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The Stroh formalism
for elastic surface waves
of general profile
rspa.royalsocietypublishing.org
D. F. Parker
School of Mathematics and Maxwell Institute for Mathematical
Sciences, University of Edinburgh, Edinburgh EH9 3JZ, UK
Research
Cite this article: Parker DF. 2013 The Stroh
formalism for elastic surface waves of general
profile. Proc R Soc A 469: 20130301.
http://dx.doi.org/10.1098/rspa.2013.0301
Received: 9 May 2013
Accepted: 27 August 2013
Subject Areas:
applied mathematics, wave motion
Keywords:
generalized Stroh formalism, surface waves,
arbitrary waveform, conjugate harmonic
functions, transfer matrix, Hilbert transform
Author for correspondence:
D. F. Parker
e-mail: [email protected]
The Stroh formalism is widely used in the study of
surface waves on anisotropic elastic half-spaces, to
analyse existence and for calculating the resultant
wave speed. Normally, the formalism treats complex
exponential solutions. However, since waves are
non-dispersive, a generalization to waves having
general waveform exists and is here found by various
techniques. Fourier superposition yields a description
in which displacements are expressed in terms of
three copies of a single pair of conjugate harmonic
functions. An equivalent representation involving just
one analytic function also is deduced. Both these show
that at the traction-free boundary just one component
(typically the normal component) of displacement
may be specified arbitrarily, the others then being
specific combinations of it and its Hilbert transform.
The algebra is closely related to that used for complex
exponential waves, although the surface impedance
matrix is replaced by a transfer matrix, which better
embodies the scale invariance properties of the waves.
Using the scale-invariance property of the boundaryvalue problem, a further derivation is presented in
terms of real quantities.
1. Introduction
For analysis of the structure and existence of guided
waves at the surface of an elastic half-space or at
the interface between two such uniform half-spaces
of general anisotropy the formalism due to Stroh [1]
has proved useful, as evidenced by the extensive
treatments by Barnett & Lothe [2,3], Chadwick &
Smith [4] and Ting [5] and more recently through
extensions to other geometry by Fu et al. [6]. Since
much of this work concerns non-dispersive guided
wave solutions (i.e. with wave speed independent
of wavelength), a generalization is readily available.
2013 The Author(s) Published by the Royal Society. All rights reserved.
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Using a standard notation for linear elasticity in which ui (i = 1, 2, 3) are the components of the
displacement vector u, while cijkl are the elastic coefficients and x1 , x2 and x3 are the cartesian
..................................................
2. A generalized Stroh formalism for elastic surface waves
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Indeed, any instance of non-dispersive waves should immediately suggest that the governing
equations define no natural length scale.
The usual Stroh formalism treats only sinusoidal (i.e. complex exponential) solutions to the
governing partial differential equations and boundary conditions. However, since the work of
Friedlander [7] and Chadwick [8], it has long been known that a representation of guided
surface and interface waves of general form is available in terms of a single analytical function
of a complex variable bounded in an abstract half space (or, equivalently, in terms of a pair of
harmonic conjugate functions). This account generalizes the Stroh formalism to waveforms
of arbitrary profile and shows how features usually associated with the real and imaginary
parts of the complex exponential wave have close parallels with those of the real and imaginary
parts of a single analytic function or, equivalently, of a harmonic function and its harmonic
conjugate. It also helps in clarifying how much freedom there exists in assigning the profile
of an arbitrary surface wave—a property that applies in other physical situations without a
length scale, such as piezoelectric surfaces, elastic interfaces and a tangential discontinuity in
magneto-hydrodynamics.
