- Email: [email protected]
Teaching ocean wave forecasting using computer-generated visualization and animation—Part 2: swell forecasting
Computers & Geosciences 28 (2002) 547–554
Teaching ocean wave forecasting using computer-generated visualization and animationFPart 2: swell forecasting$ Dennis J. Whitford* Department of Oceanography, U.S. Naval Academy, 572M Holloway Road, Annapolis, MD 21402-5026, USA Received 9 November 2000; received in revised form 20 March 2001; accepted 2 April 2001
Abstract This paper, the second of a two-part series, introduces undergraduate students to ocean wave forecasting using interactive computer-generated visualization and animation. Verbal descriptions and two-dimensional illustrations are often insufficient for student comprehension. Fortunately, the introduction of computers in the geosciences provides a tool for addressing this problem. Computer-generated visualization and animation, accompanied by oral explanation, have been shown to be a pedagogical improvement to more traditional methods of instruction. Cartographic science and other disciplines using geographical information systems have been especially aggressive in pioneering the use of visualization and animation, whereas oceanography has not. This paper will focus on the teaching of ocean swell wave forecasting, often considered a difficult oceanographic topic due to the mathematics and physics required, as well as its interdependence on time and space. Several MATLABs software programs are described and offered to visualize and animate group speed, frequency dispersion, angular dispersion, propagation, and wave height forecasting of deep water ocean swell waves. Teachers may use these interactive visualizations and animations without requiring an extensive background in computer programming. Published by Elsevier Science Ltd. Keywords: Wave dynamics; Oceanography; Education; Wave height; MATLABs
1. Introduction If we do not make undergraduate science teaching more understandable, we run the risk of making permanent our nation’s downward spiral in mathematics and science enrollments (Green, 1989). Many students choose not to major in science because of the difficulty of material (Brand, 1995). Fortunately, the introduction of computers in the geosciences provides a tool for addressing this problem. Cartographic science and other disciplines using geographical information systems have been especially aggressive in pioneering the use of
$ Code available on server at http://www/iamg.org/ CGEditor/index.htm *Fax: +1-410-293-2137. E-mail address: [email protected] (D.J. Whitford).
0098-3004/02/$ - see front matter Published by Elsevier Science Ltd. PII: S 0 0 9 8 - 3 0 0 4 ( 0 1 ) 0 0 0 5 9 - 0
computer-generated visualization and animation, whereas oceanography has not. The forecasting of ocean waves is an example of a difficult mathematics- and physics-related oceanography topic which poses difficulty to undergraduate students due to its interdependence on time and space. Computer-generated visualization and animation offer an ideal medium to clarify these concepts. The MATLABs software described in this article was designed specifically to take advantage of previous pedagogical research, discussed in Part 1 (Whitford, 2002) of this two-part series, which showed that learning is improved when computer-generated visualizations and animation are used. Part 1 was written to visualize and animate development and comparison of wave spectra, wave interference, and forecasting of sea conditions. This paper, Part 2, builds on that work and provides visualization and animation examples to
548
D.J. Whitford / Computers & Geosciences 28 (2002) 547–554
improve understanding of the mathematics- and physicsbased phenomena of group speed, frequency dispersion, angular dispersion, propagation, and wave height forecasting of ocean swell waves.
over 11,000 km (nearly 6000 nautical miles). This sorting process, occurring as a function of period or frequency, is called frequency dispersion. 2.3. Wave group speed
2. Basic concepts 2.1. Background The concepts of wave form, sea, swell, fetch, individual wave celerity (c), and wave spectra were addressed in Part 1 (Whitford, 2002). Sea, swell, and fetch relationships are illustrated in Fig. 1. 2.2. Frequency dispersion When individual waves leave their generating area, they are no longer wind-driven forced waves. These swell waves, or swell, become free waves whose individual speeds in deep water can be determined by their period from: c ¼ 1:56T;
ð1Þ
where T is wave period in seconds. These swell have considerable energy which they can transport over very long distances. Within a fetch area, there are many different waves of varying period and wavelength. Each of these waves have different speeds related to their different periods. Thus, these waves will sort themselves by different speeds, and therefore periods, and move out in sorted groups called wave groups or wave trains. Specific wave trains have been tracked from storms south of New Zealand all the way to the shores of Alaska, a distance
Frequency dispersion dictates that swell travel in groups and not as individual waves. These wave groups travel at different speeds than individual waves. This is a critical concept in understanding swell wave dynamics and one which deserves careful attention for the introductory student. If one tosses a stone in a pond, one can observe that the furthest outward-bound wave of the resultant wave group will disappear as the group advances. Simultaneously, a wave will join the group from the initial disturbance area. Thus, the individual waves inside the group move faster than the group as a whole. The group speed (cg ) for surface gravity waves can be expressed as (Airy (1845) and further explained in Kinsman (1984), Dean and Dalrymple (1991), Knauss (1997), or Sorensen (1997), among others): c 2kh cg ¼ 1þ ; ð2Þ 2 sinhð2khÞ where k equals 2p=L and is the radian wave number (m@1), L is wavelength (m), and h is water depth (m) measured from the still water line to the bottom. Applying a deep water approximation (large h) to Eq. (2) and relating to Eq. (1), yields: c cg ¼ ¼ 0:78T ðm s@1 Þ or cg ¼ 2:81T ðkm h@1 Þ; ð3Þ 2 which states that in deep water, the group speed is also dependent on wave period, but is 12 the individual wave speed. Group speed is the speed of dispersive swell waves propagating in deep water. Because of this period dependence, wave groups of the same period will disperse linearly as depicted in Fig. 2. This group phenomena is called frequency dispersion.
