# Forces of Nature: Using Earth and Planetary Science for Teaching Physical Science

## CONTENTS OF CURRICULUM UNIT 08.05.06

## Wave Math

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## LESSON PLANS

### [1] Plan one: 45 minutes

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Topic: Ocean Surface Waves In Deep Water
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Objective: SWBAT define and analyze Forming Deep Water Surface Waves

Lesson Starter: Surface Waves are generated by wind. Offshore storms generate winds which blow on the surface of the sea and create ripples, similar to the ripples in a cup of coffee or tea form when you blow on it to cool it down. The wind can be seen on weather maps as low pressure areas. The more tightly positioned the isobars, or meteorological contour lines of equal or constant pressure on the weather map, the stronger the winds blow. Consequently, small Surface Waves or capillary Surface Waves are initially generated in the same direction that the wind is blowing.

As the wind continues to blow and the Surface Waves generated remain under the influence of a constant wind the smaller Surface Waves will increase in size. The stronger and longer the wind blows the greater the effect on these ripples and the larger they will become, eventually creating a Larger Swell. Initially the Surface Waves will just be small but these will soon increase in size because the wind more effectively impacts the small Surface Waves than the calm flat sea surface.

While the size of Surface Waves is dependent on the generating wind speed, the seas are only considered
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fully formed
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once the largest Surface Waves that can be generated for a specific wind speed have been attained.

All generated Surface Waves have a signature, presenting differing speeds and wave periods. The longer period Surface Waves move faster and overtake the shorter period, slower Surface Waves. As these Surface Waves propagate, traveling farther away from the original wind source, they start to organize themselves into Swell lines:
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Wave Trains
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form. These sets of Surface Waves inevitably reach the beach as
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Ground Swell
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, since the original wind source that generated them has ceased to influence them.

*
Lesson:
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For deep water, the relationship between speed and wavelength is given by the formula:

l = g x t x t / (2 x pi) l = t x c for all kinds of waves, substitute in above equation: t x c = g x t x t / (2 x pi) c = g x t / (2 x pi) or t = c x 2 x pi / g or t = c x 0.641 (s)

where t= wave period (sec), f= wave frequency, l= wave length (m) and pi=3.1415... to calculate c and l from wave period t (in sec): c = t x 1.56 m/s= t x 5.62 km/hr = t x 3.0 knot

l = 1.56 x t x t (meters)

Thus waves with a period of 10 seconds travel at 56 km/hr with a wave length of about 156m. A 60 knot (110 km/hr) gale can produce in 24 hours waves with periods of 17 seconds and wave lengths of 450m. Such waves travel close to the wind’s speed (97 km/hr). A tsunami travelling at 200 m/s has a wave period of 128 s, and a wave length of 25,600 m.

The general formula required for these calculations is

c =

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Lesson Review:
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FIGURE A. ORBITAL MOTIONS: Wave Particle Displacement.

*
Lesson:
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Apply the formula for a circular orbit and calculate:

(x - h)
^{
2
}
+ (y - k)
^{
2
}
= r
^{
2
}
or
**
x
^{
2
}
+ y
^{
2
}
= r
^{
2
}
**
.

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Lesson Closer:
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Summarize for students that surface waves could theoretically travel much faster on larger planets, in media denser than water.
**
**

### [2] Plan two: 45 minutes

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Topic: Ocean Surface Waves In Shallow Water
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Objective: SWBAT define and analyze Forming Shallow Water Surface Waves

Lesson Starter: The relationship between wave speed or phase velocity and depth of long surface waves in shallow water is computed by the formula

c x c = g x d x (p2 - p1) / p2 or c x c= g x d for water/air

where

c= wave speed, g= acceleration of gravity (9.8066 m/s/s), d= wave depth or upper layer depth, m), p2= density of water (=1) and p1= density of air (= 0.00125).

