DATA LIBRARY | REFERENCE COLLECTION H.o. Got SH QODS HOLE OCEANOGRAPHIC INSTITUTIC’s WIND WAVES AND SWELL PRINCIPLES IN FORECASTING Prepared for the Hydrographic Office, U. S. Navy by The Scripps Institution of Oceanography University of California La Jolla, California 3 | BI if | WS | H.O. Misc. 11,275| ce entaeoo TOeE0 MUNN, TW 1OHM/ 1814 WIND WAVES AND SWELL PRINCIPLES IN FORECASTING Table of Contents INTRODUCTION SURFACE WAVES IN WATER General discussion Waves of very small] height Deep-water waves of moderate and great height Interference of waves; short-crested waves; white caps EMPTRICAL KNOWLEDGE OF WIND WAVES AND SWELL Measurements of waves and swell Comparison of measured and computed values Empirical relationships between wind and waves GROWTH Of WIND WAVES DECAY OF WAVES Waves advancing into regions cf calm Effect of following or opposing winds Distance from which observed swell comes; travel time; velecity of wind which produced the swell TH STATE OF THE SEA FPORECASTING OF WAVES FCR SHORT FETCHES FCRECASTING OF SWELL 9 6 Determination of wind, fetch, and duration Wind direction Wind velocity fetch Duration of wind Determination of highest wince waves Determination of the swell Waves advancing througk regions of caim Following or opvosing winds affect of following or cpposing winds Remarks on forecast Example FORECASTING Of THE STATE OF TH SEA APPENDIX: WAVES ENTERING SHALLOW WATER: BREAKERS AND SURF WIND WAVES AND SWELL PRINCIPLES IN FORECASTING Prevared for the Hydrogravhic Office, U. S. Navy by The Scripps Institution of Oceanography University of California INTRODUCTION Study of the problem of forecasting sea and swell was started at the request of the Army Air Forces and is being continued under the direction of the Hydrogravhic Office, U. S. Navy. Four vroblems of forecasting are involved: (1) forecasting the length and height of the swell in the open sea, (2) forecast- ing the swell reaching exposed or partially exposed anchorages, (3) forecasting the height of breakers and the amount of surf on any given beach, and (4) forecasting the state of the sea in any civen ocean area. The first problem involves two steps: (a) de- termination of height and period of the waves which emerge from any given wind area and which may arrive as swell on a distant coast, (b) determination of the travel time and the decrease of height of the waves as they proceed from the wind area. For the second and third problems an additional factor is involved, namely, the determination of the transformation of the waves as they enter into shallow water and wash the beach. The fourth problem involves two steps: (a) determination of the highest waves found under giv- en wind conditions and (b) establishment of the relation of these waves to the state of the sea as described by a scale such as the Doweilas Sea Seale. This manual deals with the generation of waves by wind and with the travel of waves in deeo water after they have left the POSLOMS OF SoicoOde viekals5 Wiswlaiocls ecuee Ceseirillsscl stoic Cloweieiul ies mye the characteristics of wind waves by means of.data from adequate, consecutive synoptic weather maos and for forecasting swell off coasts. Relationships between waves and the three important variables, wind at the sea surface, fetch (the stretch of water over which the wind blows), and duration (the length of time the wind nas blown) are discussed. Verifications and interovretations of the empirical laws developed by various observers of waves are given, together with gravhs for use in forecasting wind waves and swell. In order to use the granhs most effectively their »hysical significance and limitations must be clearly understood. Forecasts should therefore not be atteupted until the forecaster has studied the first part of the paper which describes the processes leading to the growth and decay of waves. Tests of the wethod made to date indicate that swell forecasts can be made with about the saine certainty as that of Most meteor- OGQLOLLCGLL WOCSCAS OSS LicOLMOS wIC Cleiess lies MOQ Mijooiwweias wOrwr wide forecasting of swell because considerable tiie elapses between tre generation of waves in distant storm areas and their arrival at the coast. Thus, after exverience has been gained, it 1s possible to forecast swell several days in advance. Forecasts of the state of the sea, on thevother hand), must (be sbasied) ina param onmord@ =m osrpnc weather maps und cannot be prepared for periods longer than those for which these maps can be considered valid. It is contemplated that a more comprehensive edition of this manual will be issued in the near future. This will contain meth- ods for determining the transformation of waves in shallow water and for forecasting surf from synoptic weather data or from observa- tions of waves offshore. SURFACE WAVES IN WATER General Discussion AV Waves) described by ats Venebhs is ije) the horizontal distance from crest to crest or trough to trough (see fig. 14), and bi iu smc ohne hy ic. bnew ver tically dustance som atacoued to crest. A wave is furthermore characterized by its period, I, i.e. the tiine interval between the appearance of two consecutive crests ab a given position. Ener pote aaa WATER LEVEL WATER LEVEL Figure 1. Surface waves. A. Profile of wave. B. Advance of wave, showing the wave profile at the EINES % 6 O, % = Wik, Elael lo W/Aa il wey bi Wy the wave has advanced one half wave length, L/2. A wave may be standing or progressive, but this discussion deals with vrogressive waves only. In a progressive wave, if the length and energy are constant, the wave height is the same at all localities and the wave crest appears to advance with a certain velocity (fig. 1B). During one wave period, T, the wave crest advances one wave length, L, and the velocity of the wave, C, is therefore defined as zal Hl The motion of the water particles depends on the wave length and the depth of the water. In general, it can be stated that the advance of the wave form is caused by convergences and diver- sences of the horizontal motion. In front of the crest the motion is converging and the surface is rising, but behind the crest the motion is diverging and the surface is sinking. By energy of the wave is always understood the average energy over one wave length. The energy is in part potential, Ey. asso- ciated with the displacement of the water particles abone ok below the level of equilibrium, and in part it is kinetic, E,, associated with the motion of the particles. In surface waves half the energy is present as kinetic and half as potential. The total average energy per square foot is E = 1/8 2 pH’, where g is the accelera- tion of gravity and pis the density of the water. For a 10-foot high wave the total average energy is 800 foot-pounds per square foot. Since g and p can be considered constant the energy per unit area in a wave is vroportional only to the square of the wave height. For the total energy per unit width along a wave length it is necessary to multiply the energy per unit area by the wave length. Waves of Very Small Height By waves of very small height are understood waves for WALGlA WAG weeLO C1 M@iedlrs wo LeVian a6) W/KOO (oie Wess Bas simplest wave theory deals with such waves, the form of which can be represented by a sine curve (see fig. 3). In water of constant depth, d, such waves travel with the velocity c= /g_L_ tanh 279 270 L where g is the acceleration due to gravity. If d/L is large, that is, if the wave length is small com- pared to the depth, tanh 27 d/L approaches unity and one obtains x) * tm These waves are called deep-water waves. If d/L is small, that is, if the wave length is large com- pared to the depth, tanh 271d/L approaches 27d/L and one obtains — These waves are called shallow-water waves. In general, waves have the character of deev-water waves when the depth to the bottom is greater than one half the wave length (d=>L/2). However, for shallow-water waves the depth must be less than one twenty-fifth of the wave length (d 2))o A water varticle at the sea surface remains at the surface throughout its orbit. A water particle at a given ‘averace wWdepth below the sea surface is farthest from the surface when it moves in the direction of wave progress. In a low shallow-water wave the vertical motion of the particles is negligible and the horizontal motion is independent of depth. The particles move back and forth, following nearly straight lines. In a deep-water wave only half of the energy advances with wave velocity, whereas in a shallow-water wave all the energy advances with wave velocity. The reason for this difference is that in a deep-water wave only the potential energy varies period- ically and advances