3. Method 3.1. Wave spectra
Fig. 1. Fetch box. Wind interacts with ocean surface inside fetch box to create a sea, or sea waves. Swell are those waves which leave the fetch box. Angles y3 , y4 , and swell propagation angle (a) are used to forecast swell at forecast point. R0 is distance from middle of leeward edge of fetch box to forecast point.
For swell forecasting, I have selected the Pierson et al. (1960), or PNJ, forecasting method, but have substituted a more modern description of a wave spectrum called the JONSWAP spectrum (Hasselmann et al., 1973, 1976). More modern methods of sea and swell forecasting employ large-scale computers using complex spectral methods (e.g. WAM Group, 1988; Koman et al., 1994), the comprehension of which would be beyond an undergraduate student. I specifically selected the PNJ method because it is relatively simple, pedagogically intuitive, and provides reasonable results. I have also
D.J. Whitford / Computers & Geosciences 28 (2002) 547–554
549
Fig. 2. Single frame from the program ‘‘frequency dispersion.m:’’ (a) animates how swell waves with 5, 10, and 15 s periods disperse, or spread out, due to their different speeds as function of time. (b) animates same waves and adds visualization of their combined wave heights as function of downwind distance from fetch box and of time.
selected the most common fetch situation, a stationary fetch box. Other fetch cases not addressed include those associated with moving storms and forecasting swell inside a fetch box after the wind has ceased.
forced to leave from the windward edge of the fetch box) by the minimum amount of time (i.e. waves leaving when the storm stops) and set that equal to the wave group speed of Eq. (3): cg ¼ 2:81T ¼
3.2. Swell period Since forecasting implies conditions at a future time, we define time variables as tstart =elapsed time in h since the storm started; tend =elapsed time in h since the storm ended; and, for understanding the animation computer loop in the computer software, tob =forecast observation time. (Note that in the case of a stationary fetch box, tob ¼ tstart . This would not necessarily be the case for other fetch situations.) We now must determine the largest (Tbig ) and smallest (Tlittle ) swell wave periods which could be present at a specific forecast point due solely to time and distance. We use the basic concept that speed=distanceCtime to formulate our answers. To determine the fastest wave (i.e. Tbig ) that could possibly reach a forecast or observation point from the fetch box (Fig. 1), we divide the longest possible distance to travel (i.e. waves that are
longest possible distance R0 þ F ; ¼ shortest possible time tend
ð4Þ
where R0 is the distance (in km) from the center of leeward boundary of the fetch box to the forecast point, and F is the distance (in km) from the windward edge to the leeward edge of the fetch box. Eq. (4) can be rewritten as: Tbig ¼
R0 þ F : 2:81tend
ð5Þ
To determine the slowest wave (i.e. Tlittle ) that could possibly reach a forecast point from the fetch box due solely to time and distance, we divide the shortest possible distance (i.e. waves leaving the leeward edge of the fetch box) by the maximum possible time (i.e. waves leaving as soon as the storm started): cg ¼ 2:81T ¼
shortest possible distance R0 : ¼ tstart longest possible time
ð6Þ
550
D.J. Whitford / Computers & Geosciences 28 (2002) 547–554
Eq. (6) can be rewritten as: R0 Tlittle ¼ : 2:81tstart
ð7Þ
Note that Tbig and Tlittle are used only to place bounds on reasonable solutions to the problem and are not actual physical phenomena. Since Part 1 described how to determine the range of significant sea wave periods (Tlower to Tupper ) that have physically and actually been produced by the wind, and we have now determined the range of swell wave periods that could possibly have reached the forecast point of interest due solely to time and distance (Tlittle to Tbig ), we logically select the range of swell periods which are actually present at the forecast point by choosing the smaller of Tupper and Tbig and the larger of Tlower and Tlittle , and designating these periods as T1 and T2 . Therefore, the range of swell periods present at the forecast point is T1 to T2 . 3.3. Energy dispersion
where y3 and y4 are measured in degrees from downwind extensions of the fetch box sides and are positive for angles measured clockwise from those extensions to lines connecting the box corners to the forecast point (Fig. 1). This directional spreading function represents the percentage of the total energy leaving the fetch box which arrives at the forecast point due solely to angular dispersion. For comparison, these three directional spreading functions are illustrated in Fig. 4. 3.4. Swell wave height
The energy at the forecast point is less than that originally produced by the wind and found in the fetch box. This energy dispersion is due to frequency and angular dispersion. We can envision the energy loss due to frequency dispersion by noting that the original sea contains all the energy between Tlower and Tupper ; whereas the swell situation only contains the energy between T1 and T2 (Fig. 3). Angular dispersion occurs as energy is spread out or dispersed over a larger area. We quantify this dispersion using a directional spreading function, DðyÞ, which is always p1. One of the earliest forms of a directional spreading function was a simple cosine squared relationship (St. Denis and Pierson, 1953): DðyÞ ¼
sophisticated directional spreading function which takes into account the width of the fetch box. This method determines the percentage of waves reaching a fixed point from each corner of the downwind fetch box width using (Pierson et al., 1960): 1 y4 sinð2y4 Þ Dðy3 ; y4 Þ ¼ 100 þ þ 2 180 2p 1 y3 sinð2y3 Þ þ ; ð11Þ @ þ 2 180 2p
2 cos2 ðaÞ; p
The combined effect of frequency dispersion and angular dispersion on reducing the original wave energy density can be determined using:
Sswell ¼ Sð1=T1 Þ@Sð1=T2 Þ ½Dðy3 ; y4 Þ; ð12Þ or in an alternative form: Sswell ¼ ½Sð f1 Þ@Sð f2 Þ½Dðy3 ; y4 Þ;
ð13Þ
where the first term on the right-hand side of either Eq. (12) or (13) is the cumulative energy density from the original sea wave energy density spectrum found
ð8Þ
where a, the swell propagation angle, varies from @901 to +901 (see Fig. 4). A more complex formulation for the directional spreading function is (Mitsuyasu et al., 1975): a Dð f ; aÞ ¼ DðsÞ cos2s ; ð9Þ 2 where DðsÞ ¼ ( s¼
22s@1 G2 ðs þ 1Þ ; p Gð2s þ 1Þ
sm ð f =fp Þ5
for f pfp ;
sm ð f =fp Þ@2:5
for f > fp ;
) ;
ð10Þ
where G is the mathematical gamma function, fp is the frequency of the spectrum peak, and sm is given a representative wind wave value determined by Goda (1985) to be 10. I have elected to use a more
Fig. 3. Fully developed sea wave spectrum with Tupper 2Tlower defining range of significant periods occurring in fetch box, and T1 2T2 defining range of swell wave periods at forecast point due to frequency dispersion. Reduction in energy from original spectrum within T1 2T2 is due to angular dispersion. Shaded area is swell energy density.
