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Results from an instrumental study of the attenuation of ocean waves with depth
454
Ba(;ro~
N.A.
C h a n g e s o f w a v e travel direction due to the influence o f wind
\
..\.
H.',a,ro,.
( cntral I-orecasting Institute.
~l, c a , , d o t , tvO I~1¢~."~ .I ( 5 ) ' ~2 t) 8t2
IF "rite wave tr;.t,,¢l direction doer, not COillCltJC ~.~,itll the direi.t|on of the vector of ~,illd ',titbit." it is natural to suppose that sofne force hlts arisen which. ~,,here the wind is ctmstant in direc:i,m finally " leada " the direction of v,,~t: tr,t'.el il~lO tt~e direction of `,~ind `,clocit5 l'his prot.c~s .,,~ first studied by Kashin and U~hako~a ]1 ;. I-ol this purpose they used data flolll v,eather ,rap.. in the Atlantic. They found that the angle betv, ccn the direction o f w a v e travel and :he `,`,ind dire-: i>: where the wind `,~as corISlal'~t, decreases aco.m.ling tt) the e,~ponenlial la,a, '1'
'I>,, U
L:
where the parameter A depends ,an the initial ,,alue of the angle 4,. This expression rettects ,~;,,:. the general tendenc.,, It is possible to obtain much better empirical data by l.:lking the formL0,,
where the cons.tallt A Ltlso depcm.b, on the initJ~tl angle, ttowe~cr it ts dltlicult to pro`,e this l,,'JiH:tit,: Here we ',a,ill It.,, to obtail~ the laxs o f ,.va,,e travel dire,'tion change due to the influence of the x. i~t.~ with various as,,ttll~ptiol~-, o f a :,oral-empirical natt.Lr¢. [.el /-. stand fl)r the full energy of vca,,e movenmnt. We consider that under the Jnl:luen,...¢ :,I .. wind of xelocit,. I the totai energy,, o f `,va,,e ino`,enJent, l:. during the time. At '.',ill increase 3./:
/.., 1" .\t
The energy, L. and its increase, AE are scalar quantities. However the) are formed by tile t~,:~.. ment of water particles t,pon vertical planes orientated in a definite way. If the v, ind direction ,!t~., not coincide v.ith the direction of wave Ira',el, then the ~,elocities of water particle ino,,c~,c;~: conditioning the increase of energy, AE. will also fail to coincide with the general direction of ,,a..,' travel. This ,,ariance finally causes the change in wave travel direction to that of the v, md. Apparently the energy increase is due to the movement of the water constituents in a directu ;, perpendicular to the basic wave travel and will be : ( &/:-b,
k: ( l/sin ~b)" ~ t
Without entering at the m o m e n t into the details of the mechanism o f the wave travel chang,: ,,~ direction, we will simply assume that the increase o f the deflection angle will be proportional t ~ i h,: volume o f energy, i.e. .3@ . k.l."-'sina (bAt. There should always be a minus sign in the right hand side o f this formula, since owing to the deflection the angle ,~ always decreases. In this formula we take to the extreme case. and obtain the main equation in the form: d,~ - - •.....
K sin a 4),
dt
where K symbolizes (to avoid introducing new symbols) k2 V2 o f the preceding formula. E q u a t | o , (3) describes only the process o f wave direction change and does not enter into the process o f their change o f energy, since their overall energy does not change, or alters in such a way that it is .,~.o, • We shall suppose that the vector of wind velocity coincides with the direction of ~,ind. I~ order to simplify the terminology o f this paper, we will refer the same meaning to the "' direction o f wind.'"