In §2, the standard Stroh treatment [9] of elastic surface waves is reformulated in terms of a
displacement gradient vector and the traction vector, since together they satisfy two coupled firstorder partial differential equations. This emphasizes the scale invariance [10,11] of the governing
boundary-value problem, the fact of which causes sinusoidal surface waves to be non-dispersive,
with deformation distribution scaling inversely with the wave number k. (It may be noted that
Stroh [1] expressed all displacements and three stress functions as linear combinations of just three
analytic functions, but in treating surface waves he restricted attention to complex exponential
functions). The account is largely parallel to standard treatments, but leads to replacement of
the surface-impedance matrix by a transfer matrix, which is simply related to it. In §3, Fourier
superposition of wavelengths shows how general disturbances travelling uniformly at specified
speed v may be written in terms of the real and imaginary parts of three analytic functions which
decay away from the boundary of a half space. Moreover, imposing the boundary condition
of zero traction shows that the three analytic functions are identical apart from scaling by the
components of a (complex-valued) eigenvector and produces algebra simply adapted from that
as in [9] for complex-exponential waveforms. Indeed, the speed v of surface waves must make a
transfer matrix singular, a condition which has almost identical form to the equivalent condition in
[9] for the surface-impedance matrix. Also, a representation for displacements within the travelling
wave is obtained, expressing them as a linear combination involving three differently mapped
copies of a single pair of conjugate harmonic functions. In §4, the representation is reinterpreted to
emphasize that these waves may have one component of displacement at the boundary (usually
the normal component) arbitrarily specified. Choosing this as the boundary data for the harmonic
function which is the real part of the required analytic function then completely determines that
analytic function, since it must decay away from the boundary of a half-plane. Since the boundary
values of the two conjugate harmonic functions are related as Hilbert transforms, the
new representation shows that, in general, at most one component of displacement at the
boundary may be chosen as a localized function (i.e. having compact support). This illustrates
the, now familiar, observation that surface-guided waves have both elliptic character and
hyperbolic character.
Section 5 concludes with some comments on how this representation may be generalized to
piezo-electric waves and to interface waves, including evanescent cases.
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cijkl = cklij = cjikl
and the ellipticity condition cijkl ξij ξkl > 0 for all non-zero, real, symmetric tensors ξ . Surfaceguided waves travelling parallel to Ox1 at speed v additionally have displacement components
ui (x1 − vt, x2 ) depending only on x1 − vt and x2 and decaying as x2 → +∞. Within these waves,
üi = v 2 ui,11 , so that the governing system (2.1) becomes
Tik uk,22 + (Rki + Rik )uk,12 + (Qik − ρv 2 δik )uk,11 = 0
and
0 < x2 < ∞
ti ≡ Tik uk,2 + Rki uk,1 = 0
on x2 = 0,
(2.2)
together with
uk → 0
as x2 → +∞.
(2.3)
Here, following usual practice [9] in using the Stroh formalism, the coefficients Tik ≡ ci2k2 ,
Rik ≡ ci1k2 and Qik ≡ ci1k1 have been introduced. Then, when again following normal practice by
treating the traction components ti as primary variables, equation (2.2)1 becomes
ti,2 + [Rik uk,2 + (Qik − ρv 2 δik )uk,1 ],1 = 0.
(2.4)
Normally, the system is analysed to derive an impedance relation between ti and ui but, observing
that equation (2.1)1 is homogeneous of degree 2 in derivatives, while the boundary condition
(2.1)2 is homogeneous of degree 1, we use as supplementary variable the displacement gradient
vector p ≡ u,1 , rather than u (noting that the surface wave problem is scale-invariant, a feature very
significant in nonlinear theory (Hunter [10] and Parker and Hunter [11])). Then, using matrix
notation in which T has elements Tij , etc., the definition (2.2)2 of the traction vector t on each
plane x2 = const.
t = Tu,2 + RT p
(2.5)
may be rearranged as a formula for the remaining deformation-gradient components as
u,2 = T−1 t − T−1 RT p.
(2.6)
This may be substituted into (2.4) in its matrix form t,2 + [Ru,2 + (Q − ρv 2 I)p],1 = 0 to give
t,2 + [RT−1 t + (Q − ρv 2 I − RT−1 RT )p],1 = 0.
(2.7)
Since equation (2.5) may be differentiated to give t,1 = Tp,2 + RT p,1 , equation (2.7) should be
accompanied by
p,2 = T−1 t,1 − T−1 RT p,1
to form a first-order system of equations for the two vectors p and t.