D.J. Whitford / Computers & Geosciences 28 (2002) 547–554
551
Fig. 4. Comparison of angular dispersion formulations. Swell propagation angle (a) is measured from mean wind direction in fetch box, ‘‘fw ’’ is fetch box width, and R0 is distance from midpoint of leeward fetch edge to forecast point.
between T1 and T2 , or f1 and f2 , respectively, and Sswell is the resultant swell wave energy density at the forecast point. Sswell is represented by the shaded energy density between T1 and T2 in Fig. 3. Swell wave heights are then determined from energy density using a Rayleigh distribution (Longuet-Higgins, 1952) just as they were for sea wave height forecasting in Part 1, eg. most frequent wave height: Hfreq ¼ 2:0 ðSswell Þ1=2 ¼ 2:0s;
root-mean-square wave height: ð16Þ
significant wave height Hs ¼ 4:0 ðSswell Þ1=2 ¼ 4:0s; ð17Þ and average of highest one-tenth waves: H1=10 ¼ 5:1 ðSswell Þ1=2 ¼ 5:1s;
To illustrate these complex topics, I have used MATLABs, version 5.3, to create illustrations and animations. The software code may be downloaded by Internet from http://www.iamg.org/CGEditor/ index.htm. These programs are interrelated and build upon concepts animated in Part 1. The code includes documentation for user understanding and potential modification. Program descriptions are as follows.
ð14Þ
average wave height Havg ¼ 2:5 ðSswell Þ1=2 ¼ 2:5s; ð15Þ
Hrms ¼ 2:8 ðSswell Þ1=2 ¼ 2:8s;
4. Computer-generated visualization and animation
ð18Þ
where s is the standard deviation of the demeaned water surface displacement.
4.1. Wave spectrum The program ‘‘wave spectrum.m’’ is a MATLABs function program which is called by the other programs and thus must be resident on your local machine to execute the programs described in this paper as well as those in Part 1. Its input parameters are spectrum type (PM or JONSWAP), wind speed (m s@1), the limiting condition (i.e. fetch- or duration-limited), and the value of that limitation in km or h, respectively. It outputs frequency, energy density values, Tupper , Tlower , and the area under the spectral curve. It uses Carter’s (1982) formulation for fetch- and duration-limited conditions.
552
D.J. Whitford / Computers & Geosciences 28 (2002) 547–554
The program can easily be modified to use a different wave spectrum, other than PM or JONSWAP, without adversely affecting the other visualization programs.
and wave height are arbitrary. Fig. 2 illustrates the situation at tob =15 h. 4.4. Swell forecast
4.2. Group speed The program ‘‘group speed.m’’ illustrates how the speed of the wave group is equivalent to 12 the speed of individual waves in deep water. The program is initiated by striking the space bar. The program then generates a 15 s movie on an x2z Cartesian plot where one can visually discern the wave group speed being slower than the individual wave celerity by a factor of 1/2. Fig. 5 illustrates a snapshot from the movie. 4.3. Frequency dispersion The program ‘‘frequency dispersion.m’’ displays two plot boxes using an x2z Cartesian plot. The top box initially illustrates that three wave groups of 5, 10, and 15 s periods are present in the fetch box. One may then time step, in increments of 5 h, by striking the space bar. Note that as time progresses, the groups get farther apart. The bottom box adds additional complexity by illustrating initially, i.e. tob =0 h, the combined wave height effect of three sea waves of periods 5, 10, and 15 s on the sea surface within the fetch box. One can then time step from 0 to 20 h, in 5 h increments, by striking the space bar to observe how these swell waves propagate downwind at different speeds and their combined effect on the sea surface, both within and downwind of the fetch box. The values of wave period
Fig. 5. Single frame from the program ‘‘group speed.m’’ animating an amplitude-modulated wave formed by superposition of two sinusoidal waves of different wave number as function of time. Individual waves travel twice as fast as group waves for this deep water example.
Continuing to build upon previous concepts, the program ‘‘swell forecast.m’’ has three options: ‘‘box’’, ‘‘2-d’’ (i.e. two-dimensional wave height forecast), and ‘‘3-d’’ (i.e. three-dimensional wave height forecast). The user can then specify the wind speed, fetch, and duration. (a) The ‘‘box’’ option illustrates the downwind propagation of swell boxes of 5, 10, 15, and 20 s periods as they propagate and disperse, both by frequency and angular spreading, downwind 7201 from the predominant surface wind direction measured inside the fetch box. The user can time step, in increments of 10 h, from 10 to 120 h, by striking the space bar. Fig. 6 illustrates the situation at tob =40 h. (b) The ‘‘2-d’’ option animates the combined effect of frequency and angular dispersion on swell height propagation downwind from the fetch box. The user can time step, in increments of 10 h, from 10 to 120 h, by striking the space bar. Fig. 7 illustrates the situation at tob =80 h. (c) The ‘‘3-d’’ option transforms the two-dimensional animation into three dimensions for the same input parameters. Fig. 8 illustrates the situation at tob =100 h.
