We assume a two-layered model, and a P wave is excited in the upper layer.

Transcription

1 3.4 Crtcal angle We assume a two-layered model, and a P wave s ected n the uer layer. When a P wave mnges on a horontal boundary, the ncdence angle for the transmtted P wave n the second medum s gven by (from the Snell s law) sn [sn ( / )] (3.4.) When the velocty n the lower layer, there s an ncdence angle of c (crtcal angle) that maes the ray n the lower layer roagates along the nterface, 90 (3.4.) sn (3.4.3) c / Once the ncdence angle eceeds the crtcal angle, whch s a stuaton called ostcrtcal ncdence, no transmtted lane wave ests n the lower layer total nternal reflecton. Mathematcally, the P-wave otental for the lower layer has a -deendent real eonental term, e( ), nstead of a urely magnary eonental term, e( ). The dslacement n the lower layer s not a roagatng lane wave, but occurs as an evanescent wave that travels along the nterface and decays away from the nterface. If the S velocty n the lower layer s greater than the P wave velocty n the uer layer, there ests a second crtcal angle: sn (3.4.4) c S / no transmsson of P and S waves beyond the nterface 3.5 Snell s law for SH waves

2 The dslacement of SH waves n the uer layer s u y (,, t) B e[ ( t )] + B e[ ( t + )] (3.5.) ncomng wave reflected wave The dslacement of transmtted wave n the lower layer s u (,, t ) B e[ ( t )] y (3.5.) From the Snell s law sn j sn j c (3.5.3) The crtcal angle for SH waves s sn j c (3.5.4) 3.6 Ray arameter and slowness A useful way to charactere a wave s ray ath s va ts ray arameter,, the recrocal of the horontal aarent velocty, sn c v (3.6.) Consder a P wave roagatng n the - lane:

3 3 )] ( e[ )] ( e[ e )] ( e[ s t t t t η (3.6.) where the slowness vector s defned by ), ( η s (3.6.3) c η (3.6.4) η cos (3.6.5) Thus, the magntude of the slowness vector s η cos sn + + s (3.6.6) s ˆ (3.6.7) The ray arameter,, s called horontal slowness, and η s called the vertcal slowness. The slowness for S waves s η s ˆ ), ( (3.6.8) c j η cos (3.6.9)

4 Homewor 3.6. Derve eq. (3.6.9) usng the Snell s law. The ray arameter (or, horontal slowness, ) s the same for the ncdent, reflected and transmtted waves, whereas the vertcal slowness deends on the medum and the wave tye. s et constant for any rays that are roduced at nterfaces. s constant along any ray ath, no matter how many transmssons, reflectons, or conversons the ray has undergone. 3.7 Wavegudes If sesmc waves are generated n the low-velocty medum, total nternal reflecton wll tra much of the sesmc energy n the low-velocty channel, whch acts as a wavegude. The seed of sound n seawater s roortonal to both temerature and ressure. The combnaton of temerature decreasng wth deth and ressure ncreasng wth deth roduces a low-velocty regon nown as the SOFAR (Sound Fng and Rangng) channel at a deth of ~000 m meters. Rays leavng a source n the channel at angels u to ± from the horontal are nternally reflected. 4

5 The SOFAR channel transmts sound very effcently to long dstances (e.g., T wave). The seed of sound waves n the channel s used to study of changes n ocean temerature. 3.8 Fermat s rncle and geometrc ray theory Fermat s rncle: The ray aths between two onts are those from whch the travel tme s an etremum, a mnmum or mamum, wth resect to the nearby ossble aths. Snell s law s derved from the Fermat s rncle. 5

6 a + ( b ) + c T ( ) + (3.8.) v v dt ( ) d v sn v sn sn v v a + sn v v 0 b ( b ) + c (3.8.) Geometrc ray theory s an aromaton to the solutons of the elastc equaton of moton that descrbes the generaton and roagaton of sesmc energy. Ray theory has two major lmtatons: ) Geometrc ray theory does not drectly rovde nformaton about wave amltudes. ) Geometrc rays fal to descrbe the wave s behavor. 3.9 Huygens rncle and dffracton In some alcatons treatng roagatng waves as geometrc rays fals to elan what we observe. Waves bend or dffract around the earth s core and so reach laces to whch Snell s law redcts no ray ath. Although ray theory says that no energy s transmtted when a wave s ncdent on an nterface 6

7 at an angle greater than the crtcal angle, some energy s n fact transmtted. Huygens rncle: Each ont on a wavefront s consdered to be a Huygens source that gves rse to another crcular wavefront. A straght wavefront generates subsequent straght wavefronts. Reflected and transmtted wavefronts: 7

8 Dervaton of Snell s law: Proagaton dstance of reflected hase from the nterface: AC' vt Proagaton dstance of transmtted hase from the nterface: AD vt AC AC / sn v t (3.9.) / sn AC AD / sn v t (3.9.) / sn From eqs (3.9.) and (3.9.), we have vt sn vt sn sn sn v v (3.9.3) Dffracton: Huygens rncle also elans the henomenon of dffracton, n whch waves bend around obstacles. 8

9 The dffracton elans the observaton of shear waves around the core at far dstance, although the shear waves can not ass through the lqud outer core. If the slt has wdth d, then waves from ether sde of the slt wll be out of hase by 90 and so nterfere destructvely at dstance D when the ath dfference s a half wavelength. The amltude wll be ero at a dstance 0 from the mddle of the slt: λ d0 d snθ (3.9.4) D when we assume D >> d. Homewor 3.9. Derve eq. (3.9.4). The amltude decays from ts mamum at θ 0 to ero at D / d. 0 λ A sohstcated analyss shows that the amltude vares as (Fg..5-8, bottom) sn ς ς where πd ς (3.9.5) λd The functon has a central lobe of wdth 0 and a seres of decreasng sde lobes. -The dffracton deends on the rato of the wavelength to the slt wdth. λ ) f d <, the sde lobes vansh ) f an obstacle s less than half a wavelength wde, waves mngng on t are nsenstve to the detals of ts structure. ) f the slt s very wde comared to the wavelength, dffracton occurs only at the slt s edges. - Smlar effects occur when wavefronts encounter a crcular (or shercal) obstacle. 9

10 - Geometrcal ray theory redcts that no energy wll arrve behnd the obstacle, and a hole n the wavefront wll develo. -In realty, snce the wave dffracts around the shere, the ga s closed and connected. waveform annealng Homewor 3.9. Dscuss whch one, ether slow- or fast-velocty heterogenety, may be dentfed relatvely well from traveltme observatons consderng the Fermat s rncle and waveform annealng effect (Huygens rncle). Then, dscuss what features would be well dentfed n the Earth, and what features may be dffcult to be dentfed from sesmc traveltme observatons? 0