Systems (2.7) and (2.8) may be written in a form similar to that used by Fu et al. [6] as
∂
∂
p
N1 N2
p
=
,
N3 NT1 ∂x1 t
∂x2 t
(2.8)
(2.9)
..................................................
where ρ is the density, ti (i = 1, 2, 3) are the components of traction on any surface of constant x2 ,
where commas denote partial derivatives and superposed dots denote derivatives with respect
to time t. Here, the elastic coefficients satisfy the symmetry conditions
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coordinates, the governing field equations and boundary conditions for disturbances in the
homogeneous half space x2 > 0 are
⎫
cijkl uk,lj = ρ üi 0 < x2 < ∞ ⎬
(2.1)
and
ti ≡ ci2kl uk,l = 0 on x2 = 0,⎭
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For specified v, the possible values for p are either real or occur as complex conjugate pairs, while
the solutions (2.10) decay as x2 → +∞ only if Im[p] > 0 for k > 0 and Im[p] < 0 for k < 0. Any real
values for p correspond to disturbances (2.10) which are plane waves inclined to the Ox1 axis and
so do not decay away from x2 = 0. They cannot contribute to a surface-guided wave, but do not
occur for 0 ≤ v < v̂ [1,2,9], where v̂ is the limiting velocity defined by v̂ ≡ minφ {vb sec φ} in which
vb is a speed of a bulk mode propagating in the direction of the unit vector (cos φ, sin φ, 0) within
the Ox1 x2 plane. Then, those for which Im[p] > 0 may be labelled as
p = p(q) = α (q) + iβ (q) ,
q = 1, 2, 3
(2.12)
with α (q) and β (q) their real and imaginary parts and with β (q) > 0 (so that the remaining
eigenvalues and eigenvectors are p(q)∗ = α (q) − iβ (q) ≡ p(q+3) and c(q)∗ = c(q+3) , say). This ensures
that, for each q = 1, 2, 3 and for each positive wave number k, the solution to equation (2.9)
Y(q) ≡ Re c(q) exp ik(x1 − vt + p(q) x2 )
= Re c(q) exp ik(x1 + α (q) x2 − vt) exp(−kβ (q) x2 )
(2.13)
is sinusoidal in x1 and decays as x2 → ∞. Moreover, the superposition
Y(x1 , x2 , t) = Re
3
dq c(q) exp ik(x1 − vt + p(q) x2 )
(2.14)
q=1
is a travelling sinusoidal solution, for all complex constants dq (q = 1, 2, 3). This may be split, as in
(2.10), into displacement gradient vectors and traction vectors as
p = Re
3
dq p̂(q) exp ik(x1 − vt + p(q) x2 )
and
q=1
t = Re
3
(q)
dq t̂
exp ik(x1 − vt + p(q) x2 ).
q=1
Thus, for all choices of the vector d ≡ (d1 , d2 , d3 )T and all choices of speed v(<v̂), the displacement
gradient vector and traction at the surface are related through
p(x1 , 0, t) ≡ Re Pd eik(x1 −vt)
and t(x1 , 0, t) ≡ Re Cd eik(x1 −vt) ,
where the matrices P and C are formed from the upper and lower halves p̂(q) and t̂
(q = 1, 2, 3) as
P ≡ (p̂(1) p̂(2) p̂(3) )
and
(1) (2) (3)
C ≡ (t̂ t̂ t̂ ).
(2.15)
(q)
of c(q)
(2.16)
Then, using the transfer matrix defined as L ≡ CP−1 shows how the surface traction t(x1 , 0, t) may,
for any d, be directly related to the surface displacement gradient vector p(x1 , 0, t) given by (2.15)
through t(x1 , 0, t) = Re LPd eik(x1 −vt) . This transfer matrix L depends upon v. If, for some value v,
it is singular, the traction will vanish when p(x1 , 0, t) is given by (2.15), where Pd is chosen as the
null vector of L (so that d is the null vector of C). The condition det L = 0 selecting v is analogous
to the condition that the surface-impedance matrix is singular [12] and, moreover [2,12] is known
to define a unique speed v of Rayleigh waves for arbitrary orientations of wave propagation in
arbitrary anisotropic materials.
..................................................
with k real, where c is an eigenvector and p the corresponding eigenvalue of the 6 × 6 real matrix
N arising in (2.9) and defined by
N 1 N2
c = pc.