5. Concluding remarks Recent pedagogical research has shown that computer-generated visualization and animation can improve
Fig. 6. Single frame from program ‘‘swell forecast.m’’ using ‘‘box’’ option. Program animates swell period boxes as they frequency and angular disperse downwind from fetch box as function of time. Thick line on left-hand side of plot represents leeward edge of fetch box.
D.J. Whitford / Computers & Geosciences 28 (2002) 547–554
553
student performance. This paper, Part 2 of a two-part series, describes visualization and animation of traditionally difficult oceanographic concepts such as ocean swell wave group speed, frequency and angular dispersion, propagation, and wave height forecasting. Three instructional programs are presented which satisfy the following learning objectives: * *
*
*
Fig. 7. Single frame from program ‘‘swell forecast.m’’ using ‘‘2-d’’ option. Program animates two-dimensional swell wave heights as they vary based on time and distance from fetch box. Thick line on left-hand side of plot represents leeward edge of fetch box.
*
*
Explain the difference between sea and swell. Recognize why group speed is different than individual wave speed and that wave group speed is 12 individual wave speed in deep water. Visualize the difference between individual wave group speed and group wave speed. Discern the relationship between period and wave celerity for individual waves and wave groups. Recognize the concept of energy dispersion and its two components, frequency and angular dispersion. Visualize frequency and angular dispersion as a function of time.
Fig. 8. Single frame from program ‘‘swell forecast.m’’ using ‘‘3-d’’ option. Program animates three-dimensional swell wave heights as they vary based on time and distance from fetch box. Thick line on left-hand side of plot represents leeward edge of fetch box.
554 *
*
*
*
*
*
* *
D.J. Whitford / Computers & Geosciences 28 (2002) 547–554
Visualize the effect of frequency dispersion on wave height. Recognize that frequency and angular dispersion cause a reduction in wave energy. Determine the range of periods present at a forecast point using the PNJ method of swell forecasting. Understand the mathematical and physical relationships of the PNJ method of swell forecasting. Recognize the relationship between wave energy and wave height using a Rayleigh probability distribution. Visualize swell propagation and changing wave height as a function of time in both two and three dimensions. Use numerical simulations to model physical science. Use technology resources that demonstrate and communicate curriculum concepts.
These animations also satisfy multiple education standards developed by the National Academy of Science (National Research Council of the National Academy of Science, 1995) cited in Part 1. Readers can use the code in explaining these difficult oceanographic topics during their classroom instruction or as part of a laboratory exercise. The computer code is provided to any interested person.
References Airy, G.B., 1845. On Tides and Waves. Encyclopedia Metropolitan, London, pp. 241–396. Brand, D.L., 1995. Those students who could have but didn’t: early attrition from college science. Journal of College Science Teaching 24, 180–182. Carter, D.J.T., 1982. Prediction of wave height and period for a constant wind velocity using the JONSWAP results. Ocean Engineering 9 (1), 17–33. Dean, R.G., Dalrymple, R.A., 1991. Water Wave Mechanics for Engineers and Scientists. World Scientific, Singapore, 353pp. Goda, Y., 1985. Random Seas and the Design of Maritime Structures. University of Tokyo Press, Tokyo, 323pp. Green, K.C., 1989. A profile of undergraduates in the sciences. American Scientist 77, 476.
Hasselmann, K., Barnett, T.P., Bouws, E., Carlson, H., Cartwright, D.E., Enke, K., Ewing, J.A., Gienapp, H., Hasselmann, D.E., Kruseman, P., Meerburg, A., Muller, P., Olbers, D.J., Richter, K., Snell, W., Walden, H., 1973. Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Report to the German Hydrographic Institute, Hamburg, Germany, 95pp. Hasselmann, K., Ross, D.B., Muller, P., Sell, W., 1976. A parametric wave prediction model. Journal of Physical Oceanography 6, 200–228. Kinsman, B., 1984. Wind Waves. Dover Publications Inc., New York, 676pp. Knauss, J.A., 1997. Introduction to Physical Oceanography. Prentice-Hall, Upper Saddle River, NJ, 309pp. Koman, G.L., Caverleri, L., Donelen, M., Hasselmann, K., Hasselmann, S., Janssen, P.A.E.M., 1994. Dynamics and Modelling of Ocean Waves. University Press, Cambridge, UK, 532pp. Longuet-Higgins, M.S., 1952. On the statistical distribution of the heights of sea waves. Journal of Marine Research 11, 245–266. Mitsuyasu, H., Tsai, F., Subara, T., Mizuno, S., Ohkusu, M., Honda, T., Rikiishi, K., 1975. Observations of the directional spectrum of ocean waves using a cloverleaf buoy. Journal of Physical Oceanography 10, 286–296. National Research Council of the National Academy of Science, 1995. National Science Education Standards. National Academy Press, Washington, DC, 272pp. Pierson Jr., W.J., Neumann, G., James, R.W., 1960. Practical Methods for Observing and Forecasting Ocean Waves by means of Wave Spectra and Statistics. Naval Oceanographic Office, H.O. Pub., Vol. 603, Washington, DC, 284pp. Sorensen, R.M., 1997. Basic Coastal Engineering, 2nd edn. Chapman a Hall, New York, 301pp. St. Denis, M., Pierson, W.J., 1953. On the motions of ships in confused seas. Transactions, Society of Naval Architects and Marine Engineers 61, 280–357. WAM (Wave Model) Development and Implementation Group, 1988. The WAM modelFa third generation ocean wave prediction model. Journal of Physical Oceanography 18, 1775–1810. Whitford, D. J., 2002. Teaching ocean wave forecasting using computer-generated visualization and animation F Part 1: sea Forecasting. Computers a Geosciences 28(4), 537–546.
