Sopot THE EFFECT OF INTERNAL WAVES ON THE FLUCTUA TIONS

Zygm unt K L U S E K
Polish A cad em y of S cien ces
Institute of O cean ology — Sopot
TH E E FFECT O F IN T E R N A L W A V E S O N TH E F L U C T U A ­
T IO N S O F T H E N O IS E F IE L D IN T H E S E A
Contents:
1. Introduction, 2. Basic assum ptions o f the m odel, 3. Theory, 4. Re­
sults of calculations, 5. Conclusions; Streszczenie; References.
1. INTRODUCTION
In recen t years h ydroacou stics specialists have show n particular inte­
rest in the e ffe c t o f internal w aves on the propagation o f acoustic w a ­
ves in the sea.« Internal w aves in the seas have been foun d to cause ran­
dom fluctuation s o f the acoustic field o f point sources. A ccord in g to
the hypothesis put forw a rd in [1], internal w aves m ay also cause flu c ­
tuations o f the natural noise field w hen observations are m ade b y
means of h ydroacou stic arrays of n arrow directional characteristics.
In this paper the e ffe c t o f internal w aves on the observed values
o f noise intensity at the input o f h ydroacou stic arrays is estim ated
in the approxim ation of geom etrical acoustics. This prob lem was consi­
dered on a
determ inistic
m odel
fo r
sound v e lo city
distri­
bution characteristic o f the B altic Sea during the sum m er and w inter
seasons. D espite being m u ch sim plified, the m odel enables th e exten t
o f the phenom ena to be estim ated.
2. BASIC ASSUMPTIONS OF THE MODEL
L et us assum e that sea consists o f th ree layers d iffe rrin g in sound v e ­
locity. The low er and upper layers are h om ogeneous and have a con ­
stant sound v e lo city (Ci and c2, resp ectively), w h ilst in the interm ediate
layer the v e lo city varies lin ea rly w ith the depth from Cj to c3. L et us
also assum e that the depth of the sea is infinite and the noise sources
are distributed u n ifo rm ly over its surface. The directional ch aracte­
ristics o f an elem en tary source is assum ed in the com m on ly accepted
form of
8 — O ceanologia Nr 14
G 2( f p )
-
COS2 m ( f p )
w h ere cpe is the angle m easured fro m the norm al v ector to the surfa­
ce and directed into the sea, and m — 1, 2, 3. The values of m depend
on the w in d v e lo city and observed interval of noise fre q u en cy o f the
noise. It is observed that these values increase w ith the noise freq u en cy
and w ind v elocity . The interm ediate laye:r (the so-called p y cn oclin e) p ro­
vides a sinusoidal internal w ave (the first m ode) w ith a length X and
am plitude A (Fig. 1). A cco rd in g to the foreg oin g assum ption the- sound
v e lo city in the m edium can be described as follow s:
c ix.z]
=
Cl
z < §i(x)
c2= c j i + r( z - çn(X)]
| 2l x ) >z a E^lx)
z > |2(x )
, C3
where
^
(x)
(1)
=
h1+ A cos ( 2 jt/ x
x
(j)
+
( t ))
(2 )
l 2(x) = h 2+ A, cos (2 Tf/-x x + (J) ( t i l
and the gradient o f sound v elocity , T, can assum e both negative and
positive values.
It is assum ed that the receiv er is zero level and the z axis is
directed to the surface.
z
Fig.
1. Schem atic
diagram
o f the sound
presence o f an internal w ave.
Ryis. 1. Sch em at przyjętego rozkładu
fa li w ew nętrznej.
velocity
prędkości
distribution
dźw ięku
w
in
the
m orzu
sea in
w
the
obecności
3. T H E O R Y
L et a selected acoustic ray (Fig. 1) leave the receiver situated at point
(x°k, 0, 0) at an azim uthal angle o f 0°u (the angle betw een ,the
Ox axis and p ro je ctio n o f the acoustic ra y on the yOx surface) and the
polar angle cp°u (the angle betw een the Oz axis and the acoustic ray).