with the wave form, but in a shallow-water wave both potential and kinetic energy vary veriodically and both advance with the wave form. These laws can also be stated by say- ing that the energy advances at a rate which, in a deep-water wave, equals half the product of energy and wave velocity, whereas in a Shallow-water wave it equals the product of energy and wave velocity. DIRECTION OF PROGRESS TT) Paar eee! | OO d | | | Figure 2. Movement of water particles in a deep- water wave of very small height. The circles show the paths in which the water particles tiove. The wave profiles and the positions of a series of water parti- cles are shown at two instants which are one quarter of a period apart. The full-drawn, nearly vertical lines indicate the relative vosSitions of water par- ticles) which Iie exactly on-vertical lines when the CES Ole WINE wicOUiela Oi WINS WEHYS IOVS) Eval Wis) Glelsiase| lines show the relative positions of the same particles one quarter of a period later. Deep-water Waves of Moderate and Great Height By waves of moderate and great height are understood waves for which the ratio of height to length (H/L) is from 1/100 to 1/25 and aeGasomenly/ 215 itiously/a/ arlene Citpiaviclaya meu Cit @rcMeOtL hie Sem wanes Can not be represented by a sine curve. For waves of moder- ate height the form closely approaches the trochoid, that is’, Ae GWuAVS Wiaslela WS ClesSCGiceilloacl iy Cl jOOLmMG OM 2 CGISeE Winwela wos below a flat surface (fig. 3). Waves of great height deviate from the trochoid; the troughs are wider and flatter and the crests narrower and steeper. The wave form becomes unstable when the ratio H/L equals 1/7. LINE ALONG WHICH DISC ROLLS Figure 3. Profile of a trochoidal wave (full-drawn lines) and of a sine wave (dashed lines). The wave velocity increases with increasing steepness (in- creasing values of H/L), but the increase of velocity never ExC@oas 2 weie O©Smlo> IMAI) uses S Het CASA WO ed NP NEI EN I TOONS) ee radii of which decrease rapidly with depth. The particle velocity is not uniform but is greatest when the varticles GOOG) WSUS ME) WOO Oi Wl Creloaim (siony aia Wisi wine GClaweSoe wali Ox wave progress), with the result that the particles upon com- pletion of each nearly circular motion have advanced a short distance in the direction of progress of the wave (fig. 4). Con- sequently, there is a mass transport in the direction of progress of the wave. The mass transport velocity (u') at the sea surface is expressed by the formula, Figure 4. Orbital motion during two wave periods of a water particle in a deep-water wave of iioderate or great height. In two wave periods the forward dis- placement equals 2u'T. The velocity is appreciable for high, steep waves but is very small for low waves of long period. Mass transport in waves has received little attention in previous work because in most practical applications it is sufficient to consider the water particles as ioving in circles regardless of the wave height. In order to understand the growth of waves through wind action, however, it is necessary to take the mass transport velocity into account. Interference of Waves; Short-crested Waves; White Caps When waves of different heights and lengths are vresent simul- taneously the appearance of the free surface becomes very compli- cated. At Some voints tie waves are opposite in phase and there- fore tend to eliminate each other, whereas at other points they coincide in phase and reinforce each other. As a Simple case, consider two trains of waves which have the same height and nearly the same velocity of progress. Owing to interference, groups of waves are formed with wave heights rough- ly twice those in the component wave trains, and between the wave groups are regions in which the waves nearly disappear (fig. 5A). Analysis shows that these groups advance with a velocity which is nearly equal to one half of the average velocity of the two trains. AS another example; consider the simultaneous presence of long, low swell and short but high wind waves. The resultant pattern is illustrated in Figure 5B from which it is evident that the short, high waves dominate to such an extent that the presence of the SWE sl SmoODsicumedr So far, the discussion has dealt only with long-crested waves, that is, waves with very long straight crests and troughs. Waves can, however, also have short, irregular crests and troughs. In the presence of such short-crested waves the free surface shows a series of alternating "highs" and "lows", as indicated in Figure 6. This figure illustrates the topography of the sea surface, "highs" being shown with full-drawn lines and "lows" with dashed lines. 1) Figure 5. Wave vatterns resulting from interfer- ence. A. Interference of two waves of equal height and nearly equal length, forming wave groups. B. In- terference between short wind waves and long swell. Figure 6. Short-crested waves. L = wave length, IY & ©wesw IeiMaeul. IL ; White caps are formed by the breaking of relatively short waves which often appear as "riders" on longer waves (fig. 5B). Such short waves may grow so rapidly that their steeoness reaches the critical value H/l = 1/7 and they break. If interference occurs long waves may attain this steepness and break. EMPIRICAL KNOWLEDGE OF WIND WAVES AND SWELL Measurements of Waves and Swell Wind waves are defined as waves which are growing in height under the influence of the wind. Swell consists of wind-generated waves which have advanced in- to regions of weaker winds or calms and are decreasing in height. So far, the discussion of surface weves has dealt mainly with waves which appear as rhythmic and regular deformations of the surface. Because of interference, the formation of snoreeerested waves, and the breaking of waves there is, however, little regularity in the appearance of the sea surface, particularly when a strong wind blows. Although individual waves can be recognized and their heights, periods,-lengths, and velocities measured, such measurements are extremely difficult and comparatively inaccurate. The lengths of most waves and the heights of low waves are likely to be underestimated, while the heights of large waves are general- ly overestimated. Wave heights above 55 feet are extremely rare, yet the literature contains many reports of waves exceeding 80 feet in height. Such errors are vrobably due to the complexity of the sea surface and the movement of the ships from which measurements are made. 12 Reliable measurements of wave height, H, are so diffi- CMbtunabe eA Cnemoll tic me poOmbtied = Valwes) se pimresient) crude estimates. The height of a large wave is estimated as the eye height of the observer above the water line when the Shlpens. on even keel in the trough of the wave, provided that the observer sees the crest of the wave coincide with the horizon. The height of a small wave is estimated dir- ectly, using the dimensions of the ship for comparison. On board a siiall ship the height of waves which are more than twice as long as the ship can be recorded by a micro- barograph. Mas WWE SwsLOG, W, SLi WE MOASUIPSC Iiy awSCOIwGlAMsZ wins time interval between successive appearances (on a wave crest) of a well-defined patch of foam at a considerable distance from the ship. In order to obtain a reliable value, observations should be made for several minutes and averaged. The wave length, L, can be estimated by comparing the ship's length with the distance between two successive GPESUS 5 Wans preocsdwuice Leads CO WaAGSieiweiiiad wesvllas., ladiomwevere-, because it is often difficult to locate both crests relative to the ship and because of disturbance caused by the move- ment of the ship. he velocitylon aheswame, ©) Can pe found by cecording the time needed for the wave to run a weasured distance along the side of the ship and by applying a correction for the ship's sveed. 13 Comparison of Measured and Computed Values Theory indicated that velocity, length, and period for deep- water waves are interrelated by the formulae = Be wie = pele ° Surette z= 2, -/2£7 =) 2h C= a z/6 a= Sg3 8 nae mo Te "ape oa Wawa G ain Isaiows, Ih sind wesw, ial WY iia Sooo G= 1.34. /E = 3.03 T = 00555 C= 5.12 1 tS 0.42Q/L = On23 6 Thus, if one characteristic is measured the other two can be com- puted, and if two or three are measured the correctness of the theory as applied to ocean waves can be checked. Comparisons of measured and computed values have given satisfactory results, in- dicating that wind waves and swell in deep water do have the char- acteristics described above. In general, the conclusion that the ratio H/L always remains less than 1/7 is also confirmed by obser- vations, as waves of this or greater steepness are very rarely reported. Empirical Relationships between Wind and Waves Observations of waves have not Ted to clear-cut conclusions about the empirical relationships between the wind and waves. The following nine approximate relationships have been proposed by various workers: 14 1. Maximum wave height and fetch. For a given wind velocity the wave height becomes greater the longer the stretch of water (fetch) over which the wind has blown. Even with a very strong wind the wave height for a given fetch does not exceed a certain maximum value. For fetches larger than 10 nautical miles it has eee eo IL [F where caine represents the maximum probable wave height in feet been observed that with very strong winds and F is the fetch in nautical isiles. 