Teaching ocean wave forecasting using computer-generated visualization and animationFPart 2: swell forecasting$ Dennis J. Whitford* Department of Oceanography, U.S. Naval Academy, 572M Holloway Road, Annapolis, MD 21402-5026, USA Received 9 November 2000; received in revised form 20 March 2001; accepted 2 April 2001
Abstract This paper, the second of a two-part series, introduces undergraduate students to ocean wave forecasting using interactive computer-generated visualization and animation. Verbal descriptions and two-dimensional illustrations are often insufficient for student comprehension. Fortunately, the introduction of computers in the geosciences provides a tool for addressing this problem. Computer-generated visualization and animation, accompanied by oral explanation, have been shown to be a pedagogical improvement to more traditional methods of instruction. Cartographic science and other disciplines using geographical information systems have been especially aggressive in pioneering the use of visualization and animation, whereas oceanography has not. This paper will focus on the teaching of ocean swell wave forecasting, often considered a difficult oceanographic topic due to the mathematics and physics required, as well as its interdependence on time and space. Several MATLABs software programs are described and offered to visualize and animate group speed, frequency dispersion, angular dispersion, propagation, and wave height forecasting of deep water ocean swell waves. Teachers may use these interactive visualizations and animations without requiring an extensive background in computer programming. Published by Elsevier Science Ltd. Keywords: Wave dynamics; Oceanography; Education; Wave height; MATLABs
1. Introduction If we do not make undergraduate science teaching more understandable, we run the risk of making permanent our nation’s downward spiral in mathematics and science enrollments (Green, 1989). Many students choose not to major in science because of the difficulty of material (Brand, 1995). Fortunately, the introduction of computers in the geosciences provides a tool for addressing this problem. Cartographic science and other disciplines using geographical information systems have been especially aggressive in pioneering the use of
$ Code available on server at http://www/iamg.org/ CGEditor/index.htm *Fax: +1-410-293-2137. E-mail address: [email protected] (D.J. Whitford).
0098-3004/02/$ - see front matter Published by Elsevier Science Ltd. PII: S 0 0 9 8 - 3 0 0 4 ( 0 1 ) 0 0 0 5 9 - 0
computer-generated visualization and animation, whereas oceanography has not. The forecasting of ocean waves is an example of a difficult mathematics- and physics-related oceanography topic which poses difficulty to undergraduate students due to its interdependence on time and space. Computer-generated visualization and animation offer an ideal medium to clarify these concepts. The MATLABs software described in this article was designed specifically to take advantage of previous pedagogical research, discussed in Part 1 (Whitford, 2002) of this two-part series, which showed that learning is improved when computer-generated visualizations and animation are used. Part 1 was written to visualize and animate development and comparison of wave spectra, wave interference, and forecasting of sea conditions. This paper, Part 2, builds on that work and provides visualization and animation examples to
548
D.J. Whitford / Computers & Geosciences 28 (2002) 547–554
improve understanding of the mathematics- and physicsbased phenomena of group speed, frequency dispersion, angular dispersion, propagation, and wave height forecasting of ocean swell waves.
over 11,000 km (nearly 6000 nautical miles). This sorting process, occurring as a function of period or frequency, is called frequency dispersion. 2.3. Wave group speed
2. Basic concepts 2.1. Background The concepts of wave form, sea, swell, fetch, individual wave celerity (c), and wave spectra were addressed in Part 1 (Whitford, 2002). Sea, swell, and fetch relationships are illustrated in Fig. 1. 2.2. Frequency dispersion When individual waves leave their generating area, they are no longer wind-driven forced waves. These swell waves, or swell, become free waves whose individual speeds in deep water can be determined by their period from: c ¼ 1:56T;
ð1Þ
where T is wave period in seconds. These swell have considerable energy which they can transport over very long distances. Within a fetch area, there are many different waves of varying period and wavelength. Each of these waves have different speeds related to their different periods. Thus, these waves will sort themselves by different speeds, and therefore periods, and move out in sorted groups called wave groups or wave trains. Specific wave trains have been tracked from storms south of New Zealand all the way to the shores of Alaska, a distance
Frequency dispersion dictates that swell travel in groups and not as individual waves. These wave groups travel at different speeds than individual waves. This is a critical concept in understanding swell wave dynamics and one which deserves careful attention for the introductory student. If one tosses a stone in a pond, one can observe that the furthest outward-bound wave of the resultant wave group will disappear as the group advances. Simultaneously, a wave will join the group from the initial disturbance area. Thus, the individual waves inside the group move faster than the group as a whole. The group speed (cg ) for surface gravity waves can be expressed as (Airy (1845) and further explained in Kinsman (1984), Dean and Dalrymple (1991), Knauss (1997), or Sorensen (1997), among others): c 2kh cg ¼ 1þ ; ð2Þ 2 sinhð2khÞ where k equals 2p=L and is the radian wave number (m@1), L is wavelength (m), and h is water depth (m) measured from the still water line to the bottom. Applying a deep water approximation (large h) to Eq. (2) and relating to Eq. (1), yields: c cg ¼ ¼ 0:78T ðm s@1 Þ or cg ¼ 2:81T ðkm h@1 Þ; ð3Þ 2 which states that in deep water, the group speed is also dependent on wave period, but is 12 the individual wave speed. Group speed is the speed of dispersive swell waves propagating in deep water. Because of this period dependence, wave groups of the same period will disperse linearly as depicted in Fig. 2. This group phenomena is called frequency dispersion.
3. Method 3.1. Wave spectra
Fig. 1. Fetch box. Wind interacts with ocean surface inside fetch box to create a sea, or sea waves. Swell are those waves which leave the fetch box. Angles y3 , y4 , and swell propagation angle (a) are used to forecast swell at forecast point. R0 is distance from middle of leeward edge of fetch box to forecast point.
