Analysis on lambs problem

Analysis on Lamb’s Problem by Zhang Xiongtao Thesis submitted to The National University of Singapore for the degree of Doctor of Philosophy 2014 Abstract We implement the master relationship in [12], Laplace-Fourier path in [13], and determinant of a surface wave in [14] together to form a LY algorithm and apply this algorithm to solve the Lamb’s problem completely. We obtain an explicit solution formula for the Lamb’s problem in the space-time variable x-t. The solution formula is given in terms of the fundamental solutions of the d’Alembert wave equations in 3-D and 2-D by the Kirchhoff’s formula and Hadamard’s formula . Complicated 2D-3D coupling wave structures on the surface present in the surface wave solution formula. This shows that the wave structures given in the paper are much richer than the Rayleigh wave discussed in the original articles, [19, 9]. Further computation and estimates of the solution formula would also be discussed in this article and then gain results consistent with the theory in seismology. ii Acknowledgements I would like to thank my parents, for their supports of my study and work. I would like to express my gratitude to my supervisor, Professor Shih-Hsien Yu, for his invaluable guidance. I would also like to thank my friend and co-worker, Haitao Wang, for his inspiring supports and joint-work. iii Contents Derivation of Solution Formula 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 LY Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Master Relationship, boundary condition, and matrix (Sij )6×3 . 24 1.5 Characteristic-non characteristic decomposition . . . . . . . . . 26 1.6 Realization of decomposition . . . . . . . . . . . . . . . . . . . 29 1.6.1 Inversion of the Rayleigh Wave . . . . . . . . . . . . . 29 1.6.2 Proof of Lemma 1.3.7. . . . . . . . . . . . . . . . . . . 31 Computation and Estimates 34 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2 Fundamental Solution of Elastic Equation . . . . . . . . . . . . 37 2.3 Solution in Half Space . . . . . . . . . . . . . . . . . . . . . . 42 2.4 Poisson Solid and Solution Behavior on the Surface . . . . . . . 46 2.4.1 Initial Impulsion absorption . . . . . . . . . . . . . . . 48 2.4.2 Initial Impulsion Restricted on the Boundary . . . . . . 50 2.4.3 Initial Impulsion in Interior . . . . . . . . . . . . . . . . 70 iv C ONTENTS Conclusion and Discussion 79 3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2 Difficulties and Future Work . . . . . . . . . . . . . . . . . . . 80 Bibliography 80 v C HAPTER Derivation of Solution Formula 1.1 Introduction The presence of a unbounded boundary in a multi-D space domain will completely change the natures of problems without any boundary completely such as the compressible Euler equation, compressible and incompressible NavierStokes equation, Maxwell equation, etc Without understanding the basic wave natures around boundary, the general practice “to find robust estimates” among the researchers in the modern PDE for initial value problem may fail. The necessity for a new input to gain insights on the wave natures around the boundary arouses. Such an input would become a sharp tool to help the general practice to continue when an initial-boundary value problem encountered. A good candidate for such a new input is the construction of an explicit solution formula of the Green’s function for a constant coefficient problem in a half space domain. The advantage of an explicit solution formula of the Green’s function is that one can represent the solution of a linear or nonlinear problem by the Duhamel’s