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Waves - Seismology - Lecture Slides, Slides of Geology

Punjab Engineering CollegeGeology

In these Lecture Slides, the Lecturer has illustrated the following key points : Waves, General, Solution, Wave, Velocity, Locus, Passage, Points, Same, Phase

Typology: Slides

2012/2013

Uploaded on 07/19/2013

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seri_66 🇮🇳

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Waves…+
• Waves+are+soluKons+to+the+wave+equaKon:+
– In+1D:+
– General+soluKon+in+1D:+
– x*ct:+“phase”+of+the+wave+soluKon+
– wavefront:+locus+of+points+with+the+same+phase+
• SoluKon+by+separaKon+of+variables:+
!2u
!x2(x,t)=1
c2
!2u
!t2
u(x,t)=f(x!ct)+g(x+ct)
c+is+wave+velocity+
u(x,t)=X(x)T(t)
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Download Waves - Seismology - Lecture Slides and more Slides Geology in PDF only on Docsity!

Waves…

  • Waves are soluKons to the wave equaKon:
    • In 1D:
    • General soluKon in 1D:
    • x-­‐ct: “phase” of the wave soluKon
    • wavefront: locus of points with the same phase
  • SoluKon by separaKon of variables:

2

u

! x

2

( x , t ) =

c

2

2

u

! t

2

u ( x , t ) = f ( x! ct ) + g ( x + ct )

c is wave velocity

u ( x , t ) = X ( x ) T ( t )

u ( x , t ) = X ( x ) T ( t )

2

u

! x

2

( x , t ) =

c

2

2

u

! t

2 c 2 1 X d 2 X dx 2 = 1 T d 2 T dt 2 Depends only on x Depends only on t => both sides are equal to a constant (-­‐ω^2 ) We obtain two coupled ordinary differenKal equaKons: d 2 X dx 2

! 2 c 2 X = 0 d 2 T dt 2 +! 2 T = 0 SoluKons: X ( x )^ =^ Ae i! x / c

  • Be ! i! x / c T ( t ) = Ce i! t
  • De ! i! t

f=1/T frequency

k=ω/c =2π/λ= wavenumber – units m

-­‐ At fixed posiKon: At fixed Kme:

In 3D – coordinates (x,y,z) plus Kme t

 Scalar wave equaKon:

 Vector wave equaKon:

!( ! x , t ) = A exp i (! t " " k !

" x ) Where A is a scalar and (^) | ! k ! |= " ! ! !( ! x , t ) = ! B exp i (! t " ! k !

! x ) Where B is a vector and (^) | ; α and β are wave velociKes ! k ! |= " !

Body waves and surface waves

  • The seismic waves generated by an earthquake have

two ways of propagaKng across the earth:

  • Along the surface: surface waves
    • Amplitude decays with depth
  • Through the body of the earth: body waves Along the surface: cylindrical wavefront. At Kme t its area is 2πvtz=2πxz. If energy is conserved: the energy at distance x from the source is proporKonal to 1/x, and the amplitude of the wave therefore to x -­‐1/ The wavefront is spherical, its area at Kme t is: 4πv^2 t^2 =4πx^2. Energy decays as x-­‐2^ and amplitude as 1/x Seismic rays : perpendicular to wavefronts

Seismic wave equation

  • We apply F = ma
  • Forces
    • Stresses:
      • Force per unit area
        1. Tangential stresses
        1. Normal stresses
    • Body forces See Appendix 2 of Fowler

ma = F is then: Where:

  • u i is displacement in direction i
  • f i represents body forces per unit volume (i-th component) In general, body forces include source term and gravity term
  • Strains….. (seismo notes) ij j i^ i=1,2, i f t u = + ∂ ∂ 2 , 2 ρ σ

Strain= infinitesimal deformation: In 1D : Strain is change in length divided by original length In 2D: 1) change in length: ! u = ( u , v , w ) e xx

! x +! u!! x ! x

! u ! x

" u " x e xx = change in length of LM original length of LM = ! x + ! u ! x ! x "! x ! x = ! u ! x