2. Wave velocity and fetch. At a given wind velocity the wave velocity increases witn increasing fetch. 36 Wave height and wind velocit The height in feet of the ereatest waves with high wind velocities has been observed tc be about 0.& of the wind velocity in knots. If the entire range of wind velocities is considered, the observed data conform to H = 0.026 UM where U revresents the wind velocity in knots. 4}. Wave velocity and wind velocity. Although the ratio of wave velocity to wind velocity has been observed to vary from less than 0.1 to nearly 2.0, the average maximum wave velocity apvarently slightly exceeds the wind velocity when the latter is less than about 25 knots, and is somewhat less than the wind velccity at higher wind speeds. 5. Wave height and duraticn cf wind. The time required to develop waves of maximum height corresponding to a given wind 1D increases with increasing wind velocity. Observations show that with strong winds high waves will develoo in less than LZ MOwes , 6. Wave velocity and duration of wind. Although observa-~ bional data are inadequate, it is Known that for a given fetch and wind velccity, the wave velocity increases rapidly with time. 7. Wave steepness. No well established relationship exists between wind velocity and wave steepness, that is, the ratio of wave height to Length. This ais orobably due to the fact that wave steepness is not directly related to the wince velocity, but depends upon the stage of development of the wave. The stage of development), Om age of (the wave, can be conventently expressed by the ratio of wave velocity to wind velocity (C/U), because during the early stages of their formation the waves are short and travel with a velocity much less than that of the wind, while at later stages the wave velocity may exceed the wind velocity. In order to establish the probable relation between wave steep- ness and wave age all wave observations were examined which appeared to be consistent with certain basic requirements and HO Wintela WeULMes Cie Il, ih (one ( Cie BE), Euscl W Weise ieeeord sel. The corresponding values of H/L and C/U were plotted in a dia- eram (fig. 7). The scattering of the values is no greater than MROWLLG| INE Ehcorxowmeol, COOMSilClSigiias while) EACSee Giciceies| Cl MSASpECSiMSinws - There apvears to be a definite relationship between the steep- ness and the age of the wave. This relationship, shown by the curve in Figure 7, plays an important part in the theoretical discussion. 16 o oa N a x|- a bp WAVE STEEPNESS,4, IN PERCENT uw Nn 1.0 Ma 1.2 1.3 1.4 5 1.6 1.7 1.8 ie) al 2 BS) 4 5 6 7 8 K:) WAVE AGE, < Figure 7. Relation between wave steepness as expressed by the ratio wave height to wave length, H/L, and wave age as expressed by the ratio wave vel- ocity to wind velocity, C/U. Observed values shown In? OL1eCLOS ¢ Se WecreascmoOtmlelehbOmeswelel. | Lhe hea this som ssweliide= creases as the swell advances. Roughly, the waves lose one-third of their height each time they travel a distance in miles equal to their length in feet. 9. Increase of period of swell. Some authors claim that the veriod of tne swell remains unaltered when the swell ad- vances from the generating area, wiereas others claim that the veriod increases. The greater amount of evidence at the present time indicates that the veriod of the sweil increases as the swell advances. 17 GROWTH OF WIND WAVES A knowledge of the height, velocity, and direction of progress of wind waves is necessary if their arrival as swell at a distant coast is to be vredicted. Direct observations of these wind waves are rarely available, but their height and veriod can be determined from consecutive synoptic weather maps if the relationship between wind and waves is known. In the area of wave Pomme ttton the highest waves present at any time depend upon the wind velocity, the stretch of water over which the wind has blown (the fetch), the length of time the wind has been blowing cver the fetch (the duration of the wind), and the waves which were vresent when the wind started blowing (the state cf the sea). These four factors can all be determined if a sequence of weather maps is available showing the meteorological conditions OVE wa OGSEIAS CMs sdeomyels Oi, Seay LZ Cre Zh Inewics, WHSese@ mej must be based on a sufficient number of ships' observations to make possible the plotting of fairly accurate isobars from which winds may be deteriined. In the tropics wind observations imust be avail- able from ships or exposed stations on islands. In middle and higher latitudes direct wind observations on ships will serve as checks on wind estimates from the isobars. Thus, with adequate weather maps at one's disposal, an estimate of the wind waves can be made if accurate relationships between wave height and wind velocity, fetch, and duration are known. Such accurate relationships have not been developed in the past because of the inadequacy of observational data on waves, but they can be 18 determined theoretically from a consideration of the wind energy available for wave forination if tre fundamental assumption is made that the velocity (pericd) of a wave always increases with time. The area in which waves are formed is called the generating area. In such an area waves receive energy from the wind by two orocesses, by the push of the wind against the wave crests and by the pull or drag of the wind on the water. ihe eneney transite by push depends upon the diftterence between wind velocity and wave velocity. If the waves advance with a speed much less than that of the wind the vush is great, but if the two velocities are equal no energy is transferred. If the waves travel faster than the wind they receive no snergy by push but on the contrary they meet an air resistance comoarable to the air resistance against a traveling automobile. The effect of the push of the wind or of the air resistance against the wave depends on the wave form. There enters, therefore, a fundamental coefficient which is related to the degree to which the wave is streamlined and which is called the "sheltering coefficient." The determination of this coefficient is necessary for an exact evalua- tion of energy transfer by push. The pulling force of the wind always acts in the direction of the wind. It is the same at the wave crest and the wave trough but the effect differs. Energy is transferred from the air to the water (the movement of the surface layer is speeded uv) if the sur- face water moves in the direction of the wind, but energy is given WY) off from the water to the air (the movement of the surface water is slowed down) if the surface water moves against the wind. If wind and waves move in the same direction the water particles move in the direction of the wind drag while at the crest, but against the drag when in the trough (see fig. 2). In the absence of a mass transport velocity the particle velocities at the Bee and the trough are equal but in opposite directions, so that the effect of the pulling force of the wind at the wave crest is ex- actly balanced by the effect at the wave trough. In the presence of a mass transport velocity, however, the forward motion at the crest is greater than the backward motion in the trough (sestjeey dh) and a net amount of energy is transferred to the water. No satis- factory explanation of the growth of waves can be given without assuming a transfer of energy due to the wind pulling at the water particles; and this fact is the best ergument for the presence of a mass transport velocity in ocean waves. Since the pulling force cf the wind over the ocean is known, the energy transfer from the air to the water by wind drag can be computed with considerable accuracy from the theoretical values for uiass transport velocity given on page 9. iven when the wave velocity exceeds the wind velocity, the effect of the wind drag remains nearly the same because it depends unon the difference between wind