For swell forecasting, I have selected the Pierson et al. (1960), or PNJ, forecasting method, but have substituted a more modern description of a wave spectrum called the JONSWAP spectrum (Hasselmann et al., 1973, 1976). More modern methods of sea and swell forecasting employ large-scale computers using complex spectral methods (e.g. WAM Group, 1988; Koman et al., 1994), the comprehension of which would be beyond an undergraduate student. I specifically selected the PNJ method because it is relatively simple, pedagogically intuitive, and provides reasonable results. I have also
D.J. Whitford / Computers & Geosciences 28 (2002) 547–554
549
Fig. 2. Single frame from the program ‘‘frequency dispersion.m:’’ (a) animates how swell waves with 5, 10, and 15 s periods disperse, or spread out, due to their different speeds as function of time. (b) animates same waves and adds visualization of their combined wave heights as function of downwind distance from fetch box and of time.
selected the most common fetch situation, a stationary fetch box. Other fetch cases not addressed include those associated with moving storms and forecasting swell inside a fetch box after the wind has ceased.
forced to leave from the windward edge of the fetch box) by the minimum amount of time (i.e. waves leaving when the storm stops) and set that equal to the wave group speed of Eq. (3): cg ¼ 2:81T ¼
3.2. Swell period Since forecasting implies conditions at a future time, we define time variables as tstart =elapsed time in h since the storm started; tend =elapsed time in h since the storm ended; and, for understanding the animation computer loop in the computer software, tob =forecast observation time. (Note that in the case of a stationary fetch box, tob ¼ tstart . This would not necessarily be the case for other fetch situations.) We now must determine the largest (Tbig ) and smallest (Tlittle ) swell wave periods which could be present at a specific forecast point due solely to time and distance. We use the basic concept that speed=distanceCtime to formulate our answers. To determine the fastest wave (i.e. Tbig ) that could possibly reach a forecast or observation point from the fetch box (Fig. 1), we divide the longest possible distance to travel (i.e. waves that are
longest possible distance R0 þ F ; ¼ shortest possible time tend
ð4Þ
where R0 is the distance (in km) from the center of leeward boundary of the fetch box to the forecast point, and F is the distance (in km) from the windward edge to the leeward edge of the fetch box. Eq. (4) can be rewritten as: Tbig ¼
R0 þ F : 2:81tend
ð5Þ
To determine the slowest wave (i.e. Tlittle ) that could possibly reach a forecast point from the fetch box due solely to time and distance, we divide the shortest possible distance (i.e. waves leaving the leeward edge of the fetch box) by the maximum possible time (i.e. waves leaving as soon as the storm started): cg ¼ 2:81T ¼
shortest possible distance R0 : ¼ tstart longest possible time
ð6Þ
550
D.J. Whitford / Computers & Geosciences 28 (2002) 547–554
Eq. (6) can be rewritten as: R0 Tlittle ¼ : 2:81tstart
ð7Þ
Note that Tbig and Tlittle are used only to place bounds on reasonable solutions to the problem and are not actual physical phenomena. Since Part 1 described how to determine the range of significant sea wave periods (Tlower to Tupper ) that have physically and actually been produced by the wind, and we have now determined the range of swell wave periods that could possibly have reached the forecast point of interest due solely to time and distance (Tlittle to Tbig ), we logically select the range of swell periods which are actually present at the forecast point by choosing the smaller of Tupper and Tbig and the larger of Tlower and Tlittle , and designating these periods as T1 and T2 . Therefore, the range of swell periods present at the forecast point is T1 to T2 . 3.3. Energy dispersion
where y3 and y4 are measured in degrees from downwind extensions of the fetch box sides and are positive for angles measured clockwise from those extensions to lines connecting the box corners to the forecast point (Fig. 1). This directional spreading function represents the percentage of the total energy leaving the fetch box which arrives at the forecast point due solely to angular dispersion. For comparison, these three directional spreading functions are illustrated in Fig. 4. 3.4. Swell wave height
The energy at the forecast point is less than that originally produced by the wind and found in the fetch box. This energy dispersion is due to frequency and angular dispersion. We can envision the energy loss due to frequency dispersion by noting that the original sea contains all the energy between Tlower and Tupper ; whereas the swell situation only contains the energy between T1 and T2 (Fig. 3). Angular dispersion occurs as energy is spread out or dispersed over a larger area. We quantify this dispersion using a directional spreading function, DðyÞ, which is always p1. One of the earliest forms of a directional spreading function was a simple cosine squared relationship (St. Denis and Pierson, 1953): DðyÞ ¼
sophisticated directional spreading function which takes into account the width of the fetch box. This method determines the percentage of waves reaching a fixed point from each corner of the downwind fetch box width using (Pierson et al., 1960): 1 y4 sinð2y4 Þ Dðy3 ; y4 Þ ¼ 100 þ þ 2 180 2p 1 y3 sinð2y3 Þ þ ; ð11Þ @ þ 2 180 2p
2 cos2 ðaÞ; p
The combined effect of frequency dispersion and angular dispersion on reducing the original wave energy density can be determined using:
Sswell ¼ Sð1=T1 Þ@Sð1=T2 Þ ½Dðy3 ; y4 Þ; ð12Þ or in an alternative form: Sswell ¼ ½Sð f1 Þ@Sð f2 Þ½Dðy3 ; y4 Þ;
ð13Þ
where the first term on the right-hand side of either Eq. (12) or (13) is the cumulative energy density from the original sea wave energy density spectrum found
ð8Þ
where a, the swell propagation angle, varies from @901 to +901 (see Fig. 4). A more complex formulation for the directional spreading function is (Mitsuyasu et al., 1975): a Dð f ; aÞ ¼ DðsÞ cos2s ; ð9Þ 2 where DðsÞ ¼ ( s¼
22s@1 G2 ðs þ 1Þ ; p Gð2s þ 1Þ
sm ð f =fp Þ5
for f pfp ;
sm ð f =fp Þ@2:5
for f > fp ;
) ;
ð10Þ
where G is the mathematical gamma function, fp is the frequency of the spectrum peak, and sm is given a representative wind wave value determined by Goda (1985) to be 10. I have elected to use a more
Fig. 3. Fully developed sea wave spectrum with Tupper 2Tlower defining range of significant periods occurring in fetch box, and T1 2T2 defining range of swell wave periods at forecast point due to frequency dispersion. Reduction in energy from original spectrum within T1 2T2 is due to angular dispersion. Shaded area is swell energy density.