*
Lesson:
*
The formula states that wave velocity increases with wave depth and the relative difference in density. For an ocean depth of 4000m, a wave’s celerity or speed would be about SQR (10 x 4000) = 200 m/s = 720 km/hr. Students should determine that these wave velocities are comparable to the velocities of commercial jet aircraft.

The general formula required for these calculations is

c =

*
Lesson Review:
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FIGURE B. ORBITAL MOTIONS: Wave Particle Displacement.

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Lesson:
*
Apply the formula for an elliptical orbit and calculate:

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Lesson Closer:
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Summarize for students that surface waves could theoretically travel much faster on larger planets, in media denser than water.

### [3] Plan three: 90 minutes

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Topic: Ocean Surface Waves
*

Objective: SWBAT define and analyze what determines the size of Swell

Lesson Starter: Three main factors determine the size and quality of a wave in open sea:

- 1.
Wind Speed:The faster the wind speed the larger the wave.- 2.
Wind DurationThe longer the wind blows the larger the wave.:- 3.
FetchThe greater the wind-affected area the larger the wave.:

Once Surface Waves are no longer influenced by wind they begin to forfeit their energy. Surface Waves will travel as far as they can while being decreased by friction from the sea bed and obstacles in their path, such as a big island.

There are different factors affecting the wave size at a certain surf break:

- · Swell Direction
:Will the cresting Surface Waves break open in the same direction as the current swell direction.- · Ocean Floor: A swell coming straight from deep sea up onto a reef will generate big, barreling Surface Waves. Whereas a long, shallow ledge up to the shore will slow down the Surface Waves and deplete their energy, rendering Surface Waves with less power.
- · Tide
:Some Surface Wave formations are totally tide dependant.

### On The Coastline: Shoaling Alters Surface Waves

As a swell approaches the coastline and comes into contact with the sea floor the waves will start to dissipate. Some of the wave’s energy is lost through contact with the sea floor. The shallower the water becomes the slower they move. As they slow down they have to squash together, shortening their wave period. This process is called
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shoaling
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and an increase in wave height results. The steepness of the sea floor gradient is directly proportionate to the resulting increase of Surface Wave height. This pronounced increase in wave height starts to happen at depths of approximately one half of the wavelength.

### Breaking Waves

As the wave moves into increasingly shallow water the bottom of the wave decreases speed. There comes a point where the top of the wave overtakes it and starts to spill forward and the wave starts to break. A wave will start to break when it approximately reaches a water depth of 1.3 times the wave height.The type of wave that is produced is dependent on different factors:

- · Type Of Swell
- · Wind Direction
- · Slope of Sea Bed
- · Sea Floor Features

### Type Of Swell

Ground Swell creates a more dramatic wave. The longer wavelength waves will move quickly and get into shallow water before starting to break. The breaking waves will be steeper and faster. Whereas Wind Swell will tend to break in deeper water and with less force and configuration.

### Wind Direction

An
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Offshore
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wind is capable of generating more prominent waves. The wind blows against the top part of the wave and delays the top from overtaking the bottom part. These Surface Waves break later than they normally would in calm conditions, exhibiting huge plumes of spray blowing back over the top of the wave.

Whereas an
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Onshore
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wind will have the opposite effect: the onshore wind pushes the top of the wave forward causing the wave to break before the normal breaking depth is attained. Surface Waves tend to be less uniform and with less amplitude.

### Slope Of The Sea Floor

The contrast to the gently sloping sea floor is a steep slope or a reef. The swell approaches the beach / reef at a greater speed. From the general formula required for these calculationsc =

it can be seen that the wave “jacks up” due to the rapid change in depth creating a higher wave. The breaking depth is reached much later that on the gently sloped bottom. The top of the wave quickly overtakes the bottom and pitches forward (often taking the inexperienced surfer with it). The waves created by the rapid change in depth are much steeper and hollower, forming the well-known tube.

Reef breaks such as Pipeline in Hawaii are examples of this type of break.

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**