principle in terms of the Green’s function so that the singular structure (in the space-time variable) of the Green’s function around the boundary will pass to the solution. This may give the sufficient ansatz structure around boundary so that one can focus on how to obtain sharper estimates for the linear or nonlinear problems encountered. Thus, explicit formulae in the space-time C HAPTER 1: D ERIVATION OF S OLUTION F ORMULA variable might be very useful. Our primary interest is to develop a general methodology to obtain an explicit solution formula of the Green’s function for a half space problem. For the sake of making this paper interesting to the majority of mathematical disciplines in science, we choose a rather classical unsolved mathematical problem, which is commonly known in mathematics, physics, and engineering communities, to demonstrate the effectiveness of the new methodology, the LY algorithm, which is a structured program for a general class of PDEs. The details of the algorithm will be given in Chapter 2. We choose the Lamb’s problem as a source of ideas to practice the LY algorithm. The Lamb’s problem is an initial-boundary value problem for a linear elasticity problem in 3-D half space with a free boundary condition. This problem is an important mathematical model to study the natural phenomenon, “earthquake”. The free boundary value problem for linear elasticity was initiated by Lord Rayleigh. In [19], he investigated the motion of waves on the surface by considering a linear elastic equation for an isotropic elastic medium in a threedimensional half-space R3+ with a free boundary condition at x = 0; and in [9] Lamb proposed the initial boundary value problem: ∂2 u = ∇ · λ(∇ · u)I + µ(∇u + ∇u T ) , ρ ∂t x ≡ ( x, y, z) ∈ R3+ ≡ {( x, y, z) : x > 0, y, z ∈ R}, t ≥ 0,     (2µ + λ)∂ x λ∂y λ∂z  0       u(0, y, z, t) = 0 , µ∂ µ∂ y x         µ∂z µ∂ x    u( x, 0) = Φ( x) (1.1.1)   ∂t u( x, 0) = Ψ( x), where u = u( x, t) ∈ R3 is the displacement vector; and Φ and Ψ are the given C HAPTER 1: D ERIVATION OF S OLUTION F ORMULA initial data. The elastic properties of isotropic materials are characterized by density ρ (constant) and Lamé constants λ > and µ > 0. Instead of solving the initial boundary value problem (1.1.1), Rayleigh considered a special solution u( x, y, z, t) = e pt−rx−i f y−igz v( f , g, p) (a wave train solution in y-z plane) to fit the boundary condition, the speed of surface wave motion was obtained in terms of the wave numbers ( f , g) in y-z plane and the Lamé constants, i.e. p = Ω( f , g, λ, µ), which is a dispersion relationship. This surface wave motion was named after him as the Rayleigh wave in physics; and indeed such a surface wave motion is a generic physics phenomena. In [9], Lamb continued to investigate the structure of the solution of the initial boundary value problem for (1.1.1) in the transform variables and related the Rayleigh wave to the phenomenon in seismology, the earthquake. This problem became a well-known problem, the Lamb’s problem, in the seismology, geophysics, mechanical engineering, etc One can find related references for the Rayleigh wave and the Lamb’s problem in research articles and textbooks in physics, geophysics, mechanical engineering such as [10, 1, 17, 2, 4, 8, 11, 3, 18, 20, 22]. The system (1.1.1) is a hyperbolic system in 3-D half space domain. Though there were many works for linear hyperbolic systems in half space domain, for examples, [6, 16, 21], the definite structures of the surface wave for the system (1.1.1) were never been obtained before 2011. The first key step towards this definite