Changes of wave travel direction due to the influence o f wind
455
reflected in deflection angle 4. This would be one o f the extreme points o f view. The other point of view may be formulated thus : on the sea surface there are often observed different systems of waves, which are especially distinct, when they are of different lengths. These observations may suggest that if the wind suddenly changes direction, two systems of waves will be observed at sea, the older of which will decline by dissipation of energy, while a new one will develop corresponding to the new direction of the wind. When, and only when, both these processes reach a determinate stage in their development (i.e. when the old wave system has declined almost to zero), is it possible to speak of the formation of a new direction of wave travel. This would be the other extreme point of view. But do these two points of view actually differ so much ? In the first place, the wind almost never changes direction suddenly, and in the second, the waves are certainly capable of changing their direction of travel, almost without altering their energy. This occurs when the waves move into shallow water, in rounding obstacles and in refraction. From these considerations it seems wholly natural to assume, that when a change of wind direction takes place, some part of the energy of wave movement in the old direction is diverted to the formation o f waves in the new direction. This gradual process is in fact described by equation (3). We will consider coefficient K in formula (3) as the quantity of the constant. This is entirely admissible especially if the somewhat hypothetical character of the equation itself is taken into account. The integral of this equation apparently ~ill be : c tan 4 - " Kt -r constant or after the constant has been determined : tan 4 =
tan 40
.
(4)
1 bKt'tan4o From Kashin and Ushakova's data for a wind o f constant direction (independently of its speed) was obtained K = 0-3331/hr For this value we obtain the following quantities for the angles o f wave travel direction where the direction of the wind is constant (Tab& I).
Table 1. Change of angle (deg) between the wind direction and the direction of wave travel for the three groups with differing initial angles Time (hr)
Range of initial angle I0~-20 ° Computation 20o--40 °
Computation 40o.-60° Computation
No. of cases 12 14 14 33 33 54 54
!
8 9"5 17 15"8 24 20"7
6 7'2 10 10-2 12 12'0
18
5 5'6 8 7.5
Average wind velocity (m/see)
329
13
70
14
13
14
8-5
In the first line for each of the three groups showing the initial value of angle 4 are given by Kashin and Ushakova's data; the second line gives the results of calculation on the basis o f formula (4). It can be seen that for the first group (the most numerous and therefore the most reliable) of the variance observations do not exceed 1-5. Apparently, this may be considered wholly admissible, considering the great inexactitude o f the initial data (angles were determined to within I0 °) and the possible inaccuracy o f the theory. The greatest divergence is to be observed in the third group (the least reliable), but it is only slightly in excess of 3°, which, apparently, ought to be considered entirely admissible.
456
N.A.
BAGROV
In t h e case o f a varying wind we m u s t replace e q u a t i o n (3) by the following equation d4,__.
• K sin ~ 4, .i ~
dt
where a is the coefficient derived from a z i m u t h A o f the wind velocity vector M
a
dd
!dr
" l
T h e sign before d i m e n s i o n a is c h o s e n in accordance with the c h a n g e o f wind direction in relatlo~ to t h e wave travel direction. If the w i n d ' s c h a n g e o f direction t e n d s to coincide with the direcuo~ o f the wave travel, t h e n the m i n u s sign s h o u l d be used, since angle 4, decreases o n a c c o u n t o f thi, change, while in the opposite case the plus sign s h o u l d be used. In general, w h e n the wind velocity c h a n g e s arbitrarily in direction, a n d coefficient K depend:~ o n t h e time (which, evidently c o r r e s p o n d s to the c h a n g e o f velocity in m a g n i t u d e ) e q u a t i o n ~5~ i~ n o t expressed in quadratics. This study is confined to the simplest cases. If the wind speed c h a n g e s only in m a g n i t u d e , t h e n in e q u a t i o n (5) a =: 0, a n d coefficient K ~ ,J~ be s o m e function o f the time. In this case it could be expressed in the form : tan ~o ...... T ....
