A detailed comparison of theoretical wave spectra and wave forecasting methods

134 I)eutsche t-Iydrographische Zeitschrift. Band 10, Heft 4. 1957

dar t iber vorl iegen werden, in welcher Weise das W i n d s e e s p e k t r u m u n d die Energie im See- gang ve to W i n d und yon der Tempera turd i f fe renz sowie schlieBlich anch yon der W i n d d a u e r a n d ve to F e t c h abhgngen, k a n n mi t einer wesent l iehen Verbesserung der h indcas t s und ebenso der Seegangsvoraussagen gerechnet werden.

Acknowledgements . The senior au thor wishes to express his app rec i a t i on to t he Woods Hole Oceanographic I n s t i t u t i o n for mak ing possible his v is i t to W o o d s Hole in F e b r u a r y and March, 1957. The au thors also wish to t h a n k the U. S. H y d r o g r a p h i c Office for the i r ass is tance in p rov id ing the necessary wea the r char ts a n d power spec t ra of fifteen wave records. This research was sponsored in p a r t b y the U. S. N a v y Office of N a v a l Research under con t rac t wi th the W o o d s Hole Oceanographic Ins t i tu t ion .

Sehrffttum B r a c e l i n , P., 1952 : Observing, forecasting and

reporting ocean waves and surf. Naval Wea- ther Service. Mem. No. 147/52.

B r e t s c h n e i d e r , C. L., 1952: The generation and decay of wind waves in deep water. Transact. Amer. Geophys. Un. 33, 381-389.

C a r t w r i g h t , I). E., and L o n g u e t - l c I i g g i n s , M. S., 1956: The stat is t ical distr ibution of the maxima of a random function. Prec. Roy. See. A, Vol. 237, 212-232.

I ) a r b y s h i r e , J., 1952: The generation of waves by wind. Prec. Roy. See. A, 215, 299.

L o n g u e t - H i g g i n s , M. S., 1952: On the sta- t ist ical distr ibution of the heights of sea waves. J. Mar. l~es., u 11, No. 3.

N e u m a n n , G., 1953: On ocean wave spectra and a new method of forecasting wind-genera ted sea. Beach Eros. Bd., Techn. Mere. No. 43, I)ec.

P i e r s o n , Jr . , W. J., 1954: An interpretat ion of the observable properties of sea waves in terms of the energy spectrum Of the Gaussian record. Trans. Amer. Geophys. Un. 35, 747-757.

P i e r s o n , Jr . , W. J., G. N e u m a n n , and 1~. W. J a m e s , 1953, 1955: Practical methods for observing and forecasting ocean waves. New York Univ., Coll. Eng., Dept. Meteorol. & Oceanogr., Techn. l%ep. No. 1 ; U. S. Navy Dept., I-Iydrogr. Office Publ. No. 603, Wash.

1%o11, I-I. U., 1952: l~ber Gr613enuntersehicde der Meereswellen bei Warm- und Kal t luf t . I)tseh. t tydrogr . Z. 5, 111-114.

S i l v e s t e r , 1~., 1955: Practical application of I)arbyshire 's method of hindcasting ocean waves. Appl. Science 67 261-266.

Eingegangen im September 1957

S v e r d r u p , I-t. U., and M u n k , W. It . , 1947: Wind, sea and swell. Theory of relation for forecasting. U. S. Navy Dept. , Hydrogr . Off. Publ. No. 601.

T u c k e r , M. J., 1956a: A shipborne wave recorder. Transactions of the Ins t i tu te of Naval Architects, Vol. 98, 236-250.

T u c k e r , 5{. J., 1956 b: Comparison of wave spectra. Nat. Inst . Oceanogr. (England), Int . l~eport No. A 6.

W a l d e n , H., 1953/54: Die WellenhShe neu angefachter Windsee nach Beobachtungen at lantischer Wetterschiffe und des Fischerei- schutzbootes , ,Meerkatze". Arm. Meteorol. 6, 296-304.

W a l d e n , t{., 1954: Lrber die I)finung aus einem Windfeld, welches am Beobachtung'sort in einiger Entfernung vorbeizieht. Dr. Fiydrogr. Z. 73 t90.

W a l d e n , H., 1955: I)ie t I6he der Windsee bei gleiehm~2ig zunehmendem Wind. I) t . IC[y- drogr. Z. 8, 236-241.

W a l d e n , t t . , 1955/56a: Ein neues I ) iagramm zur Berechnung des Seegangs aus den Wind- verhi~ltnissen. Ann. Meteorol. 73 213-218.

W a l d e n , I-I., 1955/56b: I) ie H6he der Windsee bei regionaler Zunahme der Windst~rke in der l~ichtrmg mi t dem Winde. Ann. 3/[eteorol. 7, 337-341.

W a l d e n , I-t., and J. P i e s t , 1957i Beitrag zur Frage des Wcllenspektrums in der Windsee. I) t . Hydrogr . Z. 10, 93.

W i l s o n , B. W., 1955: Graphical approach to the forecasting of waves in moving fetches. Beach Eros. Bd., Corps of Eng., Techn. Mere. No. 73.

A detailed Comparison of Theoretical Wave Spectra and Wave Forecasting Methods

B y G. N e u m a n n a n d W. J . P i e r s o n , J r .

(Continued from Heft 3, Band 10 (1957) I)t . I-Iydrogr. Z.)