surface wave structure was obtained in [12]. The fundamental solution was used to convert an initial-boundary value problem into a problem with an inhomogeneous boundary value problem together with zero initial data so that an intrinsic relationship among the boundary data in terms of transform variables was discovered. This step also works for the system (1.1.1). One can use the fundamental solution to convert the system into the following form: C HAPTER 1: D ERIVATION OF S OLUTION F ORMULA ∂2 λ + 2µ µ ∇(∇ · a) + ∇ × (∇ × a) = 0, ρ ρ (1.1.2a) a( x, 0) = ∂t a( x, 0) = 0, (1.1.2b) (2µ + λ)∂ x λ∂y λ∂z      a(0, y, z, t) = gb (y, z, t), µ∂ µ∂ y x     µ∂z µ∂ x (1.1.2c) ∂t  a−  and gb (y, z, t) is a given function in terms of initial data u( x, y, z, 0) and ut ( x, y, z, 0). Then, one introduces the transforms    aˆ ( x, iη, iζ, t) = F[a]( x, iη, iζ, t) ≡     a˜ ( x, iη, iζ, s) = L[a]( x, iη, iζ, s) ≡       J[a](ξ, iη, iζ, s) ≡ R2 ∞ ∞ a( x, y, z, t)e−iyη −izζ dydz, aˆ ( x, iη, iζ, t)e−st dt, a˜ ( x, iη, iζ, s)e− xξ dx. (1.1.3) Then, similar to [12], by (1.1.2a), (1.1.2b), and lim L[a]( x, iη, iζ, s) < ∞ for each given (η, ζ, s) ∈ R × R × R+ x →∞ together, one obtains an intrinsic algebraic relationship between the Dirichlet data L[a](0, iη, iζ, s) and Neummann data L[a x ](0, iη, iζ, s), which is the master relationship: M(iη, iζ, s; L[a](0, iη, iζ, s), L[a x ](0, iη, iζ, s)) = 0, where the system M is linear in L[a](0, iη, iζ, s) and L[a x ](0, iη, iζ, s). This system of linear equations and the transform for (1.1.2c) together give rise the explicit solution of (L[a](0, iη, iζ, s), L[a x ](0, iη, iζ, s)) in the transform vari- C HAPTER 2: C OMPUTATION AND E STIMATES respect to y, we have ∂y F −1 [ Sin[ci ηt] ci η ∗t Bessel [0, ηt]] is an odd function and hence we can write out the formula for y < 0. Thus we have: Lemma 2.4.13. For c2 and c3 we can derive: Sin[ci ηt] ∗t Bessel [0, ηt]] ci η t2 π + = (δ(y − ci t) + δ(y + ci t)). t2 c2i − y2 t2 − y2 2ci (c2 − 1) i (2.4.45) F −1 [ η Proof. Similarly we can deduce the results for general positive real number c L which is smaller than c2 and c3 . Lemma 2.4.14. For c2 and c3 we can derive: Sin[ci ηt] ∗t Bessel [0, c L ηt]] ci η t2 π = + (δ(y − ci t) + δ(y + ci t)). 2 2 2 2 2c ( c − c ) t ci − y cL t − y i L i (2.4.46) F −1 [ η Proof. Corollary 2.4.15. F −1 [ L −1 [ η + s2 t2 − y2 9π ]] = − √ (δ(y − 2t) + δ(y + 2t)) 2 2 8η + 2s (4t − y ) (2.4.47) Corollary 2.4.16. √ √ √ + s2 − y2 3 3η 3 3t 3π F −1 [ L −1 [ (δ(y − 2t) + δ(y + 2t)) ]] = − 2 2 8η + 2s (4t − y ) (2.4.48) 68 C HAPTER 2: C OMPUTATION AND E STIMATES Lemma 2.4.17. √ F −1 [ L 3iη η +s2 2 −1 8η +2s [ − √ √ 3i 3η 3η +s2 8η +2s2 η + s2 ]] (2.4.49) √ t2 − y2 3 3t2 − y2 =∂y ( − ) ∗(y,t) (4t2 − y2 ) (4t2 − y2 ) t2 − y2 Proof. Applying corollary 2.4.15 and 2.4.16 we obtain (2.4.49). Again (2.4.49) is only a L1 operator and we assume the initial data to be of compact support to study the time asymptotic property of this term. Lemma 2.4.18. | F −1 [ L √ 3iη η +s2 2 −1 8η +2s [ − √ √ 3i 3η 3η +s2 8η +2s2 η + s2 ]] ∗y Heaviside(1 − |y|) |≤ O(1) √ t (2.4.50) Proof. Apply lemma 2.4.17 and the (2.4.18) can be similarly proven as we did for c1 terms. Corollary 2.4.19. | √ √ √ √ √ i (7+4 3)η 3η +s2 i (1+ ) η η + s2 √ √ √ − 3η +6η +3s2 (2+ ) η +2 (3+ ) s2 ( ) F −1 [ L −1 [ η + s2 ]] ∗y Heaviside(1 − |y|) | ≤ O (1) √ t (2.4.51) Lemma 2.4.20. If the initial data is restricted on the boundary and has compact support on the boundary surface, then the behavior of the solution on the boundary would has the property below: √ √ 3 − δ(y + c t) − δ(y − c t) 1 )) ∗y Heaviside(1 − |y|) | | (u(0, y, t) − (−2 12 2c1 ≤ O (1) √ t (2.4.52) 69 C HAPTER 2: C OMPUTATION AND E STIMATES Proof. Applying corollary 2.4.19, lemma 2.4.18 and lemma 2.4.11 we can imply (2.4.52). 