tan 4' :-
I -- tan 4,o J" K d t This case in no way differs in principle f r o m t h a t considered earlier. If the wind w i t h o u t c h a n g i n g in m a g n i t u d e , c h a n g e s only in direction, provided this c h a n g e takes place with a c o n s t a n t a n g u l a r velocity, there will be o b t a i n e d the instance, where a takes the place o f various symbols, •
a ~---K-si~-4, "- t -I- c o n s t a n t
We obtain for a < K a n d 4', lying between - 90 ° a n d + 90 ~ 1 m t a n ~ -- ~ - - In - -2m m t a n 4, +
m=
t .+- c o n s t a n t
V'~-~a
(~ < K)
If s y m b o l Q is introduced : Q
_
m tan
~ -
=
e_lmt,
reran4 + = then t h e final expression m a y be represented as follows : ~I+Q t a n 4, . . . . ml --Q
c 10:
W h e r e t -~ oo, for a n y ,value o f the initial a n g l e ~0,1 ~0] < ~/2, th© value t a n ~ t e n d s to t h e ratio D e p e n d i n g o n t h e ratio K/a, t h e value o f t h e a n g l e at t -~ oo c h a n g e s f r o m 0 (where a is very s m a l l - - a very slow c h a n g e o f wind direction) to a v a l e n e a r ~/2 (where K/a a p p r o a c h e s 1). However in this case t h e value m will be vcry small, a n d t h e value Q will t e n d to n o u g h t very slowly. W h e r e a < K with a plus sign in e q u a t i o n (5) i n s t e a d o f (7) there will be obtained : a/ra.
rn
where ra z :
a 2 -- aK
or
tan
4, =
-~ tan (rot + 00),
( ~2~
Changes of wave travel direction due to the influence of wind
457
while tan 00 = m tan 4'0, ot
where 4'0 as before means the initial value of angle 4'. Equation (12) preserves its actual significance only where the angles are I~i < ,r/2. If in the equation (5) value a is preceded by a minus sign, then its integral is expressed by a formula similar to the preceeding : tan 4, = ~ tan (0o - m t ) ,
113)
m
while m 2 _ ~2 + ~ K ,
m
tan 00 = - - t a n ~ 0 .
(14)
~t
In this case the wind turns to the direction of wave travel. Equation (13) preserves its actual significance until 4, = 0. From that time further change of wave travel direction should be calculated either according to formula (10) or (12). It is apparently senseless to study more complex examples of the change of vector of wind velocity. At least at present there is no necessity to do so and the integrals of equation (5) are not expressed in quadratics. REFERENCE
[11 K~HIN, K. I. and L. L. USHAKOVA(1962) Change o f wind travel direction to that o f the wind. M e t . a n d H y d r o l . , (7).
Ba(;ro~
N.A.
C h a n g e s o f w a v e travel direction due to the influence o f wind
\
..\.
H.',a,ro,.
( cntral I-orecasting Institute.
~l, c a , , d o t , tvO I~1¢~."~ .I ( 5 ) ' ~2 t) 8t2
IF "rite wave tr;.t,,¢l direction doer, not COillCltJC ~.~,itll the direi.t|on of the vector of ~,illd ',titbit." it is natural to suppose that sofne force hlts arisen which. ~,,here the wind is ctmstant in direc:i,m finally " leada " the direction of v,,~t: tr,t'.el il~lO tt~e direction of `,~ind `,clocit5 l'his prot.c~s .,,~ first studied by Kashin and U~hako~a ]1 ;. I-ol this purpose they used data flolll v,eather ,rap.. in the Atlantic. They found that the angle betv, ccn the direction o f w a v e travel and :he `,`,ind dire-: i>: where the wind `,~as corISlal'~t, decreases aco.m.ling tt) the e,~ponenlial la,a, '1'
'I>,, U
L:
where the parameter A depends ,an the initial ,,alue of the angle 4,. This expression rettects ,~;,,:. the general tendenc.,, It is possible to obtain much better empirical data by l.:lking the formL0,,
where the cons.tallt A Ltlso depcm.b, on the initJ~tl angle, ttowe~cr it ts dltlicult to pro`,e this l,,'JiH:tit,: Here we ',a,ill It.,, to obtail~ the laxs o f ,.va,,e travel dire,'tion change due to the influence of the x. i~t.~ with various as,,ttll~ptiol~-, o f a :,oral-empirical natt.Lr¢. [.el /-. stand fl)r the full energy of vca,,e movenmnt. We consider that under the Jnl:luen,...¢ :,I .. wind of xelocit,. I the totai energy,, o f `,va,,e ino`,enJent, l:. during the time. At '.',ill increase 3./:
/.., 1" .\t
The energy, L. and its increase, AE are scalar quantities. However the) are formed by tile t~,:~.. ment of water particles t,pon vertical planes orientated in a definite way. If the v, ind direction ,!t~., not coincide v.ith the direction of wave Ira',el, then the ~,elocities of water particle ino,,c~,c;~: conditioning the increase of energy, AE. will also fail to coincide with the general direction of ,,a..,' travel. This ,,ariance finally causes the change in wave travel direction to that of the v, md. Apparently the energy increase is due to the movement of the water constituents in a directu ;, perpendicular to the basic wave travel and will be : ( &/:-b,
k: ( l/sin ~b)" ~ t
Without entering at the m o m e n t into the details of the mechanism o f the wave travel chang,: ,,~ direction, we will simply assume that the increase o f the deflection angle will be proportional t ~ i h,: volume o f energy, i.e. .3@ . k.l."-'sina (bAt. There should always be a minus sign in the right hand side o f this formula, since owing to the deflection the angle ,~ always decreases. In this formula we take to the extreme case. and obtain the main equation in the form: d,~ - - •.....