velocity and varticle velocity in the water, and in eeneral the water varticles jwove imuch more slowly than the wind even when the wave fori: moves much faster. If the wind can not transfer -aergy to the water by pulling at the water particles, 20 no satisfactory exolanation can be given of the fact that waves frequently have a higher velocity than the wind which produces them. Energy is dissipated by viscosity but the viscosity of the water is so slight that this process can be neglected. There is no evidence that energy is dissipated by turbulent motion in the wave. The chief processes which can alter the wave height or the wave velocity in desp water are therefore the push of the wind, which becomes an air resistance if the wave travels faster than the wind, and the drag or vull of the wind on the sea surface. Knowing the rate of energy transfer from the wind and the rate at which the wave energy advances (page 6) it is possible to establish a differential equation from which the relationships between the waves and wind velocity, fetch, and duration are obtained as special solutions. The equation contains three numerical constants (including the "sheltering coefficient") which have to be determined in such a manner that all the nine empirical relationships are satisfied. This can be accomplished, and at the same time discrepancies between existing empirical re- lationships can be accounted for. The growth of waves as determined in this manner is illus- trated in Figures 8 and 9 which are constructed on the assumption that a wind of a constant velocity of 30 knots started to blow over an undisturbed water surface extending for 600 or more nau- tical miles from a coast line. Figure 8 shows the height and period of the waves as functions of the distance from the coast al WIND VELOCITY 30 KNOTS 5 fe < Ww m Ww me ce) m z zy ee fo) Be go bE a ae 2 2 Wi sf 2 uJ > (2) aq [o} Zz = iz) w FETCH, F, IN NAUT, MILES Figure 8. +- Wave height and wave period as func- tions of distance from coast line at 51 to 35h after a wind of 30 knots started to blow over an undisturbed water surface. for every fifth hour after the wind started. First, small waves are formed, probably by eddies striking the sea surface. At the coast the waves remain low, but off the coast they travel with the wind and grow as they receive energy by push and oull. When the wind has blown for 5 hours one finds that with increasing distance from the coast the waves increase rapidly in height and veriod out to a distance of 35 miles. There the waves are 8.4 feet high with a veriod of 4.7 seconds. Beyond 35 miles similar waves are present but there exists a striking difference between conditions inside and beyond the 35-mile point. Inside of 35 miles ae a stscady state has been reached, that is at any given point the waves do not change, no matter how long the wind lasts, but beyond 35 wiles the waves continue to grow for a length of time which depends upon the distance from the coast. After 10 hours a steady state has been established to a distance of $5 miles, after 15 hous) ToOluamdustance on MOOlmilles: Jand Soons sin Bicure Cithe ful d— drawn and dashed curves show the steady state. Parts of the curves and the horizontal lines represent wave height and period as func- tions of the distance from the coast at 5 to 35 hours after the con- stant wind of 30 knots started to blow. The fetch shown in Figure 8 can be limited either by the pre- sence of a coast line or by the characteristics of a wind system Over the epen ocean. it may be seen from the fisure that forva given wind velocity the time needed to establish a steady state depends only upon the length of the fetch. For a given fetch this time depends, however, on the wind velocity and is longer for weak winds than for strong winds. This time is called the minimum dur- ation and is measured in hours. Plate I shows the minimum duration as function of wind velocity and fetch. Plates II and III show wave heights and periods as functions of fetch and wind velocity when the duration is longer than the iinimun. If the time is shorter than the minimum duration, the waves at the end of the fetch depend on the wind velocity and the duration in a wanner similar to that shown for a 30 knot wind in Figure 8. For practical use Plates IV and V show wave heizhts and periods as func- tions of wind velocity and duration. £3 When using Plates II to V it should be borne in mind that the curves are constructed on the assumption that a constant wind sud- denly starts to blow over an undisturbed water surface. If the wind velocity changes gradually, an average velocity has to be introduced according to rules which are discussed when dealing with the prectical applications. Also, allowances must be made for-waves that are oresent when the wind starts blowing. Some other characteristics of the growing waves are shown in Figure 9. In the upper curve the wave steepness as expressed by the rato H/lis) plotted azainst the tebceh tor la wandwotm SO isnots. The curve shows the steady state and the horizontal lines show the stage or development after 10, 20, and 30 hours. Before a Steady state has been reached, that is, when the duration is Shorter than the minimum duration, the steepness decreases with time, and when a steady state has been established it decreases Wath hebieh. In the lower curve of Figure 9 the wave age as expressed by the ratio, wave velocity to wind velocity, C/U, is plotted ageinst fetch. The wave age increases with duration before the minimum value is reached and with fetch after the establishment of a steady If the corresponding values of H/L and C/U are plotted in a gravh with wave steepness, H/L, and wave age, C/U, as coordinates Wideny sabe lll” Eodeloielhy Cia qin Clave) slat WaeqbkeS 7/. WiaalGla IWSjoreesSemgs wlae empirical data. Actually, this curve has been used for determin- ing the constants needed for carrying out all computations. By ineans of the curves in Plates II to V it can be ascertained that the ewoirical relationships 1 to 6 are satisfied. z| ef oats z wre) ul re ic: we NAUT. MILES a $ 400 9 TS | (e) ine) WAVE AGE, ¢- FETGH, F IN NAUT. MILES f 200 300 400 Figure 9. Wave steepness (upper graph), expressed by the ratio H/L, as function of distance from coast line at Loe. 20h, and 30h after a wind of 30 knots started to blow over an undisturbed water surface, and corresvonding representation of wave age (lower graph) exoressed by the ratio of wave velocity to wind vel- OGY, G/U. According to Figure 9, with a 30-knot wind the wave veloc- ity remains lower than the wind velocity at fetches of 600 miles or shorter. With increasing fetch the wave velocity would, how- ever, exceed the wind velocity and the waves would continue to grow in height but decrease in steepness. If the wave velocity exceeds the wind velocity the waves can no longer receive energy by push but will lose energy because of the air resistance they meet. They will however continue to receive energy by the pulling force of the wind and will grow in height until this gain is compensated by the loss due to air re- sistance, which occurs when the ratio C/U equals 1.45. The fetch and duration needed for reaching this stage increase rapidly with increasing wind velocity, as shown by the values in Table 1. If the fetch and the duration are longer than those listed in the table the highest possible waves will be present regardless of how much longer the wind blows. Table I Highest Possible Waves Produced by Different Wind Velocities, and Corresponding Fetches and Durations. (Ratio wave velocity to wind velocity equals 1.45, ratio wave height to wave length equals 1/45) Wind Highest waves Fetch Duration velocity Height Period (naut. m. ) (hours) (knots) (feet) (seconds) 10 26 L.8 260 25 20 OR 9.6 1040 50 30 Zou 14.4 2340 (D> LO I 55) 19.2 4150 100 50 66n2 24.0 * 6500 1.25 Waves of the character shown in Table I may be present in the trade wind regions and may be approached in the westerlies of the southern oceans. In the middle and higher latitudes of the Northern Heisphere the fetches are so short that with strong winds the wave velocity always remains less than the wind veloc- WWW 6 26 Plates II to V show only the highest waves present. These waves have traveled the entire distance from the beginning of the fetch. However, the wind can raise new waves anywhere in the fetch, and some of these may grow slowly and reach heights corresponding to the distances they travel, while others may srow Gapidly and break. These eontribute to the broken ap- pearance of the sea surface which is described as the "state of the sea." The relationship between the wind and the state of the sea is discussed later. DECAY OF WAVES