2.4.3 Initial Impulsion in Interior Now we consider the case when initial impulsion is in the interior domain i.e we suppose our initial data to be u( x, y, 0) = δ( x − x0 , y − y0 ) (2.4.53) Then on the boundary the reflection part of the solution has the formula as uˆ rb iη 2η + s2 = e x0 − η + s3 6η + √ η2 −4 √ 12η + 3s4 + 12η s2 − 3η + s2 3η η + s2 + s2 + 3s2 e x0 − √ 3η + s2 (2.4.54) Now rationalize the denominator and we can rewrite (2.4.54) as linear combination of two parts.       uˆ rT =             r    uˆ L = iη iη √ √ η + s2 √ ( 3η +s2 ( 2η +s2 +4η ) 2s2 √ η + s2 η + s3 (4η +s2 ) )−24 √ ( 3( η + s2 ) √ 12s2 η +s2 (4η +s2 )(8η +3s4 +12η s2 ) −24η +3s6 +2η s4 −28η s2 3η +s2 )( )( 2η +s2 (8η +3s4 +12η s2 ) ) 2η +s2 e x0 − √ 3η +s2 η + s3 (2.4.55) Then by partial fraction we can separate (2.4.55) into simpler operators: 70 e x0 − √ η + s2 η + s2 C HAPTER 2: C OMPUTATION AND E STIMATES √  √ 2 √ √ 2 x0 − η + s2   iη η +s + 3η +s e  r = −  √ ˆ u   T 8s2 η +s2         √    √ 2 x0 − η + s2 √ √ √ √  2 2  iη η +s +2 3η +s −3 3η +s e    √ √ +   24(2+ 3) η +s2 (c21 η +s2 )            T  + N2,3        √ 2 √ √ 2 x0 − η2 + s32    iη η +s + 3η +s e    uˆ rL =   8s2 η + s3             √ √ √ √ √   iη η +s2 +3 η +s2 + 3η +s2    −    24 η + s3 (s2 +c21 η )            L + N2,3 (2.4.56) e x0 − η + s3 where we have N1 and N2 as below:  √ √ 2 √ √ 2 x0 − η + s2   9iη η +s − 3η +s e   T = √  N2,3   η +s2 (4η +s2 )         √   √ √ √ √ √ x0 − η + s2   iη η +s2 −2 3η +s2 −3 3η +s2 e    √ √ √ +   8( 3−2) η +s2 (2(3+ 3)η +3s2 )       √ 2 x0 − η2 + s32  √ √  2  iη 3η +s − η +s e  L =   N2,3   η + s3 (4η +s2 )             √ √ √ √ √ √   iη −7 η +s2 +12 η +s2 +2 3η +s2 −3 3η +s2    +  √ √ 8( 3−2) η + s3 (2(3+ 3)η +3s2 ) 71 e x0 − η + s3 (2.4.57) C HAPTER 2: C OMPUTATION AND E STIMATES In the previous section we show the cancelation of c2 and c3 terms. Now for T and N L we can similarly prove that there would be no surface wave with N2,3 2,3 speed c2 or c3 . Then we can have some results without detail of proof. Lemma 2.4.21. T | F −1 [L −1 [ N2,3 ]] ∗y Heaviside(1 − |y|) |≤ O(1) √ (2.4.58) t Lemma 2.4.22. L | F −1 [L −1 [ N2,3 ]] ∗y Heaviside(1 − |y|) |≤ O(1) √ (2.4.59) t For the c1 terms we can use similar method as we did in previous section to study its time asymptotic property. Lemma 2.4.23. F −1 =2F [L −1 −1 [L [ iη −1 √ √ η + s2 + 3η + s2 − √ 24 + c21 η + s2 3η + s2 ]] √ η + s2 [ ]] √ 24 + c21 η + s2 iη t −( √ − ) y − c21 t2 √ − c21 Heaviside[ 3t > |y| > t]  −( √ − )  t2 − y2 +t − c21 + 3t2 − y2 +t  − c21 (2.4.60) Proof. This is a consequence of lemma 2.4.5. 72 C HAPTER 2: C OMPUTATION AND E STIMATES Lemma 2.4.24. F =∂ x −1 [L −1 [ iηe | x0 | − √ η + s2 ]] c21 η + s2 log c21 (t − x0 ) (t + x0 ) − 2c1 ty + x02 + y2 − c21 2c1 − ∂x log (c1 t − c1 x0 + y) (c1 (t + x0 ) + y) + x02 2c1 − c21 2x0 y + t2 − x02 − y2 − c21 t t2 − x02 − y2 + − c21 t2 + c21 − x02 − y2 (2.4.61) Proof. Lemma 2.4.25. we can apart c1 term of urT into combination of surface wave and L1 operators: F −1 [ L −1 [ = iη √ ∂x 12 + − √ η2 √ + s2 3η +2 √ 24 + + s2 −3 3η + s2 e x0 − √ η + s2 ]] η + s2 c21 η + s2 log c21 (t − x0 ) (t + x0 ) − 2c1 ty + x02 + y2 √ ∂x 12 + 2c1 − c21 log (c1 t − c1 x0 + y) (c1 (t + x0 ) + y) + x02 2c1 − c21 + N1T (2.4.62) 73 C HAPTER 2: C OMPUTATION AND E STIMATES Proof. Apply lemma 2.4.23 and lemma 2.4.24 one have √ x η + s2 + 3η + s2 − 3η + s2 e −1 −1 F [L [ √ 24 + η + s2 c21 η + s2 √ x0 − √ η + s2 iη 3e −1 −1 =2F [L [ ]] √ 24 + c21 η + s2 iη √ t − ∂y ( √ − ) y − c21 t2 − ∂y ( √ − ) + ∂y ( √ − ) = √ ∂x 12 + − + √ ∂x 12 + √ t2 − y2 +t − c21 ∗(y,t) F 3t2 − y2 + t √ − c21 3t > |y| > t] ∗(y,t) F −1 ∗(y,t) F [L −1 −1 [L [ e −1 x0 − √ √ ]] −1 [ x0 − [L −1 [ e x0 − √ η + s2 η + s2 ]] η + s2 η + s2 √ e η + s2 ]] η + s2 η + s2 ]] log c21 (t − x0 ) (t + x0 ) − 2c1 ty + x02 + y2 1 − c21 Heaviside[ − 12 + 2c1 − c21 log (c1 t − c1 x0 + y) (c1 (t + x0 ) + y) + x02 2c1 − c21 2x0 y t2 − x02 − y2 − c21 t t2 − x02 − y2 + − c21 t2 + c21 − x02 − y2 √ x0 − η + s2 √ t e − c21 Heaviside[ 3t > |y| > t] ∗(y,t) F −1 [L −1 [ ]] − ∂y ( √ − ) 2 y − c1 t η + s2 √ x0 − η + s2 1 e − ∂y ( √ − ) ∗(y,t) F −1 [L −1 [ ]] + s2 η 2 t − y + t − c1 √ x0 − η + s2 e ]] + ∂y ( √ − ) ∗(y,t) F −1 [L −1 [ η + s2 3t2 − y2 + t − c21 . (2.4.63) The first two terms with log function are surface wave components and we can 74 C HAPTER 2: C OMPUTATION AND E STIMATES denote the remainder as N1T . We can the same analysis on c1 term of urL . Lemma 2.4.26. we can apart c1 term of urL into combination of surface wave and L1 operators: √ F −1 [L −1 [− = ∂x √ η2 iη + s2 +3 η2 24 η2 + s2 + + s2 √ s2 3η + s2 e x0 − ]] + c21 η log c21 3t2 − x02 − 6c1 ty + x02 + y2 − ∂x 2c1 √ − c21 log c21 3t2 − x02 + 6c1 ty + x02 + y2 2c1 − c21 + N1L (2.4.64) 75 η + s3 C HAPTER 2: C OMPUTATION AND E STIMATES Proof. √ F −1 [L −1 [− η2 iη + s2 +3 24 − ∂y 24 − ∂y − ∂y + ∂y ]] +t − c21 − y2 +t − c21 e e η + s3 η2 + ]] s2 η + s3 x0 − η2 ∗(y,t) F −1 [L −1 [ e x0 − + ]] s2 η + s3 x0 − η2 + ]] s2 log c21 3t2 − x02 − 6c1 ty + x02 + y2 − ∂x + − y2 3t2 η + s3 s2 + c21 η ∗(y,t) F −1 [L −1 [ 24 e x0 − − c21 Heaviside[ 3t > |y| > t] ∗(y,t) F −1 [L −1 [ √ 3+2 + ∂y √ t − c21 t2 t2 + s2 ]] 24 s2 3η √ √ 3+2 − ∂y = ∂x y2 η2 + + √ iηe c21 η + s2 √ 3+2 + s2 η + s3 x0 − =2F −1 [L −1 [− η2 √ − c21 2c1 log c21 3t2 − x02 + 6c1 ty + x02 + y2 − c21 2c1 t2 − x02 − √ 3+2 24 √ 3+2 24 √ 3+2 24 y2 − c21 t √ 3x0 y 3t2 − x02 − y2 − c21 − t2 + c21 − x02 − 3y2 √ t y2 − c21 t2 − c21 Heaviside[ 3t > |y| > t] ∗(y,t) F −1 [L −1 [ t2 − y2 + t − c21 ∗(y,t) F −1 [L −1 [ 3t2 − y2 + t − c21 76 e ∗(y,t) F −1 [L −1 [ η + s3 x0 − η2 e + x0 − η2 ]] s2 η + s3 s2 ]]. + (2.4.65) e x0 − η2 η + s3 + s2 ]] C HAPTER 2: C OMPUTATION AND E STIMATES Lemma 2.4.27. For N1T and N1L we have the time asymptotic structure when the initial data has compact support in variable y i.e the initial data is of the form δ( x − x0 ) Heaviside(1 − |y|) Then N1T and N1L would has a uniformly decay rate of √1 t | F −1 [L −1 [ N1T ]] ∗y Heaviside(1 − |y|) |≤ O(1) √ | F −1 [L −1 [ N1L ]] ∗y Heaviside(1 − |y|) |≤ O(1) √ t t (2.4.66) (2.4.67) Proof. The proof is similar as what we did in previous sections. Now the we need to deal with the last two terms − + iη η2 + s2 + √ 8s2 iη η2 + s2 + √ 8s2 3η + s2 e x0 − √ η + s2 η + s2 3η + s2 η2 + e x0 − η + s3 (2.4.68) s2 These two terms cannot be studied independently because of the s2 on the denominator which represents the Newton potential. In fact the unbounded influence domain due to the Newton potential would be restricted in a cone after cancelation in the summation. 77 C HAPTER 2: C OMPUTATION AND E STIMATES Lemma 2.4.28. |F −1 [L −1 [− iη + F −1 [ L −1 [ η2 + s2 + √ 3η 8s2 iη η2 + s2 + √ e x0 − √ η + s2 ]] ∗y Heaviside(1 − |y|) η + s2 8s2 ≤ O (1) √ + s2 3η + s2 η2 + t s2 e x0 − η + s3 ]] ∗y Heaviside(1 − |y|)| (2.4.69) Proof. In fact the term s2 can be viewed as a limit case of s2 + c2 η where c = 0. Thus we can repeat the process in the previous work in lemma 2.4.26 and lemma 2.4.25 and combine our results for solution in free space in (2.4.7) to obtain the estimate in (2.4.69). Finally combining (2.4.7), lemma 2.4.21, lemma 2.4.22, lemma 2.4.28, lemma 2.4.27, lemma 2.4.26 and lemma 2.4.25 we can prove the theorem 2.4.1 and theorem 2.4.2. 