FORECASTING OCEAN WAVES 
a more accurate formulation of the transfer expression, 
no energy is added on the basis of either the Airy or 
the Stokes theory if the waves move faster than the 
wind. The validity of Stokes’ expression for mass trans- 
port in the case of waves on a rotating globe has also 
been questioned.’ 
If the waves are conservative, that is if they maintain 
their identity, the knowledge from theory of the way in 
1085 
the wind is divided in a certain fixed manner between 
the energies required to increase wave height and wave 
period. The solution of the resulting system of differ- 
ential equations leads to a relationship between wave 
steepness and wave age, 
6 = f(6) 
and, finally, to the relationships (1)—(4) above. 
(13) 
te 200 400 600 800 1!1000 1200 1400 1600 |800 2000 2200 2400 2600 2800 3006 
UNS Ss = == : rs 
aly ~ { By eee i Ly 2 
| ~ lo” ik i a = Sis . _ ~ | ine ~ 7a = at a TES: 
IGS i FS 20" |= = Pm tt Ss 5 ~ = = 16 
z < S KL SW S <A Tsk 
~ fk Al 30 Ss ~ ~ +~ 
alt = ae =) S ~ | nS aes 23 
O14 z foe a Jas ae on te 14 
za ~ : ~ = ~ ~ 
je) Se ~ Sje t ~ ~ 50 | 7 iracal fe eI cans p22 
(S) ~j08)/°>~ > > IN re = ss : ~ 
aie Or at 7 at lan oF A = 12 
Gy = iP SN ~ W 60" 3 = DiS = 
le 0.6. = ~S 4 iS, aA ; i 
2 ~ ~ ; SW = C Wy! 
= (258 (95° S| Jb asa > Nal il aes = 3 
Ww By x Pale 4 ~N ; i 3 “oS 
= iy = s 4 BHR LZ Be 
© I~ x > 0.3 N aN LE E goo | ~ 20 
= > <x S i LA a = 
S) 8 Si ~S iN Wa N = <= 8 
= v ai S N Se IZANS Ss [ad + 106" 
ae Y/\~< 2S XN TAS 30° we 
mG 7 A: s A Sale S S hs R 
uw K / Noi S - 1a © 
2 N N & N x N a3 18 
16° | | 
S » sil JS N aes 15° 
4 {| sep oes ee __ — __ LINES OF EQUAL WAVE 7 4 
\ acl ie | PERIOD, Tp ,IN SECONDS 
ne in = a Seay = T 
5 | 2) [WAVE VELOCITY AND LENGTH FOR DIFFERENT PERIODS] —— —— LINES OF EQUAL TRAVEL 
2 ETS G(KNOTS) 0 3 6 9 12 15 18 21 24 27 303336394245 4851 545760 TIME, tp, IN HOURS 2 
A 55 J T(SECONDS) 0 1 23 4567 8 9 10 Il 12 13 1415 16 17 18 19 20 LINES OF EQUAL Hp/He 
| L (FEET) 50 100 200 400 600 1000 1400 2000 ; 
) 0 
200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 
DECAY DISTANCE,D 
Fie. 3.—Wave-decay relationships in practical form for use in forecasting. 
which energy advances with the wave form leads to the 
following differential equations: 
Steady state: 
CdH E dc 
amibinaids Mla ae Wy 
Transient state: 
dH E dC 
pation: beeen (12) 
where the positive sign is used for C S U and the nega- 
tive sign for C > U. Actually the waves comprising 
sea cannot be conservative and the equations apply at 
best only approximately. 
Since the wave energy is related to the wave height, 
the two unknowns in the equations above may be con- 
sidered as wave height and wave velocity and the 
knowns as distance and time. An additional equation is 
obtained by assuming that the energy transferred from 
3. F. Ursell, ‘“The Theoretical Form of Ocean Swell on a 
Rotating Earth.”’ Admiralty Research Laboratory, A.R.L./ 
R.6/1.03.41/W, February 1947 (unpublished). 
, IN NAUTICAL MILES 
The relationship between 6 and £ is particularly use- 
ful in comparison with wave observations. Many of the 
observations available for the first checking of the fore- 
casting relationships were inadequate in that either 
duration time or fetch or both were missing. Most ob- 
servations are sufficiently complete for the computation 
of wave steepness and age. Certain constant parameters 
appearing in the equation are evaluated by fitting the 
analytical expression (13) to the available observations. 
The resulting relationship (solid line curve, Fig. 4) 
gives a satisfactory fit. 
Waves leaving the fetch and advancing as swell 
through a region of calms lose energy by the normal 
pressure effect of air resistance, but no effect of tangen- 
tial stress need be considered. The solution of the equa- 
tion expressing the energy budget of swell yields an 
increase of wave period during decay and a decrease of 
wave height, and these facts are in agreement with 
observations. 
Sverdrup [21] compares the decay of swell to the ad- 
vancement under the influence of air resistance of a 
train of impulsively generated waves and shows that 
the period increase is explicable as a result of selective 