Comparison o~ the Darbyshire and Neumann results. Agains t the resul ts g iven in P a r t 1, i t is poss ible to p u t t he work of D a r b y s h i r e [1955, 1956] and argue t h a t his resul ts are af ter all based on for ty-f ive ac tua l wave records t a k e n a t sea. To s t a t e t h a t one c o m p u t e d spec t rum ob ta ined b y p ro jec t S W O P is correct and shows a ver i f ica t ion of the theore t i ca l N e u m a n n

N e u m a n n - P i e r s o n , Comparison of Theoretical Wave Spectra and Forecasting lVfethods 135

spectrum is to imply tha t the forty-five spectra for the open sea and other spectra for Lough Neagh used by D a r b y s h i r e to obtain his results are incorrect or not representative.

I t is believed tha t the theoretical family of spectra for both fully developed seas and partially developed seas given by D a rb y s h i t e is not representative for the following reasons in addition to the point made above about possible errors in the use of gradient winds.

(1) The F o u r i e r series analyses for the deep sea eases were not corrected for the effects o f the ship-borne wave recorder developed by T u c k e r [1956b].

(2) For light winds the original data corresponded to dead seas produced by higher winds, and for high winds the data corresponded to seas which were not fully developed.

(3) Sampling variabil i ty in the data raises a question as to the accuracy of the curve fitting procedures employed.

The calibration of the Darbyshire spectra. The ship-borne wave recorder developed by T u c k e r [1956b] is an excellent instrument in conception and design. I t will prove to be an extremely valuable instrument for the study of heavy seas in deep water where other wave recording methods are impractical. Nevertheless, the theory of pressure sensors mounted in the side of the hull of a vessel is not completely understood, and attempts to get the same spectrum from a Woods Hole Oceanographic Institution wave pole record and a ship-borne recorder record even after theoretical calibrations of both devices have not been successful (Tucker [1956a]). It is understood that refined theoretical considerations have improved the situation somewhat over the results in the above reference, but that a discrepancy still exists. There may well be an unknown calibration error which would be a function of wave frequency in D a r b y s h i r e ' s results.

D a r b y s h i r e [1955] appears to have repeated the same type of error made by D a r b y - s h i r e [1952]. Cox and M u n k [1954], in comparing the N e u m a n n spectra with the spectra obtained by ] J a r b y s h i r e I[1952], based on pressure records taken off Lands End, England, in 50 feet of water, found that the high frequencies had been lost. D a r b y s h i r e [1955] appears to have failed to correct his F o u r i e r spectra at high frequencies for the fact tha t the pressure sensors of the T u c k e r instrument were located at a depth of I0 feet i~s suggested by a soon to be published work of C a r t w r i g h t and W i l l i a m s (Trans. Amer. Geo- phys. Union).

To show this effect, fig. 12 shows the spectrum obtained by project SWOP and what results when it is multiplied by exp (-2/~2z/g) when z is 10 feet (see also N e u m a n n [1955]). Also shown is the theoretical spec- t rum of D a r b y s h i r e for the appropriate surface wind. There is still

0 , 6 0 ~ cm 2

0,1 O-

K =

Fheeretlca/ specfrum for on 70,7knot wind . . . . . according fo OarbyshlPe

Spectrum compur from clah7 obfofned - - by Project Swop . . . . . Computed spectrum muir/plied

e-z ~= ~/9 for a depih o f 10 Feer

~= j j0{2~ L0,510 0,71~ ~,001 ,2~ ',S~ 1,7s ~,001 ~,25 zs0 ~,Ts ~,00 I~,~ a~0 T= 96~a2241612 8 6 ~ 3 2

Fig. 12. Observed free surface spectrum attenuated by hydrodynamic affecteompared with theoretical

D a r b y s h i r e spectrum

a discrepancy, b u t the effect of hydrodynamic at tenuation explains well over half of it. The disturbing presence of the hull Of the vessel, the effects of the double integrator in the T u c k e r instrument, and the other two points made above may well explain the rest of the differences.

These considerations also explain par t of the very odd behavior of the D a r b y s h i r e family of spectra as a function of fetch as shown in equations (2) and (26). The data used by Darbyshire really break down into two classes. One class is the deep sea data with fetches

136 Deutsche Hydrographische Zeitschrift. Band 10, Heft 4. 1957

over 50 NM; the other is the Lough Neagh class with fetches less than 16 miles. For the Lough Neagh data a pressure recorder at a depth of 3 feet was used.

Equation (2) shows that for short fetches the wave energy at those frequencies for which it is present is many times greater than it is for the open sea case.

I t is believed that most of this effect is explained on the basis of the difference in hydrodynamic attenuation at depths of 10 feet and 3 feet. The ratio of the squares of the amplitudes of the pressure variations of 3 second waves at these two depths is 0.16, for example. Since the sea surface slope variance depends on the high frequencies much more than on the low frequencies, the form of equation (26) is mainly due to failure to consider this effect and not to any real physical effect.

l~on-representative spectra. One of the most striking differences between the theoretical families of spectra due to D a r b y s h i r e and N e u m a n n lies in the way the spectra vary as functions of fetch and duration. D a r b y s h i r e does not treat the effect of duration, and in effect equation (2) states that the fetch needed to generate a fully developed sea is always 100 NM independent of the wind speed which in his data ranges from about 14 to 52 knots at the surface. Moreover, the significant height at a given fetch distance less than 100 NM is always the same fraction of the fully developed height at 100 NM for any wind speed. In contrast the tables given in P i e r s o n , N e u m a n n and J a m e s [1955] based on the work of N e u m a n n [1953] state that for a wind of 10 knots a fully developed sea is achieved if the fetch is 10 NM long and the wind has blown for 2.4 hours whereas for a wind of 50 knots a fetch of 1420 NM and a duration of 69 hours is needed. The minimum fetch and duration for a fully developed sea continuously increase with wind speed. These results are comparable to the results of S v e r d r u p and M u n k [1947] and B r e t s e h n e i d e r [1952].

