave propagation. When the current is flowing in the same direction as the waves, the wavelengths can increase while heights decrease:

Schematic of current moving in the same direction as waves

(Click here to view an animated version of this illustration.)

Where currents oppose waves, wavelengths decrease and heights increase, sometimes causing waves to break. This effect can be significant in the areas of river and harbor bars and along the Gulf Stream off the U.S. East Coast (from Florida to Cape Hatteras).

Schematic on current opposing waves

(Click here to view an animated version of this illustration.)

The magnitude of this effect is proportional to the difference between the current speed and wave velocity. As you have learned, wave period, T, is equal to wavelength, L, divided by wave celerity, C:

T = L/C

We can use these relationships to see how currents can affect waves:

Current Direction SAME AS Wave Direction Current OPPOSES Wave Direction
T = L/(C + u) ; L = T * (C + u)
T = L/(C - u) ; L = T * (C - u)
(T = Wave Period; L = Wavelength; C = Wave Celerity ; u = Current Velocity)

 

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4.2 Example

Consider the situation where 8-second, 8-foot swell from the northeast opposes a section of the Gulf Stream current flowing from the southeast at 1.5 m/s (about 3 knots).

Before entering the Gulf Stream, the swell has the following dimensions:

T = 8 seconds
L = 100 m (about 328 ft)
C = 12.5 m/s (about 24 kt)

After the swell enters the Gulf Stream, its speed is reduced by the opposing speed of the current:

C = (12.5 m/s - 1.5 m/s) = 11 m/s (about 22 kt)

Since wavelength is equal to wave period times wave celerity, the resulting wavelength within the Gulf Stream current, Lc, shrinks from 100 m to 88 m:

Lc = T * (C - u) = 8 seconds * 11 m/s = 88 meters (about 289 feet)

This represents a 12% reduction in wavelength, which would be accompanied by a corresponding 12% increase in wave heights.

Hc = H + (H * 12%) = 8 feet * 112% = almost 9 feet

What had been 8-second, 8-foot swell becomes a shorter, steeper 8-second, 9-foot swell.

 

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4.3. Exercise

In our fictional forecast area, Coast County State Park, there is a river that flows into the sea at Inlet Cove. Imagine that the speed of the current produced by this river is 1 m/s (about 2 knots). If 5-foot, 3-second waves approaches Inlet Cove from the west, what would be the resulting wave heights in the area where the incoming swell encounters the Inlet Cove current?
(Choose the best answer.)

a) About 4 feet
b) About 5 feet
c) About 6 feet

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ANSWER

In our fictional forecast area, Coast County State Park, there is a river that flows into the sea at Inlet Cove. Imagine that the speed of the current produced by this river is 1 m/s (about 2 knots). If 5-foot, 3-second waves approaches Inlet Cove from the west, what would be the resulting wave heights in the area where the incoming swell encounters the Inlet Cove current?

Answer: About 6 feet.

Three-second waves have a wavelength of 14 meters (46 ft) and a celerity of 4.7 m/s (9 kts):

L = T*C

14 m = 3 sec * 4.7 m/s

The 1 m/s (2 kt) speed reduction caused by the Inlet River current will shorten the wavelength to 11 m (36 ft):

Lc = T * (C-u)

L = 3 sec * (4.7 m/s - 1 m/s) = 3 sec * 3.7 m/s = 11.1 m

This represents a 21% reduction in wavelength and a corresponding 21% increase in wave heights, which would turn 5-foot waves into 6-foot waves:

5 ft * 21% = 6.05

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5. Coastal Effects

5.1 Breaker Run-Up

Photo of wave run-up on a narrow beach.

(Click here for an animated photo sequence showing wave run-up.)

On any beach at a given time, crashing waves will run up the slope generally about the same distance. Periodically, however, a wave will crash and run up the beach much higher than the average waves. This is called breaker run-up and results from the fact that waves represent a spectrum of heights and periods.

Vertical water rise for the highest 2% of waves is estimated to be one half the predicted breaking wave height plus 50 cm (20 inches):

R(2%) = 0.5(HB) + 0.5 m

We calculate 2% because approximately 2 out of every 100 waves during a particular high surf event will be large enough to pose a significant threat to beachgoers or cause beach erosion. But, how far this water rise will travel up a given beach depends on beach composition, water saturation levels, and slope of the foreshore, or shore area that lies between the high and low water marks. Run-up is reduced during the period of rising tide because the dry sand over which incoming waves travel absorbs more water, reducing the overall water volume and energy of incoming waves.

Breaker run-up is a very difficult factor to account for in an operational setting because there is so much natural variability in foreshore slope, both horizontally and between sections of beach exposed during high and low tides. However, in some situations, where specific forecasts are required for a beach that has very even slope, breaker run-up calculations may be operationally useful, especially when you consider that significant wave run-up can sometimes knock beachgoers off their feet, pull them into the surf, and drown them.

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5.2 Wave Set-Up

5.2.1 Introduction

Wave setup is the water rise at the coast due to breaking waves on the beach. The return flow of white water from waves that have already broken is much slower than the speed of incoming waves before breaking. The result is a piling up of water in the surf zone as wave action continually forces water up onto the beach. Wave set-up differs from run-up in that set-up is the continual piling of up water, while run-up is the transient encroachment of individual waves.

Illustration of vertical extent of wave set-up.

(Click here to view the animated version of this illustration.)

The height of this piling up is directly proportional to the breaking wave height. It can range from a few inches or less for very small waves to 10 feet or more for large breaking waves such as those produced by nor'easters, hurricanes, and other strong storms.

Wave set-up is breaker-height dependent, and hence, breaker-depth dependent as well. Set-up magnitudes generally range between 10 and 15 percent of shoaling breaker heights :

S = 10 to 15% of Shoaling Breaker Heights

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5.2.2 Example Exercise

The wave set-up approximation has also been incorporated into our Shallow-Water Wave Calculator. Open it and use it to find the answer to the following question: (HINT: Use the Set-up tab.)

1. What set-up would be expected for 20-second, 15-foot swell? (Ignore refraction effects for this example.)
(Choose the best answer.)

a) 2-4 feet (0.6-1.2 m)
b) 6-9 feet (1.8-2.7 m)
c) 10-13 feet (3-4 m)

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ANSWER

The wave set-up approximation has also been incorporated into our Shallow-Water Wave Calculator. Open it and use it to find the answer to the following question:

1. What set-up would be expected for 20-second, 15-foot swell?

Answer: 2-4 feet.

20-second 15-foot swell will produce 24-29 foot (7.3-8.8 m) breakers. Since wave set-up is approximately 10-15% of the breaker height, we could expect about 2 to 4 vertical wave set-up levels at the beach.
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5.2.3 Significance

Why is set-up important? It contributes significantly to coastal water rise and can cause coastal flooding in high surf. Also, since wave set-up is breaker-depth dependent, it is an independent factor from storm surge, which is wind-driven. When combined, the effects of wave set-up and storm surge can have devastating impacts.

For example, the storm surge for Hurricane Opal in 1995 was estimated to be about eight feet, but the actual water rise at the coast was measured to be between 21 and 25 feet. The 14+ foot difference between storm surge and actual water rise was caused by wave set-up. Therefore, it is essential to include wave set-up in any forecast for coastal impacts from large storms.

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6. Summary

Shoaling

Refraction

Attenuation and Reflection

Breaker Types

Currents, Run-Up, and Set-Up

Shallow-Water Wave Calculator

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