K sin a 4),
dt
where K symbolizes (to avoid introducing new symbols) k2 V2 o f the preceding formula. E q u a t | o , (3) describes only the process o f wave direction change and does not enter into the process o f their change o f energy, since their overall energy does not change, or alters in such a way that it is .,~.o, • We shall suppose that the vector of wind velocity coincides with the direction of ~,ind. I~ order to simplify the terminology o f this paper, we will refer the same meaning to the "' direction o f wind.'"
Changes of wave travel direction due to the influence o f wind
455
reflected in deflection angle 4. This would be one o f the extreme points o f view. The other point of view may be formulated thus : on the sea surface there are often observed different systems of waves, which are especially distinct, when they are of different lengths. These observations may suggest that if the wind suddenly changes direction, two systems of waves will be observed at sea, the older of which will decline by dissipation of energy, while a new one will develop corresponding to the new direction of the wind. When, and only when, both these processes reach a determinate stage in their development (i.e. when the old wave system has declined almost to zero), is it possible to speak of the formation of a new direction of wave travel. This would be the other extreme point of view. But do these two points of view actually differ so much ? In the first place, the wind almost never changes direction suddenly, and in the second, the waves are certainly capable of changing their direction of travel, almost without altering their energy. This occurs when the waves move into shallow water, in rounding obstacles and in refraction. From these considerations it seems wholly natural to assume, that when a change of wind direction takes place, some part of the energy of wave movement in the old direction is diverted to the formation o f waves in the new direction. This gradual process is in fact described by equation (3). We will consider coefficient K in formula (3) as the quantity of the constant. This is entirely admissible especially if the somewhat hypothetical character of the equation itself is taken into account. The integral of this equation apparently ~ill be : c tan 4 - " Kt -r constant or after the constant has been determined : tan 4 =
tan 40
.
(4)
1 bKt'tan4o From Kashin and Ushakova's data for a wind o f constant direction (independently of its speed) was obtained K = 0-3331/hr For this value we obtain the following quantities for the angles o f wave travel direction where the direction of the wind is constant (Tab& I).
Table 1. Change of angle (deg) between the wind direction and the direction of wave travel for the three groups with differing initial angles Time (hr)
Range of initial angle I0~-20 ° Computation 20o--40 °
Computation 40o.-60° Computation
No. of cases 12 14 14 33 33 54 54
!
8 9"5 17 15"8 24 20"7
6 7'2 10 10-2 12 12'0
18
5 5'6 8 7.5
Average wind velocity (m/see)
329
13
70
14
13
14
8-5
In the first line for each of the three groups showing the initial value of angle 4 are given by Kashin and Ushakova's data; the second line gives the results of calculation on the basis o f formula (4). It can be seen that for the first group (the most numerous and therefore the most reliable) of the variance observations do not exceed 1-5. Apparently, this may be considered wholly admissible, considering the great inexactitude o f the initial data (angles were determined to within I0 °) and the possible inaccuracy o f the theory. The greatest divergence is to be observed in the third group (the least reliable), but it is only slightly in excess of 3°, which, apparently, ought to be considered entirely admissible.