According to D a r b y s h i r e , the fetches in his deep sea study ranged from 50 to 400 NM and the effect of fetch was not marked after 100 miles.

Now, a fetch of 1420 NM and a duration of 69 hours for a wind of 50 knots will probably never occur. Thus, if the theoretical spectra of N e u m a n n are correct, a fully developed sea for a 50 knot wind will never occur (fortunately for ships at sea).

However, fetches in excess of a mere 400 NM and with wind durations sufficient to produce waves higher than those predicted by D a r b y s h i r e for the fully developed state can certainly occur if the N e u m a n n spectra are correct, and these heights would fill in the area between the D a r b y s h i r e curve and the N e u m a n n curve in figure 5.

A comparison of the S v e r d r u p - M u n k - B r e t s c h n e i d e r and P i e r s o n - N e u m a n n - J a m e s wave forecasting methods has recently been given byM. R a t t r a y , Jr . , and W. V. B u r t [1956] for an intense storm in the North Pacific. M. R a t t r a y , Jr . , and W. V. B u r r write that "a survey of the historical weather maps back to 1922 (U. S. Weather Bureau) showed only one other storm in the Gulf of Alaska capable of maintaining waves of this size . . . " . The wind in the generating area was between 45 and 55 knots for 33 hours, it was over 50 knots for 18 consecutive hours. The fetch was 500 NM long. The waves had a significant height of 48 feet as observed by trained USWB personnel. Both methods gave a height forecast of essentially 48 feet for the significant waves.

The waves were not fully developed, according to N e u m a n n , and yet equation (13) due to D a r b y s h i r e predicts a wave of only about 36 feet significant height even using the peak wind of 55 knots and a fully developed sea. The gradient wind at the peak of the storm was 80 knots. The observed significant height may have been in error by 5 feet or so, but an error in observation of 12 feet is not likely. Further downwind in the same storm, R a t t r a y and B u r r [1956] estimated the waves to be 55 feet in significant height. The D a r b y s h i r e theories, therefore, are likely to be in error for high winds, long fetches and long durations.

The conclusion by D a r b y s h i r e that fetches of more than 100 NM do not produce in- creased wave heights can be refuted on one other independent ground. Lake Superior has fetches of the order of 100 to 250 NM and is deep enough to permit compari~son with open sea conditions (maximum depth 393 meters). The wind speeds over this lake are comparable to those speeds observed over the Atlantic. According to D. D. G a i l l a r d [1904] the highest

N e u m a n n .P ie rson , Comparison of Theoretical Wave Spectra and Forecasting Methods 137

waves observed are of the order of 20 feet in height whereas waves in the North Atlantic where the fetch can be greater have reached 60 feet and more 2.

At high winds, these considerations explain why the frequency at which the spectrum achieves its maximum value varies as 1/l/v in the . D a r b y s h i r e spectra. Some auxiliary curves drawn on fig. 10 show what the frequency corresponding to the maximum spectral value is for the partially developed N e u m a n n spectra when the fetch according to N e u m a n n is 200 NM and the duration is unlimited, and When the durations are 24 and 32 hours and the fetch is unlimited. The curve for a fetch of 300 NM would fall, if drawn in, just above on the curve given by D a r b y s h i r e ' s spectra. These results suggest that the 45 cases studied by D a r b y s h i r e did not contain enough cases for high winds in which both the fetch exceeded 300 NM and the duration exceeded 32 hours. Consequently, for the cases obtained for high winds the waves were not fully developed.

The highest waves in the D a r b y s h i r e data, as shown in fig. 2 of his paper, correspond to a significant height of about 40 feet, although the reported value may be low due to the instrumental effects just discussed, whereas the v 2 law in fig. 5 predicts 30 feet for the significant height at this wind speed using the curve due to I ) a r b y s h i r e (a 33 percent error in prediction within the data). The P i e r s o n , N e u m a n n and J a m e s forecasting methods predict a wave of 40 feet significant height for a wind of 52 knots if it blows for 24 hours over a fetch of 400 N1V[ or more, or if it blows for more than 24 hours over a fetch of 400 N1Vf. Also for a 52 knot wind of 24 hours duration the predicted N e u m a n n spectrum will have a peak near 14 seconds, and this value agrees with the value shown in fig. 1 of D a r b y s h i r e ' s work for this wind speed.

At low winds an inverse error appears to have been made by D a r b y s h i r e. The frequencies at. which his spectra have a maximum are much lower than those given by N e u m a n n . The lowest wind speeds for which data were obtained correspond to a surface wind of 15 knots. The average wind over the area where the data were obtained is undoubtedly greater than this and it probably exceeded 15 to 20 knots on days prior to the time that the wave records were taken over fetches (for such low winds) of many hundreds of nautical miles and for durations sufficient to raise a fully developed sea. Then when the wind died down to 15 to 20 knots, the waves did not die down to a sea corresponding to this lower wind speed.

Filter 4 in P i e r s o n , N e u m a n n and J a m e s [1955] explains this phenomenon, and for the positiOn where the data were obtained, it is quite possible for the waves associated with higher winds than were present at the time of the recording to persist for one and perhaps two days.