(2.11)
Nc ≡
N3 NT1
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where, apart from the contribution ρv 2 I within N3 , the 3 × 3 submatrices N1 ≡ −T−1 RT ,
N2 ≡ T−1 and N3 ≡ RT−1 RT − Q + ρv 2 I are formed entirely from material constants. Equation
(2.9) possesses special sinusoidal wave solutions
p̂
p
ik(x1 −vt+px2 )
, where c ≡
,
(2.10)
Y≡
= Re c e
t̂
t
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RT−1 (RT − L)p̂(q) − (Q − ρv 2 I)p̂(q) = Lp(q) p̂(q) = LT−1 (L − RT )p̂(q)
for q = 1, 2, 3
so that
(L + R)T−1 (RT − L) = Q − ρv 2 I.
(2.18)
After allowing for differences of notation and upon writing M = iL, this becomes the algebraic
Riccati equation (1.2) of Fu & Mielke [9] defining the surface-impedance matrix M of [12]. This is
(q)
hardly surprising, since equation (2.10) shows that the surface traction t = Re t̂ exp(ik(x1 − vt +
p(q) x2 )) corresponds to the surface displacement u = −Re ik−1 p̂(q) exp(ik(x1 − vt + p(q) x2 )), when
(q)
= −iMp̂(q) .
In §3, the Stroh formalism for (complex) exponential waves is extended to surface waves with
general surface elevation profile u2 (x1 , 0, t) = ũ2 (x1 − vt) travelling at the Rayleigh wave speed v.
In this extension, the transfer matrix is preferred to the surface-impedance matrix, since it provides
a scale-invariant relationship between the surface displacement gradient vector p(x1 , 0, t) and the
surface traction (i.e. independent of k).
Before describing this treatment for waves with general waveform, a further analogy with [9]
is noted. Firstly, equations (2.17) are rewritten as
t̂
RT−1 (RT P − C) − (Q − ρv 2 I)P = CD and
− RT P + C = TPD = (L − RT )P.
Then, the eigenvalue problem (2.11) defining p = p(q) and p̂ = p̂(q) may be written as
{p2 T + p(RT + R) + Q − ρv 2 I}p̂ = 0,
(2.19)
which gives a sixth degree polynomial in p having real coefficients for each v 2 . When all
six roots p occur as complex conjugate pairs, selecting those with Im p > 0 and so defining
D = diag(p(1) p(2) p(3) ), equation (2.14) may be written so that
p = Re exp ik(x1 − vt)P exp(ikx2 D)d,
t = Re exp ik(x1 − vt)C exp(ikx2 D)d.
(2.20)
Note that, in (2.18), the factor T−1 (L − RT ) = PDP−1 ≡ Ê (say) has the property exp(ikx2 Ê) =
P exp(ikx2 D)P−1 (cf. equations (3.8) and (3.10) of [9]). Moreover, Ê has eigenvectors p̂(q)
with eigenvalues p(q) , so that Êp̂(q) = p(q) p̂(q) , while, by substituting from (2.18) for Q − ρv 2 I,
equation (2.19) gives
[(R + L)T−1 + p(q) I]T[T−1 (RT − L) + p(q) I]p̂(q) = 0,
which is equivalent to (3.11) of [9].
3. General waveforms and the Stroh formalism
For each q = 1, 2, 3 and each complex-valued function F(q) (k) defined on 0 < k < ∞, two real
conjugate harmonic functions φ (q) (X, Y) and ψ (q) (X, Y) may be defined through
∞
φ (q) (X, Y) + iψ (q) (X, Y) ≡
F(q) (k) eik(X+iY) dk
0
..................................................
(q)
Since p̂(q) = Peq and t̂ = Ceq = LPeq for each q = 1, 2, 3, where e1 , e2 and e3 are the standard unit
vectors and since P−1 exists, these give
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A characterization of L is available, without the need to identify the eigenvalues p(q) and the
corresponding eigenvectors of N. It is akin to that obtained by Fu & Mielke [9]. For each p(q) ,
equation (2.11) gives the pair of equations
⎫
(q)
(q)
[RT−1 RT − Q + ρv 2 I]p̂(q) − RT−1 t̂ = p(q) t̂ ⎬
(2.17)
(q)
⎭
and
− T−1 RT p̂(q) + T−1 t̂ = p(q) p̂(q) .