The ray intersects w ith the low er bou n d ary o f the internal w a v e at
a point o f co-ordin ates determ ined b y the fo llow in g set o f equations:
P °k
= A cos (2 jr /^ x l1l + <i)(t )) + h
X(1)_ X(0)
cos a
y(1)
Z (1)
co sP
cos/
(3)
w h ere cos a, cos |3, and cos y are direction cosines o f the straight line,
expressed b y angles cp and 0
cos a
= s in f cos©
cos (J = s in 1? sin©
cosy
(A)
= c o s 1?
For the sake of sim plicity, the indices i and j are om itted here and
in the follo w in g form ulae.
The trajectories o f the acoustic ra y in the interm ediate layer can
be fou n d from the eikonal equation (3).
(5 )
n2 t ^ ' z ]
T o do this, the form u la fo r the square of the refra ctiv e index, n2 (x , y,
z), is sim plified to bring it to the additive fu n ction of co-ordin ates:
n 2( x . y , z )
=
n f l x l ♦ n|( y ) ♦ n \ [ z )
One can assum e w ith considerable a ccu ra cy that
n2( x, y, z) ;
1* r ( z - | ,( x ) )
È
~
1 - 2 r ( z - ç , ( x ))
~
The solution of the eikonal equation will then assume the farm:
y
W
=
\J n 2 (x) -k~2 dx
______
z
+ ( ^ 2 + k i) dY +
\ / n | ( z Ï - k"| dz =
^
A\
= W ^x) ♦ W2(y ) + W3(z)
where
for r > 0
(8i
n*(x) = 1 - r ç,(x)
n|(z) = Tz
The Jc2! and k22 values are determined for individual acoustic rays
from conditions for direction cosines of the ray:
n
1 ' 0W, ( X)
cos a 0 = — !---------- J—
cos ¡p
n(x,z) 0x
x = x (1-1
(9)
1
n(x,z)
3W 2 (x)
3z
z=z<1'
After transformation:
k 2 = cos2a° - r i 1(x(1))
2„O
k2 = 1 - r :
(
10)
By using formulae for direction cosines, we find the equation of
the acoustic ray in the differential form:
dx
\/c o s 2a0 - f ^(x(1))
-dy
\ / k 2 -k 2
dz
\/c o s2jf0 + f zl1)- f :
(
11)
A s assumed, the direction characteristics of natural noise sources
on the sea surface are azimuthally homogeneous, hence to calculate in­
tensity coming from the direction determined by angles 0 and y, only 1
the value of the y angle at the surface is required. This can easily be t
found from the equation:
c
y = cos (z,r) = arc cos \ / n2(z)-k^ 1
n 2( x , z )
ix (2). * ( 2 >
112)
C
w here x<2> an d z(2) are co-ordin ates o f departure o f the acoustic ray
from the ju m p layer.
B y com bin in g equations (11), depending on x and z, and integra­
tion w e obtain the expression fo r the tra je cto ry of the acoustic ray:
*
-f• ( V a ^ T T - Va- r fr(x)
where
) =
J-->s^%=
==f==r
rijix)
xW B *
'
A = cos2^0 + f z1'1
B = cos2« 0 -
r ^]
The x ® an d z<2) values w ere determ ined n u m erically from eq. (13).
The noise in ten sity in the fre q u e n cy ban d df, reaching the recei­
ver from a sm all solid angle dQ — fro m a su rface w ith u n iform ly dis­
tribu ted direction al sound sources, w h en ignoring reflection s from the
bottom and losses in the m edium , can be expressed b y (9):
dll«,,,:.!!
=
W14 ’ ^
(14)
2 l r l } G 2(& l clSp) c ! l O I \ / l - l f e c o s s PF
0
0
w h ere W is the su rface density o f the spectrum o f strength o f noise
sources, c (0) is the sound v e lo c ity at the level of the receiv er; yP
was determ ined fro m the equation. The in dex p denotes param eters
referrin g to the sea surface.