The effect of such a situation is to lower the value of/max and cause the 1/]/v propor- tionality to exist for low winds also. I t also causes the spectra to have the property that the waves are traveling faster than the winds which generate them.

Also shown in fig. 10 is a curve due to D a r b y s h i r e [I952] which shows the frequency corresponding to the period where the maximum energy occurred on the spectra obtained at Land's End when they were plotted as a function of period. This curve agrees quite well with the curve due to N e u m a n n . I t is difficult to conceive of any physical reason why fmax

should vary like 1/v for one kind of data and like 1/}/v for the other kind of data other than the explanation just given.

The sampling variation in Darbyshire's results. One of the greatest sources of possible error in I ) a r b y s h i r e ' s results lies in the inadequacy of the NIO wave analyzer and the sampling variation in his data which has not been adequately treated. D. E. C a r t w r i g h t and L. J. R y d i l l [1957] in discussing the F o u r i e r analysis principle of the NIO wave analyzer write that the information contained in the F o u r i e r analysis of a wave record is not improved upon by any other method of analysis such as an autoeorrelation analysis of a record of the same length.

2 Concerning the important effect of fetches of more than 100 NM, see also H. Tho rade [1931]. ]0

138 Deutsche Hydrographisehe Zeitsehrift. Band 10, Heft 4. 1957

Al though this s t a t emen t is correct , i t misses t he po in t which is t h a t the re is ve ry l i t t le in fo rmat ion in a seven minu t e wave record as used b y D a r b y s h i r e (or for t h a t m a t t e r in a 20 minu te (or so) r ecord as usua l ly ana lyzed on the N I O analyzer) and i t mus t be i n t e rp re t ed wi th care. W i t h d ig i ta l analysis , t he t h e o r y deve loped b y T u k e y [1949] which involves com- pu t ing a covar iance func t ion a n d smooth ing i ts even cosine F o u r i e r t r ans fo rm in order to get an es t ima te of the power spec t rum is only a be t t e r way to do a F o u r i e r analys is more efficiently for a much longer r ecord (see P i e r s o n , J r . [1955]). The i m p o r t a n c e of the T u k e y m e t h o d lies in the fac t t h a t one has s imul taneous cont ro l over resolu t ion a n d sampl ing va r i a t ion depend ing on the leng t h of the record and the number of lags used.

D a r b y s h i r e used wave records which -were f rom seven to t e n minu tes 10ng. Le t us consider the seven minu te long record s . The length of t he record is 420 seconds. The F o u r i e r components will therefore have per iods of T = 420/n as n ranges f rom 1 to, say 420. The values of T for n near 28, a n d 42 are shown in fable 2.

T a b l e 2

Periods corresponding to various harmonics o~ a 420 seeond record

n n e a r 28

n 26 27 28 29 30 T 16.15 15.55 15.00 1.4.48 14.0

n n e a r 42

n 40 41 42 43 44 T 10.5 10.24 10.0 9.'76 9.55

The s t a r t ing po in t of I ) a r b y s h i r e ' s anMysis is the q u a n t i t y Hr which is equal to the square roo t of the sum of t he squares of all of the heights , H . , shown in the F o u r i e r analysis be tween T - � 8 9 and T q- �89 seconds where T is an integer.

Ordinar i ly , when a F o u r i e r analys is is car r ied out, t he resul t is in the form

2 7r nt 27r nt ~(t) = ~ an cos - - q- bn sin (45)

n=l T T

where T in th is case is 420 seconds. The q u a n t i t y H~ is t hen defined b y

z 2 2 H n 2(an 2 + b n )~. (46)

S. O. R i c e [1944] has shown t h a t a n and bn are i n d e p e n d e n t and n o r m a l l y d i s t r i b u t e d wi th var iances essent ia l ly equal to t he in tegra l over t he energy spec t rum of t he process be tween 27~(n- �89 and 2~(n ~- �89 Hn 2 is, therefore, d i s t r i bu t ed according to a Chi square dis tr i - bu t i on wi th 2 degrees of f reedom. I f the spec t rum is s lowly vary ing , successive values of Hn 2 can be a d d e d since t h e y are v i r t u a l l y independen t , and since t he Chi square d i s t r i bu t ion r ep roduces i tself on a l inear s u m m a t i o n of var iables , t he sum of m values of H~ 2 is d i s t r i bu t ed accord ing to Chi square wi th 2m degrees of f reedom.

F r o m tab le 2, I ) a r b y s h i r e ' s va lue of HT 2 for T equal to 15 seconds is d i s t r ibu ted according to Chi square wi th no more t h a n 6 degrees of f reedom since there are a t mos t 3 F o u r i e r components be tween 14.5 and 15.5 seconds; for T equal to 10 seconds HT 2 has a t mos t 10 degrees of f reedom.

W i t h 6 degrees of f reedom the value of HT 2 for T equal to 15 o b t a i n e d f rom the analys is of a seven minu t e wave r eco rd can be 2.33 t imes g rea te r t h a n the t rue va lue one t ime in

,a In order to analyze such a short record over the NIO analyzer, it was repeate d a nmnber of t imes around the circumference of the analyzer. This only serves to blur the F o u r ie r components and introduce noise due to copying, and it does not increase the amount of s tat is t ical information in the record.

