Sopot THE EFFECT OF INTERNAL WAVES ON THE FLUCTUA TIONS
Transkrypt
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. \