78 C HAPTER Conclusion and Discussion 3.1 Results According to the previous chapters we have two main results i.e theorem 1.1.1, theorem 2.1.1 and theorem 2.4.2. In chapter theorem 1.1.1 gives out surface formula without any restriction of the poisson ratio, i.e the poisson ratio can be any constant between and 0.5. Although our formula is valid in mathematical sense there are something strange when comparing our formula with the classic theory in seismology. More precisely, in case and case 2, all the three poles are on the imaginary axis, thus the bromwich integral cannot avoid any pole and this may lead to some surface wave with speed greater than body waves’, which contradicts classic seismology, which says surface wave speed should be smaller than body waves’, and observation in real life [1] [8]. Thus some further study is needed on these formulas obtained in theorem 1.1.1 to make our conclusion consistent with classic theory. In chapter we have two main tasks. Firstly, we recombine the formula in transform space and obtained solution for particular initial boundary value in theorem 2.1.1. The recombination highly simplified our formula and allow us to reverse it to time-space domain. Secondly, we explained that the terms with 79 C HAPTER 3: C ONCLUSION AND D ISCUSSION wave speed greater than body waves’ in 1.1.1 will be canceled and thus our conclusion will coincide with classic theory. Finally, we estimates our formulas in 2.1.1 and show that the c1 terms (contain the surface wave part with wave speed c1 which is smaller than body waves’ speed) would be the main part. More precisely, on the surface, the formulas in 2.1.1 will have a uniformly decay rate √1 t except for surface wave part which travels along the surface with speed c1 . It is well known in classic theory of seismology that the surface wave will be more destructive than body waves and our results are illustration of this theory. 3.2 Difficulties and Future Work There are still some difficulty remain. Firstly, we can generate the Green’s function of the Lamb’s problem but it is hard to give out a proper estimates of the formulas even in 2-D case. More precisely (in 2-D case) the interaction of 2-D waves with different speed is not clear enough. Secondly, Lamb’s problem assumes the homogeneous Lame constants i.e the elastic property of the materia is the same every where. It is natural to think of what happen if the media is inhomogeneous. One of the models to investigate the inhomogeneous problem is so called Love wave problem. In this model the half space is divide to layers, which are parallel to the boundary, with different elastic property. Thus the upper layer would be a wave guide in which Love wave would translate along the surface. The main difference between Love wave and Rayleigh wave is that Love wave has dispersion property while Rayleigh wave not, so Love wave speed is not unique. Many study on this topic try to reduce the system into eigenvalue problems and draw some conclusions with assumptions about the form of the solution [1], while, to handle with initial boundary value problem, some further study and maybe some new tools are needed. 