The noise intensity arriving fro m the ■so:L:id angle Q aind recorded
b y an acou stic antenna placed at the ipoiinit P(x<0), 0, 0) can b e obtain ed
b y sum m ing u p the dht values
1 = 1
^
a i j dl ij
( 15)
w here ay are co e fficie n ts describing the solid figu re o f the directio­
nal characteristics o f a given h ydroacou stic antenna in the freq u en cy
band df.
F or the sake o f sim plicity, in calculations, the directional chara­
cteristics o f the re ce iv e r w ere assum ed to p rovide a regular pyram id
com posed of 64 acoustic rays. In this case aij — 1. T he apex angle of
the pyram id w as 8°. A s it can be seen fro m eq. (15), transition to an o­
ther definite directional characteristic o f the h ydroacou stic antenna is
easy.
4. RESULTS AND CALCULATIONS
The results o f calculations ,of relative variations in the noise inten­
sity o f t h e . am bient-sea-noise at the input to hydroacoustic anten­
nas u nder an internal w ave w ere accom plished in approxim ation of g eo­
m etrical acoustics and refer to the ~ 1 to 20 — 30-kH z freq u en cy band.
T he low er 'lim it of the freq u en cy is determ ined b y the values of sound v e ­
lo city gradients assum ed in the m od el and actu ally occu rrin g in the
southern B altic [4], as w ell as characteristic dim ensions of in hom oge­
neities, in this particular case the thickness of the interm ediate layer.
The upper lim it is determ ined b y frequ en cies at w hich the intensity
o f therm al noises of the m edium is still sm all as com pared w ith those
generated b y dynam ic processes on the su rface o f the sea.
N um erical calculations w ere carried out fo r tw o types of sound
v e lo city p rofiles en cou n tered in the southern Baltic during the sum ­
m er and w in ter seasons. D uring these seasons internal w aves occur
(particu larly in ten sively in the Winter) on the boundary separating B al­
tic Sea w aters fro m those of the N orth Sea.
T he param eters assum ed to describe the internal w ave w ere as
fo llo w s: am plitude A = 10 m, length X — 1000 m and h2 — hi = 20 m.
F or the sound v e lo c ity distribution characteristic fo r the w inter
season, the values o f c2 = 1410 m /s and ci = 1450 m /s w ere assumed,
w hilst fo r the sum m er season C2 — 1480 m /s and ci = 1440 m /s . These
values w ere taken from a paper b y Tym ański [4] on soun d v e lo city in
the southern B altic. The variations in the noise intensity w ere
calcu lated fo r the fo llo w in g inclination angles o f the solid figu re
o f the directional characteristics to the level: =
5°, 10°, 15°,
20°, 25°, 45° and 90°. The acoustic axis o f the antenna was directed
tow ard the direction o f propagation o f the internal w ave. T hree values
o f the co e fficie n t m characterizing the d irectivity o f elem entary sources
on the surface w are assum ed, n am ely m = l,2 and 3, w hich corresp on d ­
ed to variou s frequ en cies and states of the sea surface (w ind velocities).
T he depth of im m ersion o f the h ydroacou stic antenna was assum ed to b e fixed .
T h e lo w v e lo city o f internal w aves enabled the prob lem to be treat­
ed statically. The w h ole picture c f the phenom enon is obtained b y
taking a series o f „p h otog ra p h s” o f an im m obile internal w ave shifted
to a variou s exten t in respect o f the fix e d receiver.
T he calculations show ed that the noise intensity increased w ith
the decrease in the angle o f observation at a given sound v e lo city pro­
file and directional characteristic of sources. This is due to greater chan­
I
ges in the direction o f individual acoustic rays em itted b y sources at
sm all slip angles.
A n oth er regu la rity observed is the increase in noise fluctuations
wiith increasing a n isotropy o f the noise fie ld (i. e. at higher m values).
E xem plary results o f calcu lation s o f the variations in noise intensity
during the propagation o f an internal w ave above the receiver, n orm a­
lized in relation to the m inim um value, are show n in Fig. 2. The abscis­
sa is scaled an tim e uniats and T demotes the ¡period of the internal w a ­
ve (at
the situation is repeated). The exam ple show n in Fig. 2 re ­
fers to the w inter stratification o f the w ater masses in the B altic w hen
the sound v e lo c ity is m inim al in the upper layer. The inclination angle
o f the acoustic axis o f the antenna is 5°.