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so that they satisfy the Cauchy–Riemann equations
=
∂Y
∂φ (q)
and
∂Y
6
+
∂ψ (q)
∂X
= 0 in Y > 0.
(3.1)
Using these, a general Fourier superposition of each travelling sinusoidal solution Y(q) of (2.13)
may be written as
∞
(q)
F(q) (k)c(q) exp ik(x1 − vt + p(q) x2 ) dk,
Z (x1 , x2 , t) ≡ Re
0
which may be put into the form
Z(q) (x1 , x2 , t) = a(q) φ (q) (x1 − vt + α (q) x2 , β (q) x2 ) − b(q) ψ (q) (x1 − vt + α (q) x2 , β (q) x2 )
(3.2)
in terms of the vectors a(q) and b(q) which are the real and imaginary parts of c(q) , and moreover
Z(q) satisfies (2.9). Thus, Z(q) is written in terms of a pair of harmonic conjugate functions φ (q) and
ψ (q) in which the arguments are related to x1 − vt and x2 through the real and imaginary parts of
the complex eigenvalue p(q) , with multiplicative factors defined by the corresponding eigenvector.
An alternative confirmation of this representation is to use the eigenvalue property (2.11) in
the form Na(q) = α (q) a(q) − β (q) b(q) and Nb(q) = β (q) a(q) + α (q) b(q) after differentiating (3.2) and so
to show that, for Y = Z(q) ,
∂Y
(q)
(q)
= a(q) φ,X − b(q) ψ,X
∂x1
while, since Na(q) = α (q) a(q) − β (q) b(q) and Nb(q) = β (q) a(q) + α (q) b(q) , then
∂Y
(q)
(q)
(q)
(q)
= α (q) a(q) φ,X + β (q) a(q) φ,Y − α (q) b(q) ψ,X − β (q) b(q) ψ,Y
∂x2
(q)
(q)
= N(a(q) φ,X − b(q) ψ,X ) = N
∂Y
,
∂x1
with all functions evaluated at (X, Y) = (x1 − vt + α (q) x2 , β (q) x2 ).
Each of the three solutions Z(q) of (3.2) involves a pair of conjugate harmonic functions
(q)
φ (X, Y) and ψ (q) (X, Y) which may be written in terms of a single harmonic function Φ (q) (X, Y) as
(q)
(q)
φ (q) (X, Y) = Φ,X and ψ (q) (X, Y) = −Φ,Y . Then, the general disturbance travelling parallel to Ox1
at any subsonic speed v is a superposition of these and so has a representation in terms of three
separate functions Φ (q) (x1 − vt + α (q) x2 , β (q) x2 ) each harmonic in Y ≡ β (q) x2 > 0 and decaying as
Y → +∞. It then has the form:
3
∂Φ (q)
∂Φ (q)
p
(x1 − vt + α (q) x2 , β (q) x2 ) + b(q)
(x1 − vt + α (q) x2 , β (q) x2 )
Y=
=
(3.3)
a(q)
t
∂X
∂Y
q=1
and so is a superposition of vector multiples of the three pairs of harmonic functions of
appropriate variables x1 − vt + α (q) x2 and β (q) x2 .
An alternative representation uses three complex-valued functions G(q) (X + iY) = Φ (q) (X, Y) +
(q)
iΨ (X, Y) which are analytic in Y > 0, which decay as Y → +∞ and in which Φ (q) (X, Y) =
Re G(q) (X + iY). Then, equation (3.3) may be rewritten as
Y(x1 , x2 , t) = Re
3
∂ (q) (q)
c G (x1 − vt + p(q) x2 ).
∂x1
(3.4)
q=1
It should be noted that Stroh [1] implied an equivalent relation for the three displacements and
for three stress functions in the case of general subsonic, travelling disturbances, but in treating
surface waves restricted attention to complex exponential functions. Observe that, associated
..................................................
∂X
∂ψ (q)
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∂φ (q)
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with the three analytic functions G(q) (X + iY), the material displacements u(x1 , x2 , t) which satisfy
u,1 = p take the form:
p̂(q) G(q) (x1 − vt + α (q) x2 + iβ (q) x2 ).
