First looking at the scalar potential term, suppose that the displacement is given by u = φ. If one can find a scalar φ such that u = φ. u x.

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1 7.4 Eastodynams 7.4. Propagaton of Wavs n East Sods Whn a strss wav travs throgh a matra, t ass matra parts to dspa by. It an b shown that any vtor an b wrttn n th form φ + ra (7.4. whr φ s a saar potnta and a s a vtor. hs two trms n th dspamnt fd an b xamnd sparaty. h most gnra dspamnt fd an b obtand by addng both sotons togthr. Irrotatona Wavs Frst oong at th saar potnta trm, sppos that th dspamnt s gvn by φ. If on an fnd a saar φ sh that φ, thn t foows that r 0, or r / / / ( hs ah of th trms nsd th brats s zro. Bt ths trms rprsnt rotatons of matra parts (s Eqns...0. For xamp, as stratd n Fg. 7.4., ω (7.4. x ω ω x Fgr 7.4.: a rotaton from th Hmhotz thory

2 hs r 0 an b ntrprtd as no rotaton of matra parts. A sma mnt of matra an st ndrgo norma and shar stran, bt th mnt w not rotat as a rgd body n spa. ang th dspamnt fd φ, wrtng t n ndx notaton, j φ / j, and sbstttng nto Navr s qatons, ads to th thr-dmnsona wav qaton: t, λ + μ ρ ρ E( ν ( + ν ( ν (7.4.4 hs dspamnt fd ths orrsponds to strss wavs travng at spd, asng matra to stran bt not to rotat. hs rrotatona wavs ar aso ad wavs of dataton. Eqvomna Wavs Consdr now th dspamnt fd ra. If on an fnd a vtor a sh that ra, thn t foows that 0, or + + ε + ε + ε ΔV V (7.4.5 hs th ondton that th dspamnt fd b dvrgn-fr mps that thr s no vom hang. hr an b norma strans ony so ong as thr sm s zro. ang ε / and sbstttng nto Navr s qatons thn ads mmdaty to μ ρ (7.4.6 t or th thr-dmnsona wav qaton: t, μ ρ E ρ ( + ν (7.4.7 hs dspamnt fd ths orrsponds to strss wavs travng at spd, asng matra to shar. hs qvomna wavs ar aso ad shar wavs or wavs of dstorton. In smmary, whn an vnt sh as an xposon ors, two dffrnt typs of wav mrg, rrotatona wavs whh rst n rrotatona dspamnt fds, and qvomna wavs whh rst n qvomna dspamnts. hs wavs trav at dffrnt spds.

3 7.4. Pan Wavs At a sffnt dstan from any nta dstrban, a strss wav w trav n a pan. It an b assmd that a matra parts w dspa thr para to th drton of wav propagaton (ongtdna wavs or prpndar to ths drton (transvrs wavs. t th wav trav n th x drton. Irrotatona (p / ongtdna Pan Wavs Consdr parts whh dspa n th drton of wav propagaton aordng to ( x, t. hs s an rrotatona wav sn r 0, and th strss wav s govrnd by th on-dmnsona wav qaton x t (7.4.8 hs ongtdna pan wavs ar aso ad p-wavs. x omprsson / rarfaton wavfront Fgr 7.4.: a ongtdna wav Eqvomna (s / transvrs / shar Pan Wavs Consdr parts whh dspa aordng to ( x, t. hs s an qvomna wav sn 0, and th strss wav s govrnd by th on-dmnsona wav qaton x t (7.4.9 p stands for prmary 4

