Nakamura  Kohei

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Nakamura, Kohei

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Velocity of Long Gravity Waves in the Ocean著者 Nakamura  Kohei雑誌名 Science reports of the Tohoku University. Ser.5, Geophysics巻 13号 3ページ 164-173発行年 1961-12URL http://hdl.handle.net/10097/44634 V  elocity of Long Gravity  W  ayes in the OceanGeophysical   By  KOHEI NAKAMURA Institute, Faculty of Science,  TOhoku     (Received Oct. 20, 1961)University Abstract            Effects on the velocities of oceanic waves of compressibility of the sea water,      elasticity of materials beneath the sea bed, the Corioli's force and curvature of theearth      are examined particularly in case of long period waves as tsunamis. Computationof the      dispersion curves for the phase and group velocities reveals that the usual velocitiesof      oceanic waves whose wave lengths fall within the range 100 to 10000 km are reducedat     most 1 percent owing to the above-mentioned effects. I Introduction    It is well known that the observed travel time of the initial disturbance of a tsunami is in most cases less than the one computed on the assumption that the tsunami is originated at the earthquake epicentre, and that the travel time is given by the       d formula jshwhere g is the acceleration due to gravity and h is the depth of the sea,          -1/and the integration is  be carried along the orthogonal  trajectory drawn in the re-fraction diagram. The discrepancy between the observed and computed travel times has been attributed to vast lateral extent of the deformed sea bed. The source area thus determined of the Sanriku tsunami of 1933 was in linear dimensions as large as about 600 km (N. MIYABE (1934)). Unless crustal deformations occurred in advance of the main shock, this figure seems to be incredibly large in spite of a large magnitude 8.0 of the shock, because there was no foreshock which could explain the earlier arrival of the tsunami.    Should the group velocity of long waves exceed Vet, the source area determined from the travel time differences could be diminished to some extent. It is not a little interesting to ascertain whether the group velocity increases or does not due to un-considered effects on the progressive waves.    The factors which influence the velocity of oceanic waves with very long periods may be the curvature and rotation of the earth, the compressibility of the sea water, elasticity of the bottom materials, eddy viscosity of the sea water and the oceanic currents.    H. NAGAOKA (1909) showed that the effect of the earth's curvature is to decrease the phase velocity. T. TERADA (1921) examined the influence of elasticity of the earth's crust, and suggested that the group velocity of oceanic waves may exceed by about 10 percent the standard value computed on the assumption of a rigid sea bed.    In this paper the effect of elastic bottom and compressible water will be examined on a more reasonable model than  TERADA'S. Also modifications due to the rotation andVELOCITY OF LONG GRAVITY WAVES IN THE OCEAN 165curvature of the earth, and eddy viscosity are investigated respectively. Then combined effects of these factors on the  phase and group velocities will  be exhibited in the form of dispersion curves. 2 Modification of Phase and Group Velocities due to Compressibility of Sea   Water and Elasticity of Bottom Materials    We suppose that a plane sheet of water of infinite extent is underlain by a half space of an elastic solid. For simplicity's ake, the gravity terms in the equations for the elastic solid will be ignored. The velocity equation for the waves propagated in this system have been studied by R. STONLEY (1926), J.G. SCHOLTE (1943), and T. MATSUZAWA (1950). STONLEY considered the effect of the ocean on Rayleigh waves, SCHOLTE treated a problem of generation of microseisms, and MATSUZAWA'S interest was in the modification of the velocity of tsunami.    