Waves Advancing into Regions of Calm When waves spread out frou a generating area into a region of calm only half of the energy of the wave advances with wave velocity. The consequence of this characteristic can be recog- nized by. examining a Simple example. Assume that a series of waves is foriied by rhythmical strokes of a wave machine which at each stroke adds the energy E/2 in a given locality. The first stroke creates a wave of energy E/2. In the time interval between the first and the second stroke one half of this energy, E/4, advances one wave length and one half, E/4, is left behind. The second stroke adds E/2 to the part of the energy which was left behind. On completion of the second stroke two waves are present, one close to the wave machine with an energy 3E/4 and one which has advanced one wave length with energy E/4. By re- peating this reasoning, Table II has been vrevared, showing the 27 distribution of energy in the waves after sach of the first five SipeOktes. | ANG) Slain Tid wine esi ibiime O18 wai wellolle e Cleit tinwe pattern has already developed after five strokes; the waves which have traveled the greatest distance have very little energy, the wave which has traveled half way has an energy B/2, and each of the waves closest to the machine has an energy which approaches the full amount HH. When a large number of strokes have been completed these gradations are much clearer and the distribution of energy can be represented schematically by the curve in Figure 10, which shows that the energy advances with a definite "front." At the front the wave height increases from nearly zero to nearly its full value in a distance corresponding to a small number of wave lengths, and this front advances with half the wave velocity. Table II Advance of Waves from a Wave Machine into Still Water Number of Relative energy of advancing waves strokes iL L/2 z 3/h ih 3 Ws ys Uys Ih IS/lS IU/ML F/G — L/L6 y) BIB) 26/32 MS /32 O/ 32 1/32 When applying the above reasoning to the behavior of wind waves which advance into regions of calm it is necessary to con- sider also the following facts: (1) the wave loses energy because 28 of the air resistance against the wave form, (2) the wave velocity (period) increases continuously. "FRONT " OF ADVANCING ENERGY Fisure 10. Advance of wave energy in time t from a Source INTO Sitti waiter. A viery smalieanounitn om the energy has advanced the distance Ct. The region of ravid increase, “the front," has advanced the dis- tance C t. ZR When the problem is treated analytically it is not necess- ary to introduce any new constants. The travel time of the waves and the decrease in wave height can be obtained as special solu- tions of the fundaiental equation which was discussed in the sec- tion on the growth of waves. The results cf this analysis are presented in Plate VI. The coordinates are the wave period at the end of the fetch, Tp and the distance of decay, D, that is, the distance which the waved travel through areas of calm. The main part of the graph contains two sets of curves. One set gives the factor by which the wave height at the end of the fetch, Ha must be multipvlied in order to fine the heisht o* the swell at Hie end of the distance of decay, Hp. The other set zives the travel ti.ue, Up» (in hours) for the 29 distance, D. Inset I sives the length and velocity of a deep-water wave for which the period is known. Inset Il gives the factor by which the veriod at the end of the fetch, Tp must be multiplied in OLeCGSie CO sinc aie josielOGcl Gig wine Giacl Of wae, ChiswemneSs Oi CleCeyy/, Th: Hats factor cdejoeiacls only upon the reduction factor for the wave nedela 5 il et The use of the diagrams will be described when dis- DE dd cussing the forecasting of swell. Effect of Following or Opposing Winds The effect of a following or an opposing wind on the decrease of the heisht of the swell is also found from a special solution of the fundamental equation of the "energy budget" of the wave. LG is assumed that the increase in wave velocity over the distance of decay is not influenced by following or opposing winds. Although this assumption has little basis in either theory or observation it MeO Moly MEAS cO BwppicomiimMewoly COieeSOw iceESuilos, lid tine Case OF A following wind the computed wave heights may be somewhat too high and the wave periods somewhat too low, whereas in the case of an opvosing wind the heights may be too low and the periods too high. Consistent differences between values computed on this basis and observed values may later be used to improve the theoretical aporoach. The following or ovvosing wind may blow over only a part of the distance of decay. The vroblem is to determine how much more or how imtuch less the wave height decreases in any given distance as compared to its decrease in the absence of any wind. This 30 problem can ber solved by ticans of Plate Vil, the use of which will be explained when discussing the practical procedure. Distance from which Observed Swell Comes; Travel Time; Velocity of Wind which Produced the Swell If the height and the period of the swell are observed it is possible to find approximate values of the distance to the end of the generating area from which the swell came, of the travel time of the swell, and of the wind velocity in the generating area. In Plate VIII the coordinates are the height of the swell (in feet) and the veriod of the swell (in seconds).- The plate contains three families of curves: full-drawn curves giving the distance to the generating area in nautical miles, light dashed amienes giving the travel time from the generating area in hours, and heavy dashed curves giving the wind velocity in the generating area in knots. The values which can be derived from the plate are only apvroximate because the height and period of the swell depend also upon the ratio between wave velocity and wind velocity (C/U) at Ghe end ofthe: feten. Lhe sraph us constructed for C/U = O16. corresponding to average conditions, and gives too high values if C/U is smaller and too low values if C/U is larger. However, variations ia 6/0 between 0.7 and O.9 will not introduce errors exceeding 10 per cent, but errors will also arise from inaccuracies in the observations of height and period of the swell and from lack of knowledge as to changes caused by following or opposing winds. The values read off from the graphs may therefore be 25 ver cent iid, GICICOIe Bi THE STATE OF THE SEA The preceding discussion has dealt only with long-crested waves, that is, waves with very long crests and troughs. Waves may also have wave-shaped crests and troughs. In the presence of such short-crested waves the free water surface shows a ser- 168 Of alwcrmeGing Varelas” eiacl Vhows” (mace 10) 5 Mwrcineimnere , attention has been paid only to the waves which accumulate the largest amount of energy and attain the greatest heights and longest periods. In addition a these waves, and superimposed upon them, a large variety of shorter and lower waves will also be peesent. AL wind vellocities exceeding Beaurort 3 many of these shorter waves imerease So rapidly in heleht thaiv they breaks forming white caps. It appears that at -low wind velocities a great amount of energy goes into the formation of regular long- erested waves, while at high wind velocities a large part is used in the generation of small and short-crested waves. After the waves leave the generating area, the small waves and the short-crested waves die out quickly because they contain little energy, and the long-crested waves of maximum height, which MEWS ISSA ClSelLiy Walwla alin wine jueoCSClibas CML YSIS, EIS wespoMmsiilole for the emerging swell. In the generating area, however, the brok- en appearance of the sea surface is chiefly determined by the pres- ence of the small, short-crested waves and is déeSscribed- by the term "state of the sea." For the state of the sea there have been proposed several scales of which the Douglas Sea Scale is the most widely used. 32 There exist, however, discrepancies between definitions of the term "state of the sea" and cisagreeiients as to the wave heights to be assigned to the descriptive terins. The following discus- sion appears in "Instructions to Marine Meteorological Observers." (U. S. Weather Bureau. Circulur M, 6th ed., 1938). nages 53-55: "Ordinary waves which are moving with the wind constitute the 'sea' while a relatively low, undulat- ing sea surface, with motion in a direction different from the local wind, is the 'swell.' "These definitions are not entirely satisfactory. Usually, the ocean surface is disturbed by both forms of wave motion, with the swell from distant winds crossing the local sea. The combined effect is the "sea,' while the well-defined ridges of waves sioving in a different direction from the local wind are the 4 SHE ILILSS 5 Y Mating 565 Seale? (Maile inl, Golkuiiins I, 2s etal 3)/ SHOW d ben UScd sin wCKaASsciityincs tines Cheracter some sea disturbance. In recording observations in accord- ance with this scals, 'sea' may be considered to be com- posed of swells, combined with waves produced by the winds at the place of observation. "The scale of sea disturbance is approximate, based roughly on the observer's judgment as to the height of waves." On the other hand, the "Admiralty Weather Manual," 1938, pages 5O=5llL, Sweisese "The state of the sea should be reported according to the Douglas Sea Scale (Code XIII), which is here re- produced with = table of heights of waves corresponding to the code figures" /Table III, columns 1, 2, and 4/ "... Careful distinction skould be made between sea and Swell, sea being the waves caused by the wind at the place and time of observation, while swell is wave motion due to past wind or wind at a distance. The direction from which the swell comes should be noted to nearest colpass point." 