N e u m a n n - P ierson, Comparison of Theoretical Wave Spectra and Forecasting Methods 139

T a b l e 3

Composite frequency spectrum for the local sea determined from the wave pole and the stereo data

A E /~ stereo

A E Corn- wave bined pole

Degrees of

freedom

k A E stereo

A E w a v e

pole

C ) m -

bi ned

Degrees of

freedom

7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3O 31 32 33

0.0831 0.1384 0.2392 0.3716 0.5496 0.6498 0.6208 0.5135 0.4545 0.3818 0.3272 0.3056 0.2656 0.2144 0.1894 0.1766 0.1373 0.1086 0.0984 0.0769 0.0785

0.1336 0.1137 0.0979 0.0815 0.0591 0.0491 0.0443 0.0395 0.0392 0.0420 0.0327

0.1354 0.1110 0.0982 0.0787 0.0711

22 21 30 41 50 59 68 82

128 110 122 133 345 377 401 441 458 174 174 174 174 174 174

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 6O

0.0212 0.0193 0.0200 0.0180 0.0160 0.0150 0.0140 0.0110 0.0100 0.0100 0.0090 0.0090 0.0080 0.0080 0.0O8O 0.0070 0.0060 0.0O6O 0.OO5O O.0O5O 0.0050 O.0O5O 0.0050 O.0O5O 0.OO5O 0.0040 0.0040

174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 i74 174 174

twenty arid less than 0.36 times the true value one time in twenty due to sampling variation. Moreover, the estimate of TIT 2 has a greater chance of being lower than the true value than it has of being higher. The probability is one half that HT 2 will be 89 percent of the true value or less.

With 10 degrees of freedom, the value of HT ~ for T equal to 10, can be more than 1.83 times the true value of HT ~ one time in twenty and less than 0.394 times the true value one time in twenty. Consequently, if the values of HT ~ obtained by D a r b y s h i r e for T = 10 seconds are multiplied by 2.54 and 0.55 there are nine changes in ten that the true value will be contained in the resulting range of values.

These considerations are qualitatively borne out by consideration of fig. 3 in the paper by D a r b y s h i r e [1955]. The plot for 10 seconds shows values of H~,/T for ~he same wind speed which differ by a factor of 4, and so the values of HT 2 differed by a factor of 16. The plot for 15 seconds shows values very close to zero as would be expected from ~he limited number of degrees of freedom. (at most 6).

The variability at 10 seconds appears to be even greater than would be expected on the basis of 10 degrees of freedom. The additional scatter can possibly be attributed to a combina- tion of all of the other points made above. 10"

140 Deutsche Hydrographische Zeitschrift. Band 10, Heft 4. 1957

I t was assumed by D a r b y s h i r e tha t HT*= ~[HT-I-d-2HT@HT+~] would give a smoother and moore accurate value for HT. Under such a smoothing procedure, HT* may not be exactly distributed according to a Chi sqare distribution. The scatter in H~,* should, however, be less, and not more, than the amount computed above and this is not borne out by the values as plotted.

The above discussion shows tha t the sampling variat ion in a seven minute long wave record analyzed on a period scale is itself variable and tha t the estimates of/art are highly unreliable for long periods and unreliable for moderate periods. Unless such sampling variation is fully understood and adequately treated, the results of fitting curves to such data must be doubted.

The determination of the spectrum in Project SWOP should be contrasted with the above. Each point on the spectrum plotted as a function of frequency determined by the wave pole data has 174 degrees of freedom for a frequency range from 2 ~ ( k - 1)/96 to 2~(]c + �89 This is equivalent to a weighted average of more than 87 H~ ~ values with the weights chosen so as to best preserve the shape of the spectrmn and so tha t the effective number of heights considered is 87.

The directional spectrum originally had 19 degrees of freedom for each estimate in the plane of the spectrum. After eliminating a disturbance from a distance, the points on the observed spectrum given in figs. 1, 2, and 3 as given by # = 2~k/96 as /c varies from 11 through 22, were determined from the directional spectrum. Points 23 through 27 were determined from both the wave pole spectrum and the directional spectrum. Points greater than 27 were determined from the wave pole spectrum.

The values for the spectrai estimates and their degrees of freedom are given in table 3, p. 139.

Spectral values tbr/c equal to 8, 9, and 10 are also shown. Their degrees of freedom are not shown. Due to the steep rise of the forward face of the spectrum and problems in resolution it is difficult to determine the precision of these points. There is little or no energy for k less than or equal to 7.

The wave pole spectrum agreed within sampling variation with the directional spectrum for ]c equal to l l , 12, 13, 22, 23, 24, 26, and 27. The two different spectra disagreed quite markedly at/c equal to 15. The conclusion reached was tha t the wave pole calibration was in par t in error and tha t the data based on the directional spectrum were more likely to be correct. For full details , see the repor t by C h a s e , Co te , et al [1957].

The NIO wave analyzer has proved to be a valuable instrument in the s tudy of ocean waves ( N . t ~ . B a r b e r and 1~. U r s e l l [1948]), but it is now possible to construct wave spectrum analyzers on sounder principles (see P i e r s o n , J r . [1954b], and S. S. L. C h a n g [1954]), Such instruments are now being constructed by J o s e p h C h a d w i c k (J. C h a d w i c k and S. S. L. C h a n g [1957]) of Sperry Marine Division for the s tudy of waves and ship motions. The analyzer will be capable of analyzing a much longer record and graphing a running weighted average of what is effectively H~2/4 for a choice of filters of different band widths. Since one is forced to take weighted averages of the H~ 2 vMues fl'om the NIO analyzer, it might just as well be done electronically at the start, and the results presented in terms of height squared instead of heights.