(3.5)
q=1
Then, if v is chosen so as to make C singular with d a null vector and if the analytic functions
G(q) (X + iY) are chosen to be proportional to each other, so that on the real axis they are G(q) (X) =
dq G(X) for arbitrary G(X), the surface displacement is given by
u(x1 , 0, t) = Re
3
dq p̂(q) G(x1 − vt) = Re [Pd G(x1 − vt)] ≡ ũ(x1 − vt),
(3.6)
q=1
while the surface traction vanishes, since Cd = 0 in
(q)
∂
Re
dq t̂ G(x1 − vt) = Re [Cd G (x1 − vt)] ≡ t̃(x1 − vt).
∂X
3
t(x1 , 0, t) =
(3.7)
q=1
The corresponding displacement gradient vector at all x2 ≥ 0 is
p(x1 , x2 , t) = Re
3
dq p̂(q) G (x1 − vt + α (q) x2 + iβ (q) x2 )
q=1
and so, on the surface x2 = 0, may be written as
p(x1 , 0, t) = Re p̃G (x1 − vt) = p̃+ Φ,X (x1 − vt, 0) − p̃− Ψ,X (x1 − vt, 0),
(3.8)
where p̃ = Pd, with real part p̃+ and imaginary part p̃− , is the null vector of L = CP−1 and where
Φ(X, Y) is the real and Ψ (X, Y) is the imaginary part of the single analytic function G(X + iY).
Moreover, G(X + iY) is defined throughout Y > 0 by the boundary values Φ̃(X) ≡ Φ(X, 0) of its
real part, the harmonic function Φ(X, Y), while Ψ̃ (X) = Ψ (X, 0) is its Hilbert transform HΦ̃(X) [11].
Correspondingly, the displacement at the boundary is
u(x1 , 0, t) = Re p̃[Φ(x1 − vt, 0) + iΨ (x1 − vt, 0)] = p̃+ Φ̃(x1 − vt) − p̃− HΦ̃(x1 − vt).
(3.9)
When the null vector d of C is scaled so that the second component of the null vector p̃ of L = CP−1
has the real value +1, the surface elevation has the simple representation
u2 (x1 , 0, t) = ũ2 (x1 − vt) = Re G(x1 − vt) = Φ̃(x1 − vt),
(3.10)
while the other components of displacement are
−
ũ1 (x1 , 0, t) = p̃+
1 Φ̃(x1 − vt) − p̃1 HΦ̃(x1 − vt)
and
−
ũ3 (x1 , 0, t) = p̃+
3 Φ̃(x1 − vt) − p̃3 HΦ̃(x1 − vt).
(3.11)
This shows that all three displacement components at x2 = 0 are linear combinations of ũ2 (x1 − vt)
and its Hilbert transform, the linear combinations being determined by the real and imaginary
parts of the null vector p̃. Of course, it is also possible to describe u2 (x1 , 0, t) and u3 (x1 , 0, t) in
−
terms of Φ̂(x1 − vt) ≡ ũ1 (x1 , 0, t) by the substitution Φ̂ = p̃+
1 Φ̃ − p̂1 HΦ̃. For example, this choice
gives
ũ2 (x1 , 0, t) =
−
p̃+
1 Φ̂(x1 − vt) − p̃1 HΦ̂(x1 − vt)
−2
p̃+2
1 + p̃1
,
(3.12)
since H(HΦ̂) = −Φ̂.
Moreover, equation (3.11) shows that even if ũ2 (x1 − vt) in (3.10) is chosen as a localized
function, the remaining displacement components on x2 = 0 usually will extend over (−∞, ∞).
..................................................
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u = Re
7
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4. Representation of the general travelling wave in terms of real functions
8
Y(q) = a(q) φ (q) (x1 − vt, x2 ) − b(q) ψ (q) (x1 − vt, x2 )
for q = 1, 2, 3,
in each of which the two functions φ (q) and ψ (q) decay so that |φ (q) | → 0, |ψ (q) | → 0 as x2 → ∞.