The velocity equation contains two differenttypes of dispersion, one is concerned with suboceanic Rayleigh waves, and the other is related to gravita-tional waves in water. H.  MIYOSHI (1954) treated-a problem of generation of a tsunami in compressible sea water, and showed that the phase velocity of gravity waves decreases by about 1 percent on ac-count of the compressibility of the water.    The coordinate system is shownin Fig. 1. The depth of the sea is h, the densities of water and solid are denoted by  pi and  p,, respectively, and X and  it designate the Lame's constants. The velocity of of dilatational and distorsional waves in the solid are  vs. The time factor  eico is assumed in this section.    Only plane waves are considered, and in the wat ed shallow water theory is used.    The motion of a water particle is governed 1 potential  g),                    o2p                                     aopi •22                               VI V (pi+ g—        o12a z The solution of (1) can be writtenwhere   The pressure and pectively byP2, A, /4\ Sat6d \up, US\Fig. 1. Water layer over a half        space of the elastic solid. "he velocity of sound in water and the ve]ocities  e lid re denoted respectively by  v1;  v  p and  is ction. d e ater part of the system, the linealiz-by the equation 0  . q, =  &e (A sinh  IC  z  +B cosh  fc  z)  ei  x  , Y=g/2vi,        Ic2 E  2 +Y2  .the velocity of a particle along the z-for the velocitydirection(1)(2)   (3)    (4) are givenres-166 K.  NAKAMURA p149i and  —  aa(e (5)    In the solid, the horizontal  u and vertical w displacements are written in terms of the potentials  (p and  lir,           aq, w9, +                                            (6) ax az azax The functions  (p and  'k satisfy the wave equations                    1  a2 p  V2  V2  3  t2  v21,1.a2 1P' (7)                                            at. The solutions of (7) can be written                p P eaz_itx  = Q  ei3z-  itx (8) where             a2E2 h22E2k2hcolvp, k =to/vs , (9) and Re  (a,  13)  >  0  . Substituting (8) into (6), we have                        u =— Pez+13 Q  e-itx,  w= (a  Pez—i  E Q  efiz) e-qx  .The stress components are       — IL F2 a2-P-- + a2* a2* 13'2 XCr2 93+2 [         L axaz ax. az.(10)                                                         a2                                                   +   P„[112 _9S,_+apxv2(p+2tt         2az.- az. axaz  _                                            (11) We assume hereafter that  X-   The boundary conditions are that the pressure at the sea surface is constant, and at the sea bed, the vertical velocity is continuous, the horizontal stress is zero, and the vertical stress is continuous. These conditions can be written as follows,                aa2.°, aapi =0  at  z 0  ,  aw   at  z  =—h.,               azat  11,,— 0 at  z  =  —h.,  P.,z+  661+  g  (p2—pr) w = 0. at  z Substituting (2), (5), (10) and (11) in (12), the boundary conditions     A  [g  Ic]—B  [(02—  g y]  =0     A  [K  cosh  K  h—y sinhK  hi  +B  [y cosh  K  h---  K  sinh  K  12]  P  [i  co  a  eh )]  +Q  [—co  Eeh  (Y-P)] =  0  .  P  a  e--11+Q  [—i  (2  E2—k9  e-gh]  =  0become(12)(13)VELOCITY OF LONG GRAVITY WAVES IN THE OCEAN 167    A  [—i  w  pie-74 sinh K  h]  +B  [i  co  pi  e--Th cosh K +P (2  E2  -  k2)          +  g  (p2-101))-  e--01+Q  [—i  {2  E  g  (p2—pr)  e--01  0  .Eliminating A, B, P and Q from the simultaneous equations (13), we obta racteristic equation •  [p, {4  E2 a  13—  (2E2—k2)2}+ g  (P2—P1) a  k2] [y {g K cosh  IC h  g y) sinh  K  +  K  {(co2—g y)  cos h  rc  h—g K sinh  ±CO2  k2  a[g  K  cosh  Kh—(CO2—  g  y)  sinh  K  11]  =  0  .This can be rewritten in a convenient from [(2  —   vc;   )2—  4  17  1  —  tanh  h F  ,,1 12  2 P C  V C2  V2 1 —      14 ) 1(2 — 2C121 )2 -  V4  1  (  p,  v;712^192c2 1  Vt  )  V :12,]  II  1 c2  vf  ‘v1  Ai=  ,btain the  cha-where 'eh  c2v[v2(21)2    1  C2  101   P2  1   c2  C2  1.4 4  -1/1  —1 1 C2 C22  2 pC 7)(14)       —1772v2)                        C2)(2V2 ,                   1  c=  ,  77=-Eh=27th/X,. (15) In (15), c and X represent he phase velocity and the wave length, respectively.    