Concluding comparison of theoretical families of spectra due to Darbyshire and 5~eumann. In summary, the theoretical spectra of D a r b y s h i r e do not predict high enough waves for a fully developed sea and do not agree with the results of Cox and M u n k and other statistical considerations. Moreover, various sources of error in the determination of the theoretical spectra have been pointed out. All of these sources of error, when corrected for, tend to result in changes which will give agreement with the theoretical family of spectra due to N e u r h a n n .

A comparison of various wave forecasting methods. Wave forecasting methods can con- veniently be placed in two categories depending on whether or not they give methods for forecasting solely the so-called significant height and period of the waves or methods for

N e u m a n n - P i e r s o n , Comparison of Theoretical Wave Spectra and Forecasting lYiethods ]41

forecasting the spectrum of the waves and then a few selected statistical properties of the waves such as height, characteristics, and certain representative periods.

.The S v e r d r u p - M u n k [1947] method, as later modified by B r e t s c h n e i d e r [1952], is perhaps the most well known method for forecasting waves according to the first kind of procedure. Moreover, it has been used quite extensively in the United States for many years.

Other methods of this first type .are current in Europe; for example, there is one recently developed by W a l d e n [1955/56]~

Moreover, a large volume of subsidiary papers based on the main theory has built up over the years. Attempts to account for moving fetches and for the effect of fetch width are among such papers as are also numerous papers on verification of the method and on wave statistics hindcast by the method.

Two methods for forecasting waves by means of wave spectra are in principle available. One would be based on the work of D a r b y s h i r e [1955, 1956] and the other would be based on the work of N e u m a n n [1953] and P i e r s o n , Jr., [1952]. The work of D a r b y s h i r e [1955] is not in a very practical form for a standard procedure although in principle his results are sufficient, if correct, for forecasting waves in an area of wave generation as a function of fetch and wind speed if the duration is unimportant. In contrast, the work of N e u - m a n n and P i e r s o n has been incorporated in a practical wave forecasting manual prepared by P i e r s o n , N e u m a n n and J a m e s [1955].

B r e t s c h n e i d e r [1957] in reviewing the work of P i e r s o n , N e u m a n n and J a m e s , writes that "it is only logical to make a comparison between the two methods [i. e., the P i e r - s o n - N e u m a n n - J a m e s method and the S v e r d r u p - ~ u n k - B r e t s c h n e i d e r method], since at some future date some type of spectrum method will probably replace the significant wave method for forecasting waves, now generally used for engineering purposes." M u n k [1957], commenting on the above review of B r e t s c h n e i d e r , writes that the P i e r s o n , iNeumann and J a m e s method is a "conceptual advance" and that in his opinion the Sver - d r u p - M u n k method as originally developed should be retired.

The crucial question, however, is which of the above three wave forecasting methods will most accurately describe and/or forecast the state of the sea and the characteristics of swell insofar as, say, the significant height and some characteristic period are concerned over the widest range of meteorological conditions. I f the superior method, as far as height and period are concerned, is a spectral method, and if the predicted spectra also verify, then there has been a definite net gain. For example, naval architects can only use spectral methods in studying ship motions at sea, and significant height and period concepts are useless in studying certain properties of the sea. If, however, the S v e r d r u p - M u n k - B r e t s c h n e i d e r method gives consistently better forecasts as verified against observations than a spectral method, then there is something obviously wrong with the spectral method and ultimately a better one will have to be derived.

Sea forecasts. I t has been our experience, and the experience of others who have discussed this problem with us, that the S v e r d r u p - M u n k - B r e t s c h n e i d e r and P i e r s o n - N e u - m a n n - J a m e s methods quite frequently tie as to significant height and period forecasts for a wind generated sea. The study of R a t t r a y and B u r t is an example in which there is no essential difference in the results obtained by the two methods. Arguments, such as those given by B r e t s c h n e i d e r [1957] as to the relative merits of the v 2 versus the v 2-5 law, which more or less reduce to the statement that the P i e r s o n - N e u m a n n - J a m e s method is wrong because the P ie r s o n- N e 11 m a n n- J a m e s fully developed sea does not agree with the S v e r- d r u p - M u n k - B r e t s c h n e i d e r fully developed sea, can really only be settled by measuring the waves in f u l l y d e v e l o p e d seas and measuring the effects of fetch and duration on wave generation. For example, S v e r d r u p - M u n k - B r e t s c h n e i d e r predicts waves about 55 feet high for a wind of 55 knots and a duration of 30 hours, and P i e r s o n - N e u m a n n - J a m e s predicts a wave of about 48 feet under the same conditions. I f such conditions ever occur over a long enough fetch, there is no currently available method of wave recording or wave observation which could surely differentiate between 48 and 55 feet in this case. Also the

142 Deutsche Hydrographische Zeitschrift. Band 10, tIefb 4. 1957

usual degree of uncertainty as to whether a fetch is 400 NM or 500 NM long and whether a duration is 26, 30, or 34 hours long, say, provides enough variat ion to make the predicted heights essentially agree. A twenty minute wave record which would record the waves perfectly would, for example, have a range of at least =]= 10 percent to 20 percent in the estimate of the E value from a computed spectrum, and hence the recorded significant height would be accurate to at most 5 to 10 percent for this reason alone. When questions as to the response of the wave recorder are raised in addition, the accuracy of a significant height is probably only about ~ 15 percent. Moreover, visual estimates of the significant height are even more to be questioned since the procedures recommended in P i e r s o n - N e u m a n n - J a m e s are seldom followed because they are too t ime consuming.