4 hs transvrs/shar wavs ar aso ad s-wavs. σ N σ S 0 0 x Fgr 7.4.: a transvrs wav 7.4. Vbraton Anayss A vbraton anayss an b arrd ot n xaty th sam way as n Chaptr, ony th wav spds n th D wav qatons and ar now dffrnt from th D spd E / ρ. h partar sotons, ford vbraton and rsonan thory of Chaptr an agan b appd hr. h anayss hr s approprat for thn pats nfnty wd n th x, x drtons, Fg h fgr shows ongtdna vbraton, bt on an aso hav transvrs vbraton whr th parts dspa prpndar to th x axs. x Fgr 7.4.4: strth vbraton of a pat Wavs at Bondars Pan wavs xst n nbondd ast ontna. In a fnt body, a pan wav w b rftd whn t hts a fr srfa. In ths as, on nds to sov Navr s qatons s stands for sondary 5

5 sbjt to th bondary ondtons of zro norma and shar strss at th fr srfa. Wavs of both typs w n gnra b rftd for any sng typ of ndnt wav. Smary, whn a wav mts an ntrfa btwn two dffrnt matras, thr w b rfton and rfraton. h bondary ondtons ar that th dspamnts ar ontnos and th norma and shar strsss ar ontnos, Fg E E ( ( (, ν, ρ ( ( (, ν, ρ ( x ( ( ( x, y y σ σ ( N ( ( ( N, σ S σ S Fgr 7.4.5: rfton and rfraton of a wav at an ntrfa Wavs at Bondars h wavs dsssd ths far ar body wavs. Whn a fr srfa xsts, for xamp th srfa of th arth, anothr typ of wav moton s possb; ths ar th Raygh wavs and trav aong th srfa vry mh watr wavs. It an b shown that th spd of Raygh wavs s btwn 90% and 95% of, dpndng on th va of Posson s rato. Smar typs of wavs an propagat aong th ntrfa btwn two dffrnt matras Probms. Consdr th moton π sn ( x t, 0, 0, What ar th strans n th matra? What ar th orrspondng strsss? What s th vom hang n th matra? What s th nam (or nams gvn to th typ of wav whh ass ths nd of moton?. Consdr th moton π 0, sn ( x t, 0, What ar th strans n th matra? What ar th orrspondng strsss? What s th vom hang n th matra? What s th nam (or nams gvn to th typ of wav whh ass ths nd of moton?. Drv an xprsson for th rato / n trms of th matra s Posson s rato ony. Whh s th fastr, th ongtdna or transvrs wav? 6

6 4. Show that th moton π 0, 0, os s qvomna. ( px os ( x t 5. Consdr th moton [ sn β ( x t + α sn β ( x + t ], 0, 0 ( what nd of ast strss wav dos ths nvov? (Sth th pan of th wav and ts drton of propagaton. ( what ar th strans and strsss. ( s th qatons of moton to dtrmn th wav spd. Is t what yo xptd? (v Sppos that th pan x 0 s a fr srfa. Dtrmn α. (v Sppos aso that x h s a fr srfa. Dtrmn β. 6. Consdr a pat wth ft fa ( 0 α Ωt and th rght fa ( x x sbjtd to a ford dspamnt sn fr. ( fnd th thnss-strth vbraton of th pat. What ar th natra frqns? ( Whn dos rsonan or? 7. Consdr a pat wth ft fa ( 0 th rght fa ( x x sbjtd to a traton t α os Ωt and fxd, as shown n th fgr bow. ( fnd th thnss-shar vbraton of th pat. What ar th natra frqns? ( Whn dos rsonan or?, x fxd x σ [not: assm a dspamnt ( x, t ; as wth transvrs wavs, ths w satsfy th -d wav qaton wth bng th transvrs wav spd. Us th traton to obtan an xprsson for th shar strss σ ovr th ft hand fa. Whn appyng th strss bondary ondton, yo w nd th stran-dspamnt xprsson and strss-stran aw, 7

7 ε +, ε σ μ 8. Consdr th as of α ( osωt + sn Ωt ovr th ft fa ( 0 rght fa ( x x wth th fxd. Drv an xprsson for th partar soton and show that t rprsnts rar moton of th parts n th x x pan. [hnt: vaat th partar sotons for and sparaty and thn show that + for som r (ndpndnt of tm] r 8

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