D.J. Whitford / Computers & Geosciences 28 (2002) 547–554
551
Fig. 4. Comparison of angular dispersion formulations. Swell propagation angle (a) is measured from mean wind direction in fetch box, ‘‘fw ’’ is fetch box width, and R0 is distance from midpoint of leeward fetch edge to forecast point.
between T1 and T2 , or f1 and f2 , respectively, and Sswell is the resultant swell wave energy density at the forecast point. Sswell is represented by the shaded energy density between T1 and T2 in Fig. 3. Swell wave heights are then determined from energy density using a Rayleigh distribution (Longuet-Higgins, 1952) just as they were for sea wave height forecasting in Part 1, eg. most frequent wave height: Hfreq ¼ 2:0 ðSswell Þ1=2 ¼ 2:0s;
root-mean-square wave height: ð16Þ
significant wave height Hs ¼ 4:0 ðSswell Þ1=2 ¼ 4:0s; ð17Þ and average of highest one-tenth waves: H1=10 ¼ 5:1 ðSswell Þ1=2 ¼ 5:1s;
To illustrate these complex topics, I have used MATLABs, version 5.3, to create illustrations and animations. The software code may be downloaded by Internet from http://www.iamg.org/CGEditor/ index.htm. These programs are interrelated and build upon concepts animated in Part 1. The code includes documentation for user understanding and potential modification. Program descriptions are as follows.
ð14Þ
average wave height Havg ¼ 2:5 ðSswell Þ1=2 ¼ 2:5s; ð15Þ
Hrms ¼ 2:8 ðSswell Þ1=2 ¼ 2:8s;
4. Computer-generated visualization and animation
ð18Þ
where s is the standard deviation of the demeaned water surface displacement.
4.1. Wave spectrum The program ‘‘wave spectrum.m’’ is a MATLABs function program which is called by the other programs and thus must be resident on your local machine to execute the programs described in this paper as well as those in Part 1. Its input parameters are spectrum type (PM or JONSWAP), wind speed (m s@1), the limiting condition (i.e. fetch- or duration-limited), and the value of that limitation in km or h, respectively. It outputs frequency, energy density values, Tupper , Tlower , and the area under the spectral curve. It uses Carter’s (1982) formulation for fetch- and duration-limited conditions.
552
D.J. Whitford / Computers & Geosciences 28 (2002) 547–554
The program can easily be modified to use a different wave spectrum, other than PM or JONSWAP, without adversely affecting the other visualization programs.
and wave height are arbitrary. Fig. 2 illustrates the situation at tob =15 h. 4.4. Swell forecast
4.2. Group speed The program ‘‘group speed.m’’ illustrates how the speed of the wave group is equivalent to 12 the speed of individual waves in deep water. The program is initiated by striking the space bar. The program then generates a 15 s movie on an x2z Cartesian plot where one can visually discern the wave group speed being slower than the individual wave celerity by a factor of 1/2. Fig. 5 illustrates a snapshot from the movie. 4.3. Frequency dispersion The program ‘‘frequency dispersion.m’’ displays two plot boxes using an x2z Cartesian plot. The top box initially illustrates that three wave groups of 5, 10, and 15 s periods are present in the fetch box. One may then time step, in increments of 5 h, by striking the space bar. Note that as time progresses, the groups get farther apart. The bottom box adds additional complexity by illustrating initially, i.e. tob =0 h, the combined wave height effect of three sea waves of periods 5, 10, and 15 s on the sea surface within the fetch box. One can then time step from 0 to 20 h, in 5 h increments, by striking the space bar to observe how these swell waves propagate downwind at different speeds and their combined effect on the sea surface, both within and downwind of the fetch box. The values of wave period
Fig. 5. Single frame from the program ‘‘group speed.m’’ animating an amplitude-modulated wave formed by superposition of two sinusoidal waves of different wave number as function of time. Individual waves travel twice as fast as group waves for this deep water example.
Continuing to build upon previous concepts, the program ‘‘swell forecast.m’’ has three options: ‘‘box’’, ‘‘2-d’’ (i.e. two-dimensional wave height forecast), and ‘‘3-d’’ (i.e. three-dimensional wave height forecast). The user can then specify the wind speed, fetch, and duration. (a) The ‘‘box’’ option illustrates the downwind propagation of swell boxes of 5, 10, 15, and 20 s periods as they propagate and disperse, both by frequency and angular spreading, downwind 7201 from the predominant surface wind direction measured inside the fetch box. The user can time step, in increments of 10 h, from 10 to 120 h, by striking the space bar. Fig. 6 illustrates the situation at tob =40 h. (b) The ‘‘2-d’’ option animates the combined effect of frequency and angular dispersion on swell height propagation downwind from the fetch box. The user can time step, in increments of 10 h, from 10 to 120 h, by striking the space bar. Fig. 7 illustrates the situation at tob =80 h. (c) The ‘‘3-d’’ option transforms the two-dimensional animation into three dimensions for the same input parameters. Fig. 8 illustrates the situation at tob =100 h.