In computing the dispersion curves two models are  used;    Model  A  :  h= 4.5 km,  p,— 3.47,  pi  = 1,  vs  = 4.60 km/sec,  v1  = 1.5 km/sec,    Model  B  :  h= 4.8 km,  p2  = 5.52,  p,— 1,  v8  = 6.00 km/sec,  v1 = 1.5 km/sec.    The model A represents he approximate natures of a portion of the upper few hun-dred kilometers of the mantle, and the model B represents the averaged natures of the1  00  r-  -     C/./gX  0.95   1 0  90r  0  851 080--  IL--/     4.5 le  A/5 kil7  P  3.47  a  4  60km  /.4pcA __J1  ; /02 4.8  km  5  km  ',/tr  552 6  00  fe111/A&' /03 (kin)  1•Ifl  Af!  !  1   'Fig. 2.         /0"  /05        Dispersion curve of phase velocity of gravity waves in com-       pressible water overlain by the elastic solid substratum.168 K. NAKAMURAelastic part of the earth. In usual tsunamis which have small wave heights, the elastic yielding may not be transmitted far beneath the sea bed, so that the model A seems to be more realistic than the model B. Fig. 2 shows the dispersion curves for the phase velocity c, the abscissa indicating the wave length in kilometers.    The group velocity U is computed by the formula  U=c—A,  dcldX, through nume-rical differentiation of the dispersion curve for the phase velocity, and is shown  in Fig. 3. In Figs. 2 and 3, A and B designate the two models assumed. We see that the maxima of both the phase and group velocities decrease at most by 1 percent. /00 095    T  090    085  0751 --  0701h  —065 ---  Fig, 3.-JIitz:v  ,  _  A=  45  U1= 1.5km/.14.  P-347  US=-4  60  lefiztax       7737. A t /02 A=  48lin  15t  ?,/,&'  =  552  Us--  6001Vdec  /03  2  /   /04  105 Dispersion curve of group velocity of gravity waves in com-pressible water overlain by the elastic solid substratum.3 Effect of Corioli Force    For long waves the effect of the rotation of the earth can be given by the equation (Ref. LAMB  (1932)).  v2  .'•_i_  62-4  co2  sin2  cp  =  0                                            (16)g  h where is the elevation, a- is the circular frequency,  q) is the latitude, and  co is the angular velocity of the rotation of the earth. The phase velocity c is expressed by the relation               Chc_                            ---— —1 — 46,2sin2q)X2   11/2  (17)                  Vgh- 7-t2  g hand the group velocity becomes                 U c  (02  sin2  co  X2   (18)  -  V  gh  v"  g  h  7,2  g  h  (c  Iv  gh)            VELOCITY OF LONG GRAVITY WAVES    The dispersion curves for different latitudes of the shown in Figs. 4 and 5, respectively, assuming the These figures show that the phase velocity increases, creases with the wave length. 125  --/20 //5 110 105c/iyA i 1 9' 9; 5' --÷  A(12/77)9–  60°  30°_/ 002  /03  /04     Fig. 4 Effect of Corioli's force on the             phase velocity of oceanic waves. co designates latitude, h=4.5km. 4  Effect of Curvature of the Earth    When the waves follow the zonal  ha] extremely long, the phase velocity of oc (1909), LAMB (1932))                   --1         V(2 where n= 2  7rafX, a and p are the radius    The group velocity can be written  U c  g  h  g  h/00095 095 085— 080— 075 —   /02 Fig. 5IN THE OCEAN 169 phase and group velocities are depth of the sea to be  4.5  km. whereas the group velocity de-= 30n --- 4 59-0-Sta.=90A (kin)1111A 1  1  /03  /04 Effect of Corioli's force on the group velocity of oceanic waves.  go designates latitude, h=4.5 km.1 rmonic distribution and the wave length is not  eanic waves can be written by (cf. NAGAOKAc -  3  -11/2  -—tot _1 (2  n  d-  1)  p] and the mean density of the earth, 3n  (2  n+  1)2 1   C/Vg  h 010   '  ,  ,  I,  1  Ii !  