Then by requiring that
Na(q) = α (q) a(q) − β (q) b(q)
and Nb(q) = β (q) a(q) + α (q) b(q) with β (q) > 0
(4.1)
(which is the eigenvalue problem Nc(q) = p(q) c(q) (2.11) where c(q) = a(q) + ib(q) with p(q) = α (q) +
iβ (q) and Im p(q) > 0) it is found that, for each q = 1, 2, 3, φ (q) and ψ (q) may be written in terms
of conjugate harmonic functions φ (q) = φ̂ (q) (X, Y), ψ (q) = ψ̂ (q) (X, Y) of X = x1 − vt + α (q) x2 and
(q)
(q)
(q)
(q)
Y = β (q) x2 . Furthermore, it is possible to write φ̂ (q) = Φ,X = Ψ,Y , ψ̂ (q) = −Φ,Y = Ψ,X in terms of
the conjugate harmonic functions Φ (q) (X, Y) and Ψ (q) (X, Y). By choosing these to be expressed in
terms of a single pair of conjugate harmonic functions (Φ(X, Y), Ψ (X, Y)) through
⎫
(q)
(q)
−
(q)
(q)
Φ (q) = d+
q Φ(x1 − vt + α x2 , β x2 ) − dq Ψ (x1 − vt + α x2 , β x2 )⎬
(4.2)
(q)
(q)
+
(q)
(q)
⎭
and
Ψ (q) = d−
q Φ(x1 − vt + α x2 , β x2 ) + dq Ψ (x1 − vt + α x2 , β x2 )
−
with d+
q and dq the real and imaginary parts of dq defined by the conditions LPd = 0 and with
(Pd)2 = +1(= (p̃)2 , the representation (3.3) for p and t is equivalent to (3.5) in the form
u=
3
{Re (dq p̂(q) )Φ(x1 − vt + α (q) x2 , β (q) x2 ) − Im(dq p̂(q) )Ψ (x1 − vt + α (q) x2 , β (q) x2 )}.
(4.3)
q=1
Moreover, this gives the boundary values (equivalent to (3.9))
u(x1 , 0, t) = Re (Pd)Φ(x1 − vt, 0) + Im(Pd)Ψ (x1 − vt, 0)
= p̃+ Φ̃(x1 − vt) − p̃− HΦ̃(x1 − vt)
(4.4)
+
and is consistent with u2 (x1 , 0, t) = Φ̃(x1 − vt), since (Pd)2 = +1 and where p̃ = Pd = p̃ + ip̃− .
It also recovers the relations (3.11).
..................................................
bounded in Y ≥ 0 which decays so that |G| → 0 as Y → +∞.
For arbitrary Φ̃(X), it is possible, without using Fourier superposition, to represent the
displacements u(x1 , x2 , t) within an elastic surface wave travelling at speed v and having
normal displacement at the boundary u2 (x1 , 0, t) = Φ̃(x1 − vt) in terms of Φ(X, Y) and Ψ (X, Y).
The algebra required in constructing the representation may simply be related to that
required within the construction of the standard sinusoidal (complex exponential) solutions
u = Re U(x2 ) exp ik(x1 − vt), which corresponds to Φ̃(X) = cos kX when U2 (0) = +1.