33 On the Meteor Expedition wave heights were measured from stereophotogrammetric pictures and these wave heights were compared to Simultaneous estimates of the state of the sea made by the ships' cfficers. A comparison of the two sets ae observations led to the assignient of the wave heights which are given in Table III, column 5. In view of the discrepancies between different systems for describing sea state, only a tentative assignment of wave heights (in feet) to the different terms of the Douglas Sea Scale is eiven in Table 2121, columm 6. It Should be noted that this assignment intends to relate the wave heights as obtained from Plates fk and Vi vol the terns of the Sea Scale. Gensiderabilke Weight has been given to the Meteor data and to the fact that for low waves the observed wave heights are in general too low. L& the letter Leasure as taken a nito acicount there rius mo great discrepancy between the values of the Admiralty Weather Manual (Table III, column 4) and the values introduced here (column 6). The validity of the tentative assignment can be WSSCSC [di COmipesling iweSjooOIeGs Cie wae See oO wie Se wO WellwSes derived from wind fetches and durations as determined by weans of weather maps. In columns 7 and 8 are stated the corresponding wind vel- ocities based on fetches of 500 wiles and durations of 24 hours. The frequency and direction of different states of sea in certain parts of the oceans, as well as the frequency and GhLeOEigiOid Cs Syl, cee Slyem Om Isls O, Glaesess NOS, LOs 72. CG, eiagl 15, 3h “2G 988d “6€61 6 “OU *0 "H ‘WoqdTpMog 09 Sutpaoooy °F "9947 TUM00 [BOTSOTOLOS4e,] 1BUOCTIRULE\VUT Aq peqdope sTeos oF FUTPIOOOY °9 "eTB0G Bsg SBTenog Ssyq JO suzs1 9844 09 poUstsse squstey eaem peqnduog

= - ad pesnjuog 6 O< OL< Cry << GE< OK OW << sno4tTdtosig 8 6‘S O16 97-S€ G€-Gz2 96-72 Ot-02 YUSTH Aton ie ON 8 LE-82 G2-ST WEG Og=Zil U=TH 9 i," ©) ah O€-2z SISGl- OZ Site Giles ysnoy AeA S 9°S 9°S WGI Elt—9 CUrG’S 6G 8-S — ysnoy ul uf + QI-TI O46 G°O=y 6-2 G-€ 94819 poy € E=T WY=G°E GOD. EL VUSTIS G T=0 G°0-0 a y.oousg i 0 0 0 WlBO 0 (6) 2 (8) (Z) (2) (T) g‘qiojnveg oa‘ qrozjneog m..L9, “ON 010g PUTM 200m pUuTM A JTOOTSA PUTM SOTILTOOTSA PUTM puUe SIUSTSH SAeM SUTPUCdSSTIOQ pue esq 94 JO 9a4eqg III 914eL aTeog Beg seTonog Way FORECASTING OF WAVES FOR SHORT FETCHES In areas such as the Méditsrranean and other partially enclosed bodies of water, it is often necessary to forecast waves senerated over short fetcnes wnich are dstermined entirely by coast lines and Wind direction. In this case the problem of forecasting becomes primarily a meteorological problen of forecasting the direction and velocity of the wind. If this cam be done, the wave height and waive WSieLod aAiweS Foumc irom PleawSs MLL Af wae Fewci LS Sinorweie wide LOO MEMEO ilies, feliael wie Ieee INE iw aig as Womecre, US C.0,0,0,0-0,6-0,0. 0,000.0, [a a SOOO fer Ohta To ys waa | | | | Wind Velocity | Second | Wind | | Area Wave Height XX | ee Present RS S52 O | | Wave Period | | [eee [ie ik Sr AN ne a (diate ener alee! | Special D=0 | Tp =7 Waray ae Boe i Hi a : Hp'y'sHp y', D'= 'D Sa er aig ape Keio Pesci Nanna sey cline ves rene < we Second Wind Area Special e ‘ | Case | D'=D | Hp"=Hp o'=Tof é Hp! u'=Hp.u' [Sra atl een Sa —— ore rn Se \/ KEK KKM KOKO [ : fr a al SOR KR KK | Special | ‘ | It T OOOR Case | D=0 niay ve | Moyes 5 Nscieg! | | |i oer ier | oe ll ee See a SE QOS Definitions: D Distance from end of generating area to locality for which forecast is desired. D' Distance from end of generating area to beginning of second wind area. D" Distance from end of generating area to end of second wind area. Hp! Wave height at the beginning of second wind area. Hp"'y' Wave height at the end of second wind area Hp" Wave height at the end of second wind area, if secondary wind U' were zero. (The same system of notations applies to wave periods, except that Tp" u! always equals Tp) # Note: Hp',Hp",Tp', Tp", are wave heights and periods in area of decay at distances D'and D’ from generating area. (No second wind area ). Te) 23 2k 25 26 27 28 29 30 Bil 32 33 34 35 36 Si 38 39 Table VI Summary of Quantities to be Determined when Forecasting Swell in the Presence of Following or Opposing Winds Term Distance to beginning of second wind area Distance to end of second wind area Wind velocity in second wind area (D" - D') Sign to be applied to U' Reduction factor for wave height for distance D' (U' = 0) Wave height at distance D' (U' = 0) Factor of period increase for distance D! Period at distance D' Reduction factor for wave height for distance D" (U' = 0) Wave height at distance D" (U' = 0) Factor of period increase for distance D" Period at distance D" Average period for distance (D" Ee D') Ratio wind velocity to wave period in second wind area Ratio of wave heights in second wind area (U' = 0) Correction factor to (35) Wave height at end of second wind area Reduction factor of wave height for distance (D - D") Wave height near coast Symbol Units D' Naut. m. Dy Naut. m. Uy Knots Be Hp: /Hp Hp: Feet Tp. /Tp Tp: Seconds Hon /Hp Hp Thy /Tp Tow Seconds oe Seconds Ui Knots/sec |. pao Hn ur Feet Hp yt/Epw ys Feet Hp ur ral Source Current and prognostic synoptic charts Current and prognostic synoptic charts Estimated from current and prognostic synoptic charts Following or opposing wind Plate VI, using (12) from Table IV and (21) (11), Table IV, times (25) Plate VI, Inset II, using (25) (12), Table IV, times (27) Plate VI, using IV, and (22) (12), Table (11), Table IV, times (29) Plate VI, Inset II, using (29) (12), Table IV, Average of (28) (23), considering (24), divided by (33) (30) divided by (26) Plate VIII, using (34) and (35) (30) times (36) Plate VI, using (32) and (D - D") (37) times (38) 2. Wind velocity (5). Outside the tropics the wind vellocity, over the generating area is obtained from the pressure distribution. Instead of computing i=) the gradient wind it is sufficient to compute the geostrophic wind (4) and to multiply the value so obtained by a LEOGuUcCETOMGaAchOm whieh tpealce sm niiOmmac COMMIT mul c uc Usa cil Uie Cs Omembiae LS OMEUCS o The following factors appear to be sufficiently accurate to dispose of the somewhat uncertain computation of the gradient wind: Great cyclonic curvature of isobars 0.60 Small cyclonic curvature of isobars 0.63 Straight isobars 0.65 Siaall anticyclonic curvature of isobars 0.67 Great anticyclonic curvature of isobars 0.70 The computations may have to be carried out for different parts of the fetch in order to obtain the average wind velocity in the generating area. Ships' observations should be used as a check on the computed value. A difference of not more than one on the Beaufort Scale between computed and observed velocity is A SeoLSLACuory ClmSCk ~ WalS GwieSow Ort the following or opposing wind on the wave height only has to be determined, because it is assumed that the wave period is not in- fluenced by these winds and that, consequently, the travel time remains unaltered. Travel time and wave period at the end of the distance of decay are therefore found by means of Plate VI in the manner described above. In order to determine the wave height at the end of the dis- tanee of decay, H the wax lrary quanbinnes Tasted in Fe ple vi DU? 47 have somber mound.) lines values Oil, Hp: and Hp are obtained from Plate VI by entering the graph with the period at the end of the fetch, Tp and the partial distances of decay, D'; and D" (see Table V). The corresponding veriods, Tp and Tp are obradliacd from Inset II to Plate VI in the