80 References [1] K. Aki; P. G. Richards, Quantitative Seismology, University Science Books, 2002 [2] R. Burridge, Lamb’s problem for an anisotropic half-space, The Quarterly Journal of Mechanics and Applied Mathematics, vol. 24, (1971), , 81-98 [3] L. Cagniard, Refection and refraction of progressive seismic waves, translated and revised by E. A. Flinn and C. H. Dix, (1962), McGraw-Hill, New York. [4] AT De Hoop, A modification of CagniardÂa´ ˛ rs method for solving seismic pulse problems, Applied Scientific Research, Section B, vol. 8, (1960), 349-356. [5] S. Deng; W.K. Wang; S.-H. Yu, On 2x2 hyperbolic systems, [6] L. Hörmander, The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients, Springer, 2004. [7] F. John, Partial Differential Equations, 4th ed, Spring, 1981. [8] L.R. Johnshon, Green’s Function for Lamb’s Problem, Geophys. J. R. asfr. Soc. 37, (1974), 99-131. [9] H. Lamb, On the propagation of tremors over the surface of an elastic solid, Philosophical Transactions of the Royal Society of London. Series 81 R EFERENCES A, Containing Papers of a Mathematical or Physical Character, vol. 203, (1904), pp 1-42. [10] EM. Lifshitz; LD. Landau, Theory of elasticity, Course of Theoretical Physics, Oxford , New York , Pergamon Press, 1970 [11] E. Kausel, Lamb’s problem at its simplest, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, (2012). [12] T.-P. Liu, S.-H. Yu, On boundary relation for some dissipative systems, Bull. Inst. Math. Acad. Sin. (NS), (2011), no. 3, 245-267. [13] T.-P. Liu, S.-H. Yu, Dirichlet-Neumann kernel for dissipative system in half-space, preprint. [14] T.-P. Liu, S.-H. Yu, Determinant of the linearized compressible NavierStokes equation in 2-D half space, print [15] T.-P. Liu, S.-H. Yu, 2-D viscous shock wave and Surface wave,in preparation [16] H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298. [17] J. Pujol, Elastic wave propagation and generation in seismology, Cambridge University Press, 2003. [18] M. Rahman, JR. Barber, Exact expressions for the roots of the secular equation for Rayleigh waves, Journal of applied mechanics, (1995) 62, 250. [19] L. Rayleigh, On Waves Propagated along the Plane Surface of an Elastic Solid, Proceedings of The London Mathematical Society, vol. s1-17, (1885), pp 4-11. 82 R EFERENCES [20] P.G. Richards, Elementary solutions to Lamb’s problem for a point source and their relevance to three-dimensional studies of spontaneous crack propagation, Bulletin of the Seismological Society of America, (1979) vol. 69, 947-956. [21] R. Sakamoto, E-well posedness for hyperbolic mixed problems with constant coefficients, J. Math. Kyoto Univ., 14 (1974), 93-118. [22] JR Willis, Self-similar problems in elastodynamics, Philosophical Transactions for the Royal Society of London. Series A, (1973), 435-491 83 [...]... differential equations at x = ∞, d and similarly one can identify the denominator Sij (iη, iη, s, ξ T , ξ L ) as an im- plicit balance between PDE at x = ∞ and boudary condition It is a common sense in science to make every quanitities into the same UNIT in order to make comparison Now, the only unit in the problem within our imagination is polynomial This concept leads to characteristic-non characteristic... assumption will exclude the exponential growth component in (1.3.9) It gives rise to the following Master Relationship: 15 C HAPTER 1: D ERIVATION OF S OLUTION F ORMULA Definition 1.3.2 (Master Relationship) For each (η, ζ, s) ∈ R × R × R+ , 0 = M(iη, iζ, s; D, N) ≡ Res soln(ξ, iη, iζ, s; D, N) ξ =ξ ∗ p(ξ ∗ ,iη,iζ,s)=0 Re(ξ ∗ )>0 (1.3.11) The master relationship (1.3.11) and the boundary condition (1.3.5)... together conclude Theorem 1.1.1 This gives the final composition of the explicit solution formula of the surface wave, (a(0, y, z, t), a x (0, y, z, t)) in terms of the given inhomogeneous term gb (y, z, t) 1.4 Master Relationship, boundary condition, and matrix (Sij )6×3 The master relationship (1.3.11) with ξ ∗ ∈ {ξ L , ξ T } will pose 6 equations, but there are only 3 linearly independent equations The... compressible NavierStokes equation [14] With above five components, the solution formula for any 2×2 hyperbolic system in a 2-D half-space domain was obtained in [5] with any arbitrary wellposed boundary condition In Section 2, the preliminaries materials are given In Section 3, we will give the LY algorithm to conclude Theorem 1.1.1 as the main program of the paper; and design Sections 4,5,6 as the subroutines... is the solution given by Corollary 1.1.5 with the given inhomogeneous boundary data gb (y, z, t) as the one given in (1.1.15) The ingredients of the LY algorithm in Section 3 are given in a logical order below: 1 A fundamental solution: to shift initial data to boundary data 10 C HAPTER 1: D ERIVATION OF S OLUTION F ORMULA 2 A master relationship: An intrinsic algebraic relationship among the full... solution of differential equation with zero initial data 3 Algebraic solution of the full boundary data in the transform variables 4 An algebraic characteristic-non characteristic decomposition of the symbols of boundary data: To decompose the symbols into a polynomial in ∂s ξ L /s, and ∂s ξ T /s over the ring spanned by rational functions in η, ζ, and s The denominator of the rational function gives... Well-Posedness, Master Relationship, and solution of boundary data in transform variables The function soln is rational function in ξ so that one can perform the inverse transform in the x-variable for given (η, ζ, s) ∈ R × R × R+ : eξ ∗ x Res soln(ξ, iη, iζ, D, N) ∑ L[a]( x, iη, iζ, s) = p(ξ ∗ ,iη,iζ,s)=0 (1.3.9) ξ =ξ ∗ The well-posedness assumption: For each (η, ζ, s) ∈ R × R × R+ , the solution a( x, t) satisfies... 1 2πi Re(s)=0 12 est G (s)ds C HAPTER 1: D ERIVATION OF S OLUTION F ORMULA Proposition 1.2.4 Suppose that g ∈ L∞ (0, ∞) and its Laplace transform G (s) = ∞ −st g(t)dt 0 e is a rational function of s Then, Res G (s) = 0 G (s∗ )=0 Re(s∗ )>0 s=s∗ Proposition 1.2.5 (Fourier Transform) The Fourier transform of the solutions of the d’Alembert wave equations given (1.1.11) are   ˆ sin( ξ 2 + η 2 + ζ 2 t)... ζ, s, a, and b Remark 1.3.3 In Section 4, we will give the expression of matrix (Sij )6×3 At least, we will write down the rational function S22 (iη, iζ, s, a, b) i.e d n S22 (iη, iζ, s, a, b)/S22 (iη, iζ, s, a, b) explicitly iii Characteristic-non characteristic decomposition of the symbols Sij , Determinant of Rayleigh Wave 16 C HAPTER 1: D ERIVATION OF S OLUTION F ORMULA d The symbols 1/Sij (iη,... integral would only contain c1 part and the integral along branch cut Case2 For the two poles in the left half space, one can compare their coefficients with the coefficients of the two poles in the right half space Then, as the symmetric property of these poles, the contribution of the two conjugate poles in the left half space would be canceled just like the two in the right half space Then we can conclude . 13 1.4 Master Relationship, boundary condition, and matrix (S ij ) 6×3 . 24 1.5 Characteristic-non characteristic decomposition . . . . . . . . . 26 1.6 Realization of decomposition . . . . . . practice to continue when an initial-boundary value problem encountered. A good candidate for such a new input is the construction of an explicit solution formula of the Green’s function for a constant. constant coefficient problem in a half space domain. The advantage of an explicit solution formula of the Green’s function is that one can represent the solution of a linear or nonlinear problem by the Duhamel’s