Ji t )
Fig. 2. R elative variations o f the noise inten sity below the (internal w ave at
various directional characteristics of sources.
R ys. 2. W zględn e zm iany natężenia iszumów pod fa lą w ew n ętrzn ą iprzy różnych
charakterystykach kierunkow ych źródeł.
*
R elative variations in the noise intensity b elow the internal wave,
at a fix e d observation angle o f 8 = 15° and m = 2, as a fu n ction of
the shape of the sound v e lo city p ro file considered, is show n in Fig. 3.
It is clear from this that at the same angle o f inclination o f the axis
of the antenna, noise flu ctuation s are greater during the sum m er (cur­
ve a). This is due to the fact that at the same observation angle, in w in ­
ter acoustic rays are em itted at a larger slip angle than in sum m er, ow in g
to refraction. It is, h ow ev er w orth n otin g that absolute noise intensities
recorded from a g iv en direction are greater fo r sm all 8 angles in w inter.
J( t )
Fig. 3. R elative variations of the noise -intensity below the internal w ave fo.r
sum m er (a) and winter (b) iscund velocity profiles.
Rys. 3. W zględn e zm iany natężenia szum ów pod falą w ew nętrzną przy (a) letnim
i (b) zim ow ym profilach prędkości dźwięku.
5. CONCLUSIONS
The results o f the calculations point to the possibility o f indicating in­
ternal w aves in a w ater b o d y b y m eans o f observation of an acoustic
noise fie ld o f natural origin. The flu ctuation s of noise intensity depend
on the directional characteristics o f the antenna, therm ohaline situation
in the basin, param eters of the internal w ave and sound frequ en cy.
The flu ctua tion s of in ten sity occu r reg u larly w ith a variation period
equal to the period of the internal w ave.
Zygm u n t K L U SE K
Polska A kadem ia Nauk
Zakład Oceanologii w Sopocie
W P Ł Y W F A L W E W N Ę T R Z N Y C H N A F LU K TU A C JE P O L A SZU ­
M ÓW W MORZU
Streszczenie
Na podstaw ie m odelu m atem atycznego rozpatrzono zjaw isko flu ktu acji naturalnych
szum ów m orza spow odow anych przez fa le w ewnętrzne.
M orze przedstawiono jako ośrodek trójw arstw ow y, w którym
w arstw y w ierz-
chnia i dolna są jednorodne, z prędkościam i dźwięku c 1( C2 = oonst, w arstw a p o ­
średnia z liniow ą zależnością c = c(z) jest zabuirizona sinusoidalną falą w ew nętrzn ą
(rys. 1).
Przy założeniu o izotropow ym rozkładzie źródeł szum ów na powierzchni, w y ch o ­
dząc z równania eikonału, znaleziono w zory na obliczenie natężenia szum ów poniżej
fali w ew nętrznej (12 — 14).
Przedstawiono przykłady
obliczeń
zm ian natężenia
szum ów na
w ejściu
ukie­
runkow anej anteny akustycznej um ieszczonej pod falą w ew nętrzną — dla zim owego
i letniego profilu prędkości dźwięku charakterystycznego dla B ałtyku, przy różnym
stopniu kiarunikawości pow ierzch n iow ych źródeł s,:iumów.
REFERENCES
1. F u r d u e v A . V ., S h u m y okeana, A kustika okeana, L. M . Brekhovskikh, Editor,
M oskva 1974.
2. K l u s e k
Z.,
C h a ra k terystyk i k ieru n k ow e
pola szu m ów
w
Płd. B a łtyk u , St u­
dia i M ateriały Oceanologiczne K B M P A N , 17, 1977, p. 85.
3. M a ł e c k i J., Teoria fal i układ ów a k u styczn ych , W a rszaw a, 1964;
4. T y m a ń s k i
P., A k u styk a w ód Pld. B a łtyku , Studia i M ateriały O ceanologicz­
ne K B M P A N , 7, 1973, p. 183.
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