manner described above. Having determined these quantities, Boe oh is obtained from Plate VII in the following manner: The average value of Tp: and Ton is computed and called T, The ratio between the wind velocity in the second wind area, U', and the average veriod, T, is found and is taken as vositive for a following wind and negative for an opposing wind. From Plate VII which is entered with the ratio U'/T and the ratio Hpn/Ep: a correction factor, Hn yr /Hpw> is read off. Multiply- ing Side) Pacer by How the value of he a is found. Finally, Hp ut is obtained from Plate VI by entering this gravh with the period Tp and the distance (D - D"). If the second wind area extends over the entire distance of decay or if there is only one region of calm (see Table V) the procedure is shortened, as evident from the following examples. Example 1 (Table V, special case a) Wave height at end of fetch, H 18 feet Wave period at end of fetch, ie BEY fo GO mesic Gongs Distance of decay, D. ; 600 naut. m. It is estimated that a following wind of 10 knots will blow over the entire distance of decay. The computation of the wave height at the end of the distance of decay, Hp ur is carried out as follows, using the symbols in Table VI: is} Number Syinbol Numerical Value 21 D! 0) 22 p" 600 maw. ma, (iD = 1) DD 23) U! 10 knots 29 Hp/Hp On a7 (Hp. = Hp) 30 Hy 8.5 feet 31 T)/Ty Wa27y (Thx = Ty) 32 Ty TLL 6 de seconds 33 ie Ove seconds 34 oo / Ww 0.98 36 i 5 (Ml Loh3 fusiae (29) aac (BL) 7 Dt! OD - 39 Hp ut W252 feet TMS, Wine COMMOECHOC Iieisinng Ol wine Swell as L252 sews Mine Peclod ass Vly seconds. sas) in vee vEeceding examples, and) ume travel time, 40 hours. It is probable that the method gives wave heights that are somewhat too great and periods that are too short. Example 2 (Table V, sveciai case b) Wave height at end of fetch, H amet 18 feet Wave period at end of fetch, ae oo CEOmSecomads Distance of decay, D. Ader 5 (00) meibnes im- On the basis of the subsequent weather map it is estimated that the swell will meet an opposing wind of 30 knots over the last 200 nautical miles of the distance of decay. Again using the symbols in Table VI: Number Syibol Numerical value Ad ID) LOO MEG 5 ils 22 iD" 600 WAG. Wi, (ID 5S 1D) 23) 5 Gl. (Ui -30 knots 25 Hy. /Hp 0.58 26 Hp: LO ch wee 27] Tp: /Tq gz 28 Ths 10.8 seconds 29 a ae Ook? (Bon = Hy) 30 Hp 8.5 feet Bil Wy) oe Loe7 (Mn = Ty) 32 Ty 11.4 seconds 33 ae lies seconds 3h Wy AE =2 07 35 - Hp/Hp, 0.82 36 Hy yt /Ap 0.62 39 Hy ys Jos wOSt Thus, the corrected wave height is 5.3 feeb, bub period and travel time remain unchanged. Remarks on Forecasts Estimates of the probable decrease and increase of the swell have to be based in part upon a prognosis of weather conditions. Usually the forecaster need not construct a prognostic map but can 50 base his estimate on the conditions he anticipates from his examin- ation of the weather maps. The following or the opposing winds can be estimated in a Similar manner. In order to arrive at an estimate of the rapidity with which Swell nay de out at Ts advisable to split the fetch anvo several parts and compute the swell from each. In middle latitudes a sequence of low-pressure systems, that iS, a sequence of generating areas, often travels across the oceans. It is recommended that the swell which is forecast from each genera- ting area be plotted on graph paper, using height of swell and time OmecmciniclwasmCOOrdiMicnes|.sObServcd values) Should bemcniteiced monte Same graph in order to test the accuracy of the forecasts. lin Carrying out the forecasting it may be found that several wave trains arrive at approximately the same time; in this case the resulting swell will be complicated because of interference. The greatest wave heights may eoual the sum of the heights in the indiv- idual wave trains but the average height will be that characteristic of the train having the highest waves. it apovears probable that with experience the complexity of the expected swell can be forecast. The general procedure which has been outlined should be modified according to the type of weather maps which are available and accord- ing to the experience of the forecaster. However, it should be em- phasized that the continuity of the processes must be borne in mind. Example Forecast of swell for Casablanca and vicinity, Northwest coast of Africa, November 7, 1931. Dal The forecast is based on the weather map for the North ame or Nowenlosre 7 ey IAC, ELM. (Pie, 12) eiaél om joPeced= ing maps. The weather mav of November 6 showed an elongated low-pressure area to the south of Greenland from which a cold front extended south in longitude 32° W, bending toward SW in latitude 40° N. Behind the cold front the wind was WNW with an average sneed of about 30) knots. Elo the casi, of they mcontee co— ward the coast of Spain, the wind was nearly W and the average speed about 20 knots. | On November 7 the low-pressure area and the cold front had advanced toward SSE and a well-defined generating area was pre- sent to the northeast of the Azores (fig. 12). The isobars, drawn at intervals of 5 mb, were nearly straight and in 40° N they were 1.6 degrees of latitude apart. The corresvonding geo- strophic wind was 50 knots and, with a reduction factor of 0.65, the wind at the sea surface was 32.5 knots. Ships revorted wind velocities of 8 Beaufort (30-35 knots according to the scale ad- opted by the International Meteorological Committee). The aver- aze wind velocity during the past 24 hours is found to be 29 knots, according to the rule given when the determination of the wind velocity was discussed. In selecting the boundaries of the generating area the front boundary was placed somewhat behind the cold front because of the curving of the isobars, and the rear boundary was placed where the isobars fanned out. This selection gave a fetch of 800 naut- LOCUL fit LES 5 De AREA OF DECAY LAND aes NOV. 7, 1931 ee 1300 GMT AREA OF GENERATION * TETCH [Al 7] Figure 12. Isobars over the North Atlantic on Nov. 7, 1931, at 139% G.M.T. taken from the meteor- ological charts of the Northern Hemisphere. Original observations are omitted, except a number of ships' observations of wind. A generating area and the dis- tance of decay for swell traveling toward northwest Africa are indicated. The duration was determined in the following manner: On November 6 a wind of 20 knots had blown over the generating area and had been preceded by stronger winds. The waves present on November 6, therefore, were the highest possible at that wind WELOGIGy 5, WINENS WS, ElOCOwehlide qe) Iilenes) IML foe I) Wleieyy, viene 10) ieosiw high. A wind velocity of 29 knots would need 7 to & hours to raise these waves (Plate IV) and the duration of the wind was therefore 32 hours. D3 With tnese values one obtains from Plate IL: Minimum duration, CA = /h3) laoibaes} SuLMOS WlalS GlbseEHErLCin als} Slniopeioese elatetal 7h5) Invowboss!, Ilene BOY as) wisiecl . from which one obtains: He = 18.0 feet, Tp = 9.0 seconds. The distance of decay was 600 miles. Entering Plate VI with a period of 9.0 seconds and a distance of decay of 600 miles, a travel time of 40 hours and a reduction factor of 0.47 are read off. Consequently, the swell should arrive at the northwest coast of Morocco in 40 hours, that is, on November 9 at 959%, G.M.T., with a height of 8.5 feet. From Inset II in Plate VI one finds a factor of 1.27 for the period increase, that is, the swell should arrive with a period of about 11.4 seconds. The calculations can be tabulated as follows, using the sym- IOS iin Welloile INVES Number Symbols Numerical value iL G 1.6 degrees of latitude 2 ) L0° N. 3 - straight L Ug 50 knots 5 Uy 32.5 knots (factor 0.65) 6 U 29 knots (adopted average) qT (U) 8 Beaufort 8 F 800 naut. m. 9 tg 32 hours Dy Nuiber Syiubols Numerical value ake) erie 43 hours TAL Hy 18.0 feet LZ Ty 9.0 seconds 13 D 600 naut. m. ih. H)/Hp 0.47 15 Hp 8.5 feet 16 T)/Tp Lo 27 1L7/ Ty 11.4 seconds 18 ty L0 hours 19 Ly 690 feet 20 Ch 35 knots When vreparing a forecast on the basis of this analysis it must be considered that the winds over the distance of decay can be exovected to continue to blow in the direction of progress of the swell so that the decrease in height will be less than that obtained from Plate VI. From this prognosis of the weather conditions it is estiiated that a following wind of 10 knots will be present over the entire distance of decay. According to the procedure outlined above (example 1) the swell should then arrive with a height of 12.2 feet and the veriod of the swell should remain unchanged, but this wave height may be soimewhat high and the period may be too short. Fur- thermore, the wind system causing the swell will