i T  -  —1  '  7  -ri  1--,,i '1:9X --  0061-1 E1  1  004  -   002,-  __----------)-- -002-  -004  -  1  I  '111,J  o3  hr/em)  /04 Fig. 6 Effect of cruvature of the earth on the phase        and group velocities of oceanic waves.  (19) respectively.(20)170 K. NAKAMURA    Assuming  a  —637  0 km and p  = 5.52, the effect of the curvature on the phase and group velocities is shown in Fig. 6. We see that the phase velocity decreases, whereas the group velocity increases with the wave length. 5  Effect of Eddy Viscosity    When eddy viscosity is taken into account, the velocity  1,/  gh of long waves decreases, the order of the decrease is given for example by NAKAMURA (1960) by the relation  /g  h  p 1,C viA 2-±B 2(21)  -1/2A2+ B2 where  A=  2  K (cosh 2  re+  cos 2  1c)—  (sinh 2  lc  ±  sin  2  K)  , B  —sinh 2  K  —sin  K,  C  =  2  K  (cosh  2  K  +  cos  2  K)  ,  K  h  a  /-1/1,— and  a  —271T is the circular frequency and  v is the coefficient of eddy viscosity. If we assume that  h=4.5 km and  T  =60  min., it follows that 1     The figures in (22) show that the factors considered  i 6 Combined Effect on  Ph   in Sections 2-4    The effectson the  velocit. to give the total effects, and and 8 for the model A, and  it the dispersion for the phase group velocity. In these figu the system of  incompressibh curvature of the earth are ign    Fig. 8 which shows the  ( that the maximum group  ye] q7=30° and  99=-60°. It thousandths percents due to groups velocity has the order model B, the group velocity    that the  E n  section; ase and   y of the  fi the  dispe Figs. 9 velocity,  res, the le water  o  ored. dispersion v locity  dec  (p- °    .7) 60°. It  may be  cert eddy  visc of one  (0.9998 for  v = 102 cm2/sec, 0.9981  for  v = 104cm2/sec, (22)  r0.9941 for  7, =  105  crrP/sec.  : t e effect of eddy viscosity is comparatively small, so  ctions 2-4 will be most important ones.   d Group Velocicities of the Factors discussed  y of the  factors described in the sections 2-4 are superposed e ispersive property thus obtained is shown in Figs. 7  igs.  and 10 for the model B. Figs. 7 and 9 represent locity, and Figs. 8 and 10 show the dispersion for the  res, the  broken cruves indicate usual dispersion curves for  ater  ver a rigid boundary when the roation and the  red. li rsion of the group velocity for the model A indicates  ocity  creases by about 0.7 percent for both latitudes  7  rtain that, even if the decrease by about several dy  iscosity is taken into account, the decrease in the of one  percent. It is to be noticed that, in the case of increases by few percents.Fig.VELOCITY OF LONG GRAVITY WAVES IN THE OCEAN  -11°1_  -11H1111  Ii  T'1111  I  I  !!111     1   /05'--L-u/Ii4  3,-60°  /0o  095  Y=30°  090E  -  4  5  kill  II!  -15  kil?,  sec  035/-. P  =3.47                      = 460kgi/Aw    A  (km)  0807 _ i___L,I,'fL milL  /0  /0  2  103  /04 7 Dispersion curve of the phase velocity of oceanic waves when     effects of compressible water, elastic bottom, rotation and     curvature of  .the earth are taken into account. Broken curve     represents dispersion of usual gravity waves. (Model A)171 /00       u//-4 0.95  090  ---  -  085 080   075  070 1  I. 1  1-1 /0  /02  Dispersion  curve, of  the   effects of compressible  IA  ture of the earth ar   represents dispersion of-- A=  4.5n2    1.51m/s_ec iP= 3.47 us=  460/tve4ecI ! I I I  Igroup tter,  e  take usual (/2/71)            1111Fig. 8  /03  /04   group velocity of oceanic waves when ble wa r, lastic bottom, rotation and curva-h e ken into account. Broken curve  m f ual gravity waves. (Model A)172Fig. 9Fig. 10               K. NAKAMURA        r  ;   /0,5  /.001 095,  090L  - .  4.8Iem  1  51e111/Aec  0.85 P-552                     -6.001PRAJec (ten)  080  II;L.! i;.J.  /0 /02  /03 /04  Dispersion curve of the phase velocity of oceanic waves when  effects of compressible water, elastic bottom, rotation and cur- vature of the earth are taken into account. Broken curve  represents di persion ofusual gravity waves. (Model B)  t   095—  50=                                                                                                                         -- 090  1                    4=  4.8 0.85. -1                                      — 080  075                              A (km) 070  _1_ • LI L  I I 1 1 1 111.1! _L.  f._.1..i_L1_11_  /02  /03  /04  Dispersion curve of the group velocity of oceanic waves when  effects of compressible water, elastic bottom, rotation and cur- vature of the earth are taken into accout. Broken curve  represents dispersion of usual gravity waves. (Model B)           VELOCITY OF LONG GRAVITY WAVES IN THE OCEAN 173 7 Concluding Remarks    From the results obtained in the preceding sections, it is made clear that the initial distrubance of a tsunami is propagated with the velocity little less than  V  gh. However, the change in the group velocity seems to  be too small to be considered, since the accuracy in the time keeping of the drums of mareographs is  rather bad at least in the present stage of observation. This is probably true even if the effect of ocean currents is introduced in some way. However, in future, the increase in travel time amounting to about 15  min of a trans-Pacific tsunami such as the Chile tsunami of 1960, would become important in the travel-time analysis of a tsunami.    OMOTE (1948) reexamined the marigrams for the Sanriku tsunami of 1933 and determined the central area of the tsunami source. Excess travel time of a tsunami to that of the waves assumed to be radiated from the earthquake picentre may cer-tainly be ascribed in part to a finite area of the source, but does not seem to be a definite evidence that the source area has such a dimension as to account for the whole excess time. It is needed to  study minute behaviour of the front of waves generated by finite deformations of the sea bed. Particularly,  in tsunamis of nearby origin, the epicentre being located within a few hundred kilometers off the coast, it is important to take account of the mechanism of tsunami generation and also conspicuous change in water depth coastwards.  References   1. NAGAOKA, H.  (1909): Note on the Propagation of Oceanic Waves. Tokyo Su. Buts.          Kizi, 4, 113-114.   2. TERADA, T.  (1912): On the Velocity of Sea Waves. Proc. Tokyo Phys. Math. Soc. 2, 6,          260-265.   3. STONLEY, R.  (1926)  : The Effect of the Ocean on Rayleigh Waves. Monthly Notices Roy.           Astro. Soc. Geophys.  Suppl. 1, 349-356.   4. LAMB, H. (1932) : Hydrodynamics, sixth edition, Dover publication, New York. (1950).   5. MIYABE, N. (1934) : An Investigation f the Sanriku Tsunami Based on Mareogram Data.          Bull. Earthq. Res. Inst. Tokyo Imperial Univ.  Suppl. 1, 112-126.   6. SCHOLTE, J.G.  (1943)  : Over het Verband tussen Zeegolven  Microseismen, I and II.           Ned. Akad. Wetenschap.  Verslag. Gewone Vergader. Afdeel. Natuurk., 52,         669-683. (see also Elastic Waves in Layered Media by W.M.  EwiNG, W.S. JARDETZKY         and F. PRESS  (1957).   7. OMOTE, S.  (1948): On the Central Area of Seismic Sea Waves. Bull Earthq. Res. Inst., 25,            15-19.   8. MATSUZAWA, T.  (1950): Seismology, Kadokawa Shoten, Tokyo, (in Japanese)   9.  MIYOSHI, H.  (1954): Generation f the Tsunami in CompressibleWater, I. J. Oceanorg.           Soc. Japan, 10, 1-9.  10.  MIYOSHI, H.  (1955): Second Report. Rec.  Oceanogr. Wor. Jap., Japan Science Council,          Tokyo, 2(1), 49-56.  11. NAKAMURA, K.  (1961)  : Motion of Water due to Long Waves in a Rectangular Bay of          Uniform Depth. Sci. Rep.  TOhoku Univ. 5-th Ser. Geophys. 12, 191-213.

Velocity of Long Gravity Waves in the Ocean

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Velocity of Long Gravity Waves in the Ocean

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