5. Concluding remarks Recent pedagogical research has shown that computer-generated visualization and animation can improve
Fig. 6. Single frame from program ‘‘swell forecast.m’’ using ‘‘box’’ option. Program animates swell period boxes as they frequency and angular disperse downwind from fetch box as function of time. Thick line on left-hand side of plot represents leeward edge of fetch box.
D.J. Whitford / Computers & Geosciences 28 (2002) 547–554
553
student performance. This paper, Part 2 of a two-part series, describes visualization and animation of traditionally difficult oceanographic concepts such as ocean swell wave group speed, frequency and angular dispersion, propagation, and wave height forecasting. Three instructional programs are presented which satisfy the following learning objectives: * *
*
*
Fig. 7. Single frame from program ‘‘swell forecast.m’’ using ‘‘2-d’’ option. Program animates two-dimensional swell wave heights as they vary based on time and distance from fetch box. Thick line on left-hand side of plot represents leeward edge of fetch box.
*
*
Explain the difference between sea and swell. Recognize why group speed is different than individual wave speed and that wave group speed is 12 individual wave speed in deep water. Visualize the difference between individual wave group speed and group wave speed. Discern the relationship between period and wave celerity for individual waves and wave groups. Recognize the concept of energy dispersion and its two components, frequency and angular dispersion. Visualize frequency and angular dispersion as a function of time.
Fig. 8. Single frame from program ‘‘swell forecast.m’’ using ‘‘3-d’’ option. Program animates three-dimensional swell wave heights as they vary based on time and distance from fetch box. Thick line on left-hand side of plot represents leeward edge of fetch box.
554 *
*
*
*
*
*
* *
D.J. Whitford / Computers & Geosciences 28 (2002) 547–554
Visualize the effect of frequency dispersion on wave height. Recognize that frequency and angular dispersion cause a reduction in wave energy. Determine the range of periods present at a forecast point using the PNJ method of swell forecasting. Understand the mathematical and physical relationships of the PNJ method of swell forecasting. Recognize the relationship between wave energy and wave height using a Rayleigh probability distribution. Visualize swell propagation and changing wave height as a function of time in both two and three dimensions. Use numerical simulations to model physical science. Use technology resources that demonstrate and communicate curriculum concepts.
These animations also satisfy multiple education standards developed by the National Academy of Science (National Research Council of the National Academy of Science, 1995) cited in Part 1. Readers can use the code in explaining these difficult oceanographic topics during their classroom instruction or as part of a laboratory exercise. The computer code is provided to any interested person.
References Airy, G.B., 1845. On Tides and Waves. Encyclopedia Metropolitan, London, pp. 241–396. Brand, D.L., 1995. Those students who could have but didn’t: early attrition from college science. Journal of College Science Teaching 24, 180–182. Carter, D.J.T., 1982. Prediction of wave height and period for a constant wind velocity using the JONSWAP results. Ocean Engineering 9 (1), 17–33. Dean, R.G., Dalrymple, R.A., 1991. Water Wave Mechanics for Engineers and Scientists. World Scientific, Singapore, 353pp. Goda, Y., 1985. Random Seas and the Design of Maritime Structures. University of Tokyo Press, Tokyo, 323pp. Green, K.C., 1989. A profile of undergraduates in the sciences. American Scientist 77, 476.
Hasselmann, K., Barnett, T.P., Bouws, E., Carlson, H., Cartwright, D.E., Enke, K., Ewing, J.A., Gienapp, H., Hasselmann, D.E., Kruseman, P., Meerburg, A., Muller, P., Olbers, D.J., Richter, K., Snell, W., Walden, H., 1973. Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Report to the German Hydrographic Institute, Hamburg, Germany, 95pp. Hasselmann, K., Ross, D.B., Muller, P., Sell, W., 1976. A parametric wave prediction model. Journal of Physical Oceanography 6, 200–228. Kinsman, B., 1984. Wind Waves. Dover Publications Inc., New York, 676pp. Knauss, J.A., 1997. Introduction to Physical Oceanography. Prentice-Hall, Upper Saddle River, NJ, 309pp. Koman, G.L., Caverleri, L., Donelen, M., Hasselmann, K., Hasselmann, S., Janssen, P.A.E.M., 1994. Dynamics and Modelling of Ocean Waves. University Press, Cambridge, UK, 532pp. Longuet-Higgins, M.S., 1952. On the statistical distribution of the heights of sea waves. Journal of Marine Research 11, 245–266. Mitsuyasu, H., Tsai, F., Subara, T., Mizuno, S., Ohkusu, M., Honda, T., Rikiishi, K., 1975. Observations of the directional spectrum of ocean waves using a cloverleaf buoy. Journal of Physical Oceanography 10, 286–296. National Research Council of the National Academy of Science, 1995. National Science Education Standards. National Academy Press, Washington, DC, 272pp. Pierson Jr., W.J., Neumann, G., James, R.W., 1960. Practical Methods for Observing and Forecasting Ocean Waves by means of Wave Spectra and Statistics. Naval Oceanographic Office, H.O. Pub., Vol. 603, Washington, DC, 284pp. Sorensen, R.M., 1997. Basic Coastal Engineering, 2nd edn. Chapman a Hall, New York, 301pp. St. Denis, M., Pierson, W.J., 1953. On the motions of ships in confused seas. Transactions, Society of Naval Architects and Marine Engineers 61, 280–357. WAM (Wave Model) Development and Implementation Group, 1988. The WAM modelFa third generation ocean wave prediction model. Journal of Physical Oceanography 18, 1775–1810. Whitford, D. J., 2002. Teaching ocean wave forecasting using computer-generated visualization and animation F Part 1: sea Forecasting. Computers a Geosciences 28(4), 537–546.