456
N.A.
BAGROV
In t h e case o f a varying wind we m u s t replace e q u a t i o n (3) by the following equation d4,__.
• K sin ~ 4, .i ~
dt
where a is the coefficient derived from a z i m u t h A o f the wind velocity vector M
a
dd
!dr
" l
T h e sign before d i m e n s i o n a is c h o s e n in accordance with the c h a n g e o f wind direction in relatlo~ to t h e wave travel direction. If the w i n d ' s c h a n g e o f direction t e n d s to coincide with the direcuo~ o f the wave travel, t h e n the m i n u s sign s h o u l d be used, since angle 4, decreases o n a c c o u n t o f thi, change, while in the opposite case the plus sign s h o u l d be used. In general, w h e n the wind velocity c h a n g e s arbitrarily in direction, a n d coefficient K depend:~ o n t h e time (which, evidently c o r r e s p o n d s to the c h a n g e o f velocity in m a g n i t u d e ) e q u a t i o n ~5~ i~ n o t expressed in quadratics. This study is confined to the simplest cases. If the wind speed c h a n g e s only in m a g n i t u d e , t h e n in e q u a t i o n (5) a =: 0, a n d coefficient K ~ ,J~ be s o m e function o f the time. In this case it could be expressed in the form : tan ~o ...... T ....
tan 4' :-
I -- tan 4,o J" K d t This case in no way differs in principle f r o m t h a t considered earlier. If the wind w i t h o u t c h a n g i n g in m a g n i t u d e , c h a n g e s only in direction, provided this c h a n g e takes place with a c o n s t a n t a n g u l a r velocity, there will be o b t a i n e d the instance, where a takes the place o f various symbols, •
a ~---K-si~-4, "- t -I- c o n s t a n t
We obtain for a < K a n d 4', lying between - 90 ° a n d + 90 ~ 1 m t a n ~ -- ~ - - In - -2m m t a n 4, +
m=
t .+- c o n s t a n t
V'~-~a
(~ < K)
If s y m b o l Q is introduced : Q
_
m tan
~ -
=
e_lmt,
reran4 + = then t h e final expression m a y be represented as follows : ~I+Q t a n 4, . . . . ml --Q
c 10:
W h e r e t -~ oo, for a n y ,value o f the initial a n g l e ~0,1 ~0] < ~/2, th© value t a n ~ t e n d s to t h e ratio D e p e n d i n g o n t h e ratio K/a, t h e value o f t h e a n g l e at t -~ oo c h a n g e s f r o m 0 (where a is very s m a l l - - a very slow c h a n g e o f wind direction) to a v a l e n e a r ~/2 (where K/a a p p r o a c h e s 1). However in this case t h e value m will be vcry small, a n d t h e value Q will t e n d to n o u g h t very slowly. W h e r e a < K with a plus sign in e q u a t i o n (5) i n s t e a d o f (7) there will be obtained : a/ra.
rn
where ra z :
a 2 -- aK
or
tan
4, =
-~ tan (rot + 00),
( ~2~
Changes of wave travel direction due to the influence of wind
457
while tan 00 = m tan 4'0, ot
where 4'0 as before means the initial value of angle 4'. Equation (12) preserves its actual significance only where the angles are I~i < ,r/2. If in the equation (5) value a is preceded by a minus sign, then its integral is expressed by a formula similar to the preceeding : tan 4, = ~ tan (0o - m t ) ,
113)
m
while m 2 _ ~2 + ~ K ,
m
tan 00 = - - t a n ~ 0 .
(14)
~t
In this case the wind turns to the direction of wave travel. Equation (13) preserves its actual significance until 4, = 0. From that time further change of wave travel direction should be calculated either according to formula (10) or (12). It is apparently senseless to study more complex examples of the change of vector of wind velocity. At least at present there is no necessity to do so and the integrals of equation (5) are not expressed in quadratics. REFERENCE
[11 K~HIN, K. I. and L. L. USHAKOVA(1962) Change o f wind travel direction to that o f the wind. M e t . a n d H y d r o l . , (7).