By seeking solutions in which the 6-vector Y = (p, t)T of §2 depends only upon x1 − vt and x2 ,
it follows that ∂Y/∂x2 = N∂Y/∂x1 (i.e. (2.9)). Solutions may then be constructed as a superposition
of three special solutions
rspa.royalsocietypublishing.org Proc R Soc A 469: 20130301
Corresponding to an arbitrary bounded function Φ̃(X) on (−∞, ∞) there exists a unique
bounded, harmonic function Φ(X, Y) in y ≥ 0 having boundary values Φ(X, 0) = Φ̃(X) and which
decays as Y → +∞, together with a bounded conjugate harmonic function Ψ (X, Y) satisfying
∂Ψ/∂Y = ∂Φ/∂X and ∂Ψ/∂X = −∂Φ/∂Y. The boundary values of Φ and Ψ are related as Hilbert
transforms through
1 ∞ Φ̃(s)
ds and Φ̃(X) ≡ Φ(x, 0) = −HΨ (X, 0),
Ψ (X, 0) = HΦ(X, 0) = HΦ̃(X) ≡ −
π −∞ s − X
where − denotes the principal-value integral. These are just the identities provided by the Cauchy
integral formulae relating the values on the real axis of the real and imaginary parts of an
analytic function
G(X + iY) ≡ Φ(X, Y) + iΨ (X, Y)
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5. Concluding comments
The Stroh formulation has been extended from the standard treatment of complex exponential
elastic waves on uniform anisotropic half-spaces to surface waves in which the elevation profile
is arbitrary. Much of the standard treatment readily transfers to this general case, since surface
waves on uniform media are non-dispersive (even if wave speed may depend strongly on
propagation direction). Using the (complex-valued) transfer matrix L rather than the (related)
surface impedance matrix, identifying the speed for which it is singular then constructing its
null vector p̃ = Pd and finding three vectors p̂(q) and values pq from (2.19) allows the general
displacement field to be written as (4.3). In this, it is seen that real and imaginary parts usually
associated with displacements having the respective factors cos k(x1 − vt) and sin k(x1 − vt) are
merely the factors required to multiply an arbitrary harmonic function Φ and its harmonic
conjugate Ψ , both decaying with x2 .
In fact, a more concise derivation follows from the ansatz
Y = Re
3
∂ dq c(q) G(x1 − vt + p(q) x2 )
∂x1
q=1
in terms of a single analytic function G ≡ Φ + iΨ (i.e. (3.4) with G(q) = dq G) which, when inserted
into (2.9), leads to equation (2.11) for each c(q) = c, p = p(q) . The traction-free boundary condition
t(x1 , 0, t) = 0 then gives Cd = LPd = 0 directly. It is clear from this derivation that the fact that all
derivatives in (2.9) are first derivatives (i.e. scale invariance) is crucial.
Generalizations of the current treatment to piezo-electric surface waves, Stoneley waves
and to Schölte waves are possible. All involve linear partial differential equations and scaleinvariant boundary-value problems in one half-space or two adjacent half-spaces. The eigenvalue
problems determining the wave speed and deformation field of piezo-electric surface waves were
computed in [13,14] in the context of nonlinear evolution effects. The cases both of an electrically
earthed boundary and of free space adjoining the material were studied. For Stoneley waves (at
the interface between two elastic solids) and Schölte waves (at a solid/fluid interface), the fact
that general waves in many directions simultaneously have a simple representation in terms of
harmonic functions when materials are isotropic [15], indicates that, in anisotropic cases, waves
propagating in one direction with arbitrary waveform may be treated by the generalized Stroh
formalism. The possibility of treating also evanescent cases is suggested by Parker [16].
Acknowledgement. The author thanks a referee for the observation that in 1962 Stroh used a representation which
could readily have led to these results.
..................................................
L in terms of the p̂(q) .
It may be noted from (4.4) that, in general, the three components of displacement at
the boundary are inter-related. The longitudinal and transverse components are combinations
of the normal component and its Hilbert transform, with multipliers determined as identically
those which arise for the corresponding components within a sinusoidal travelling wave. Indeed,
that standard wave solution has ũ(x1 − vt) = p̃+ cos k(x1 − vt) − p̃− sin k(x1 − vt), so that the
general surface elevation profile and its Hilbert transform are the generalizations in (4.4) of the
real and imaginary parts of exp ik(x1 − vt). Thus, the real vectors p̃+ and p̃− in (4.4) are most
readily identified from the standard complex exponential solution.
9
rspa.royalsocietypublishing.org Proc R Soc A 469: 20130301
Although derivation of (4.3) and (4.4) requires some algebra involving complex eigenvalues
and eigenvectors, the condition det L = 0 which defines the propagation speed v the matrix L
may be found directly from (2.18). The complex vectors p̂(q) ≡ p̂(q)+ + ip̂(q)− forming the columns
of P and the eigenvalues p(q) ≡ α (q) + iβ (q) may be found from (2.19) and have β (q) > 0 for each
q = 1, 2, 3. The dq are just the components of the complex vector d expressing the null vector p̃ of
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References
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rspa.royalsocietypublishing.org Proc R Soc A 469: 20130301
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