For these reasons, a proof tha t one of the methods is definitely superior to the other in an area of wave generation With high winds will require many careful and detailed studies.

For low winds in areas of wave generation, the P i e r s o n - N e u m a n n - J a m e s method appears to predict lower waves than the S v e r d r u p - M u n k - B r e t s c h n e i d e r method. This is because the presence of a dead sea from a previously higher wind is often neglected in the data used by B r e t s e h n e i d e r , and because there are frequently swells present, often quite low, which raise what might be called the background noise level..

For forecasting waves in areas of wave generation, it is believed tha t the D a r b y s h i r e methods are definitely incorrect. Strangely enough, however, more wave records taken by the same instrument at the same places in the Atlantic where the original data were obtained, would probably verify quite well against the D a r b y s h i r e spectra. The built in dead seas, the built in climatological conditions as to typical fetches and durations, and the built in instrumental and sampling errors would simply be repeated. However, in other parts of other oceans such as the North Pacific during an unusually s tormy winter as described by E. F. D a n i e l s on, W. V. B u r r and M. l :~a t t r ay [1957], and in the roaring 40's of the Southern Hemisphere where much longer fetches and durations are possible for high winds and where a previously generated dead sea of considerable height can be rebuilt by a new storm, the D a r b y s h i r e spectra will fail to describe the waves, and we venture to predict on the basis Of the above analysis tha t eases will occur in which the observed waves would be twice as high as the forecasted waves.

Wave decay. After the waves have built up in a fetch, the winds eventually must die down and the waves must die down in the fetch area and propagate as swell into other areas. I t is in this area of wave forecasting tha t the S v e r d r u p - M u n k - B r e t s c h n e i d e r method displays its greatest conceptual weakness. Without correctly applied wave spectrum concepts, dimensionless analysis of the problem leads to incorrect results.

One can search the literature of the S v e r d r u p - M u n k - B r e t s c h n e i d e r method on the problem of how to forecast what the waves will do in the area where they were generated after the wind dies down and not find any precise statements on how to proceed.

The usual qualitative explanations for what happens depend on the concept of eddy viscosity and lead to great difficulties in deciding how long the waves will last and what their properties will be. P i e r s o n , J r . [1951] has discussed some of these difficulties in at- tempting to forecast waves by the S v e r d r u p - M u n k method on the east coast of the United States.

In contrast, filter IV of P i e r s o n - N e u m a n n - J a m e s gives precise procedures verified by examples of how to forecast the waves in this case. Very rapid decreases in wave height, compared to the meteorological t ime scale after the wind dies down suddenly, are explained solely as an effect of dispersion. The work of 'T. I j i m a , T. T a k a h a s h i and K. N a k a m u r a [1956] appears to have verified several additional filter IV cases. Figs. 3-8 and 3-9 of this work as shown here in fig. 13 are an example of the spectra which would be obtained during such conditions. Note tha t the wave period will decrease and not, as the eddy viscosity concepts of S v e r d r u p - M u n k - B r e t s c h n e i d e r would suggest, increase (see T. I j i m a [1957]).

As more computed wave spectra become available, the above considerations will provide an excellent test of spectral methods as compared with significant height and period methods.

N e u m a n n - P i e r s o n , Comparison of Theoretical Wave Spectra and Forecasting Methods 143

28 I' I

26 r t

t

t i t 22 ,

!

!

!

, i " l I , ~',.

12 ! t t

I I ' 10 I .i

!Ii ' ' i I

6 l i " ,

/ i " " - . . .

/ -

N--3--8

i -

t '

i '

t i t

i ~ 58176~ i i

i i i i

/ , I _

Co~ ffJ6 ~Og ~ 0t2 ff~

I N --3--9

Fig. 13. Sequences of wave spectra numbered in their order of occurrence (After T. I j i m a , T. T a k a h a s h i , and K. N a k a m u r a [1956]

SS~ ~

Such tests will also serve to improve the spectral methods because there ]nay be some effect of viscosity which needs to be added to the filter IV technique and its exact amount needs to be found.

Swell. The swell forecasting methods of S v e r d r u p and M u n k [1947] when compared with the swell forecasting methods using wave spectrum concepts offer opportunities to detect even greater discrepancies. Moreover, the swell forecasting methods of S v e r d r u p and M u n k [1947] and B r e t s c h n e i d e r [1952] are essentially incomplete.

The early swell forecasting methods using significant height and period concepts depend more or less on a paper by S v e r d r u p [1947] which tacit ly assumes tha t all wave periods are present in an area of wave generation, and the resulting swell forecasting chart as given, for example, by W. H. Munk and R. S. Arthur [1951] has this error built into it. The result is that the period of the swell continuously increases with increasing distance of travel.

The final forecast then supposedly gives the arrival time, the height and the period of the highest swell to reach the point of observation. Since this period as forecasted for great distances does not correspond to a frequency at which there is energy actually present in the wave spectrum, no swell will arrive at the forecasted time, and it is necessary to wait quite a few hours before the swell with a considerably shorter period will arrive.

The Bretsehneider [1952] procedure for forecasting swell is quite a departure from the original Sverdrup-Munk procedure in that the effect of the length of the fetch is in-

144 Deutsche Hydrographische Zeitschrif~. Band 10, I~Ieft 4. 1957

eluded quMitatively. I t also tends to avoid forecasting swell periods three times greater than the period of the waves in the generating area.