vorobably continue to advance towards the east so that the height of the swell can be exvected to increase for some time as the distance of decay shortens. The following forecast should therefore be issued; D9 Casablanca and vicinity: On November 9 between 24 and @8@@: Swell from NW, height 8 to 12 feet, veriod ll.4 to i) seconds. Swell increasing during the day. This forecast did not need modification on November & because the weather map of the 8th showed the estimate of the following wind to be nearly correct. The following values were observed on the morning of November 9: Locality Approx. height Period Swell (feet) (seconds) from Mehedya 7, 15 NW Rabat g ILS) NW Casablanca 12 15 NW Safi 6 12 W Mogador W WZ NW The observations at Safi give consistently lower values than those at neighboring stations, possibly because the locality at which observations were made is less exposed. For the other sta- tions the forecast height of the swell was nearly correct, but at the northern stations the forecast veriod was too short. FORECASTING OF THH STATE OF THE SEA A forecast of the state of the sea must be based on the con- clusions as to the state of the sea drawn from preceding and cur- rent weather maps and upon a prognostic weather map. The procedure in using the prognostic weather map is exactly the same as that 56 which applies to the current mav. When winds, fetches, and dura- tions have been estimated the wave heights and periods in the generating areas are found in the manner described when discussing the forecasting of swell. If desired, the state of the sea may be described by a term on the Douglas Scale, according to Table III. Although the method has not yet been tested extensively, it is believed that the accuracy of the forecast will correspond to the accuracy of the prognostic map. It must again be emvhasized that success can be expected only if the continuity of the proc- esses are borne in mind. D7 Appnendix WAVES ENTERING SHALLOW WATER: BREAKERS AND SURF. A manual on forecasting breakers and surf is in vreparation. For temporary guidance the transformations of waves that enter shallow water are briefly discussed here. Consider a wave which approaches a straight coast off which the depth to the bottom increases regularly and slowly, and as- sume that in deeo water the wave crest is parallel to the coast line. At a distance from a coast at which the depth to the bottom, ad, 18 Bloowle 1/2 the wave length transformation from a deep-water wave to a shallow-water wave begins to be perceptible. The veloc- ity of progress decreases but the period remains unaltered so that the decrease in velocity avpears as a decrease in wave length. If the wave lengths in deep water, Lo» and in shallow water, Ls, are known, the depth to the bottom is obtained from the equation: tanh 27 G_ = _S Ls Oo Where the depth is less than 1/25 of in the equation is reduced to ely eas) em Ly Ge These equations have been used to determine the bottom topography from aerial photographs of waves. The wave height remains constant until a depth is reached which equals about 1/25 of the wave length in deep water. This is explained by the fact that if the effect of friction is disregarded 58 changes in wave height depend uvon changes in the rate at which energy advances. In deep water the amount of energy which advances through a cross section of the wave is 1/2 CEG: where C Alte Land E is the ttean energy of the wave per unit O Cm fo) fo) Surface area. In shallow water the corresvonding amount is Cle where oF m/ed- hit ia) Gigacey as) MOS lony loo wml TIPU CelOMN ES ielaS wave advances toward shore, 1/2 COB o = CE. Where Bo =: one 9 has 1/2 Cy = C, oney/2 Ly = L.- The corresvonding devth is = ho te RS Gar = 25 Weo5 Therefore, the wave height, which is provortional to the square root of the wave energy, is the same in deep water and in shallow water where the devth is avvroximately L,/25- The wave height however does appear higher. The steeoness of the wave has been doubled because the wave length has decreased one half. As the depth becomes less than a/25 the wave height increases rapidly and the wave length continues to decrease. When long and low swell approaches a gently sloping beach, narrow, steep crests, SSOWeIcAwSGl loy/ Lome, wilaw wie Oulelasi, EoOSEIe GO WISS El Siao@ies CaS wosideS Igo, wie DSACll, AiuGl wMeSS C1esSus SOOM MOSCOMmEe SOQ SwSSo wise wae break. It is the narrowness, however, and not the height of the crests which makes them plainly visible. The breaker height, Hy and the depth of breaking, qd, depend unon a number of factors: the steepness and direction of the waves in deep water, the slope and regularity of.the bottom, the strength and direction of local winds, and the number of wave trains present. As yet no general rules can be given, but the ratio H,/H, appears to lie between one a) and two with the smaller value referring to steep waves on Sembly Slopilias DeAcMes, Whe wWelioi© d,/H, varies between one and three, the smaller value referring to a gently sloning beach. Where a wave train apvroaches the coast at an angle the CUIRSCULOM Oi DiKOLKPSSS Cliaises Aas one waves enter shallow water. Snimicie! the velocity ws Mess! in shallow waiters ithe part of the wave which first reaches shallow water vrogresses at a slower rate than the part which is Still in deeo water and consequent— ly the wave front turns gradually until it becomes narallel to the beach. The height of the waves will be less than thab of waves which advance directly against the coast as the energy must be distributed over a greater length of beach. As a simole example consider a straight coast off whicn - Hae Gswigln COidwowne IdiMSs) Cucee joeeeiWILSil, Gali, Gly Smeicsy Cit the waves in deep water Eo Lew a, 0G wae ginyeile yWidwteia wae wave crest in deep water forms with the coast line, and let a, be the angle with the coast line where d = L/25. Where - a.), d = yf 25 the energy of the wave equals Eo cos ( a, _ and the wave height is because the wave height is vroportional to the square root Of ithe enereys) “huss the areduiciihontein. inecntehiGs sswsimcwel: be- cause even for ( a= a.) = 5° If the bottom topography is not too complicated and a good chart is available the bending of the waves can be computed, but such computations should be checked by measurements or aerial nhotographs. Methods for computations will be dealt with in the forthcoming manual of forecasting breakers and surf. 61 ait ‘. Lee -yo1egJ USATS © JO pue 94 7B SeABM STqTSsod 4yseustTy eyq ester 09 AQTOOTSA UWSsATS JO puTM eB AQ papeeau SINOU UT SWT_ “SoeAeM PUTM JO YQMOTH “TT 34Td SATIN LAVN NI 3 *HOLSS OO€! 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RS “J OS Joaas x | \ |i = vl as : ba: BSS \ UE oe $9 09 28 — SS [SS BS NK 09 ote} 06 08 OL 09 OS OV Oe Od Ol (@) suotTjzoun]s se ov *puTM JO UOTIeIND pus AITOOTSA puTM JO g potzed SAeM pue 4JUSTSU SAeM “SSAC DUTM JO UIMOTH “AT 9981 SYNOH NI *°NOlivuna OS SO CEES Cut.CRC CHOC 8¢ OlSeaN| Ov sia 4 1334 NI OF sab ‘tL ‘dOlu3d 3AVM WADA 4O SaNn —— ‘H “LHOISH SAVM AWWADZ 4JO s3Nnq — Ss: + t SEesae es = Hy AT TT] SLONY NIM ‘ALIDOT13SA GNIM LINES OF EQUAL WAVE HEIGHT, H, IN FEET — — LINES OF EQUAL WAVE PERIOD,T,IN SEC. SLONY NI‘N*ALIDOT3A GNIM Plate V 2 DURATION,t,IN HOURS Wave height and-wave period as functions Growth of wind waves. of wind velocity and short duration of wind. Plate V. Plate VI a ayo H OT481 JO uoTIOUNZ se dy ‘gover go pue qe pue ‘-y ‘Aeoep yo s0ueqgSsTp fo pus 4e potted usem1eq OTISY “TT 1eSsuUT “spoyxeid QVUSTOSTITD TOF YAsueT pue AQTOOTOA SAeM °T JoSuUT “Aeoap JO asouedistp pue yores UO) WE Ale) POH EWEN TO SwoOwaouMe sie CBhsy SWoMe a GO jowwe) aie) jowne) CChap C Migare)jo Avo) S0ue1STP JO pus 4e VUSTSY SABM WeeMIeq OTISY ‘*SeAeM Jo Keoeq “TA 99eTd SJTIN LAVN NI ‘G ‘3ONVLSIG 0082 0092 ole) A o00¢2 0002 008! 009! oov! 002) 000! 00g 009 (ole) 4 002 {e} , i “"H/"H Olle SLONX NI ‘9D “ALIDOTIA FAVM - “S nal hic Ae 1 MCE sell a =) ‘linea eae liana 4y/u qvnoda 40 sani cde SWIL T3AVYL WNOA 40 SAN ——— “8 <1 —}+——+— SYNOH NI IN FEET P= aqna93a1 04> rf, JL O1lva 4 WAVE LENGTH a 0082 0092 010} 4 0022 0002 008! 009) oor! 002! ooo! 008 009 OOr 002 (e} ‘DSS Ni “44 ‘dOl6Sd 3AVM Plate VII *sputmM SuTsoddo JO SuTMOT[TOI 04 anp T[ems Jo quStey ut eBueug “TIA 384BTd | | GNIM ONISOddO — is 1 f 4 : t 4 ike) a WM03 JO SSN eo e'O ———— —-+—_4 selon! + - z=) Se pe eae 40 MET ° o + Te Paes Hise) Pacem Vaenn *[TTems JO poTszed pue 4YUs Tey peArTasqo jo suotTqouny se (SqyOUuy UT) Bede SuTqyertoues ut AATOUTSA pPuTM pue (SsrTaAOY UT) amtTq Teaedqy ‘(°ur *qneu UT) Sewoo T[TEems YOTYM MOTIF 9o0ueISTqd “TIIA 298Td SGNOODSS Ni ‘9. ‘W3MS 3O GolMsd ey ° fo) fat) 1334 NI ‘°H ‘173MS JO LH9ISH w nN o¢ s¢ bi Wate