Both of the above methods lack a basic feature. There is no way to forecast how long the swell will last at a given point and what characteristics the swell will have, say, a day after the first high swell arrives.

Wave spectra have been studied in connection with the propagation of swell by N. F. B a r b e r and 1~. U r s e l l [1948], M u n k [1951], and P i e r s o n [1952]. B a r b e r and U r s e l l established the fact that the spectral components travel with their appropriate group velocity but they failed to formulate the problem as an initial value problem in a dispersive medium.

M u n k [1951] in studying a sequence of spectra obtained at Pendeen, England, has inter- preted these spectra in terms of a point disturbance and a rather erratically moving fetch. The winds in the fetch areas chosen by M u n k were much too low to have generated the periods shown in the spectrum, and an intense extratropical cyclone which really generated the waves has been completely ignored.

P i e r s o n, Jr. , [1952] has formulated the problem of swell forecasting as an initial value problem which considers the area extent of the waves at their source. This procedure depends critically on the spectrum at the source, but once given the spectrum at the source and the dimensions of the source the swell at any point of observation can be forecast.

There are enough differences between the two forecasting methods for great travel distances to make it easily possible to establish which is correct. Differences of a factor of two in arrival times and swell periods can easily result.

The methods given by P i e r s o n, J r . [1952] (see R o ll [1957b] for a discussion of the concepts employed) have been incorporated in the P i e r s o n - N e u m a n n - J a m e s wave forecasting method and tested for a number of storms. One is given as an example in P i e r s o n - N e u - m a n n - J a m e s . W a l d e n [1954] has compared the P i e r s o n - N e u m a n n - J a m e s method with various significant height and period methods for forecasting swell for a situation which existed on Feb. 21, 1952 and found the P i e r s o n - N e u m a n n - J a m e s method to be superior. More recently, W a l d e n [1957] has studied some swell observed off the coast of Angola in January 1955. The arrival times and the periods of the swell as forecasted by the P i e r s o n - N e u m a n n - J a m e s method, by the original S v e r d r u p - M u n k method, and by the B r e t - s c h n e i d e r method varied considerably. The P i e r s o n - N e u m a n n - J a m e s method gave a swell period near 16 seconds and a correspondingly long travel time. The original S v e r d r u p - M u n k method (Hydrogr. Off. Pub. 604) gave a swell period of more than 22 seconds, and the B r e t s c h n e i d e r method gave a swell period of 23.7 seconds with quite short travel times. The observed swell period was 16 seconds. The swell did not arrive until the t ime predicted by the P i e r s o n - N e u m a n n - J a m e s method.

At the spring meeting of the American Geophysical Union, ]957, J : E. I) i n g e r presented a series of studies showing sequences of computed swell spectra observed at Barbados. These swell spectra vary just like the spectra derived theoretically by P i e r s o n , J r . [1952].

The work of I j im a. The work of T. I j i m a [1957] is an example of the type of s tudy which is needed in developing wave forecasting methods. Pressure wave records taken at various harbors in Japan were used to compute numerous wave spectra, and the effects of hydrodynamic at tenuation due to depth were considered. Due to white noise reading error, the hydrodynamic effects appear to have been overcorrected for in some cases.

This work substantiates some of the conclusions reached herein such as the difficulty in forecasting decreases in wave height when the wind dies down by the S v e r d r u p - M u n k - B r e t s c h n e i d e r method. I t also shows numerous sequences of wave spectra which agree with the concepts in P i e r s o n - N e u m a n n - J a m e s ,

However, other sequences of wave spectra are not easily explained by the P i e r s o n - N e u m a n n - J a m e s method due to complex variations in the wind field and other factors. Also the entire problem of forecasting waves for typhoons and hurricanes is emphasized as an unsolved problem.

N e u m a n n - P i e r s o n , Comparison of Theoretical Wave Spectra and Forecasting Methods 14~5

Conclusion. The s tatus of var ious results on the de te rmina t ion of the appropr ia te family of theore t ica l wave spectra to describe wind genera ted seas and the s tatus of var ious wave forecasting methods have been reviewed. The spectra of N e u m a n n appear to describe the sea more accura te ly than other theore t ica l spectra and the wave forecast ing methods of P i e r s o n , N e u m a n n and J a m e s appear to be the most near ly correct for the widest va r i e ty of possible wave and weather s i tuat ions.

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W a l d e n , H. , 1955/56: Ein neues Diagrarnm zur Berechnung des Seeganges aus den Wind- verh~ltnissen. Annalen Meteorol. 7, 213.

W a l d e n , H. , 1956: Vorschlag zur Nnderung der Neumannschen Konstanten C bei der Berechnung der Wellenh6he aus der Wind- st/irke. Dr. t tydrogr. Z., 9~ i4.

W a l d e n , H. , 1957 : Methods of swell forecasting demonstrated with an extraordinarily high swell off the Coast of Angola. Prec. Sympo- sium on the Behavior of Ships in a Seaway. Netherlands Ship Model Basin. Wageningen.

W o o d i n g , 1%. A., 1955: An approximate joint probabili ty distribution for wave amplitude and frequency in random noise. New Zealand J . Sol. Tech. No. 36 (6), 537-544.

Eingegangen im August 1957.

Documents

A detailed comparison of theoretical wave spectra and wave forecasting methods