Wave Motion 39 (2004) 191–197

On formulas for the Rayleigh wave speed
Pham Chi Vinh a , R.W. Ogden b,∗ a Faculty of Mathematics, Mechanics and Informatics, Hanoi National University, 334, Nguyen Trai Street, Thanh Xuan, Hanoi, Vietnam b Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK

Received 23 June 2003; received in revised form 18 August 2003; accepted 25 August 2003

Abstract A formula for the speed of Rayleigh waves in isotropic materials is obtained by using the theory of cubic equations. It is expressed as a continuous function of a certain material parameter. The formula obtained by Malischewsky [Wave Motion 31 (2000) 93] is explained on the same basis and its connection with our formula is identified. © 2003 Elsevier B.V. All rights reserved.

1. Introduction A formula for the Rayleigh wave speed in compressible isotropic elastic solids was first obtained by Rahman and Barber [1] for a limited range of values of the parameter γ = µ/(λ + 2µ), where λ and µ are the usual Lamé constants, by using the theory of cubic equations. Employing Riemann problem theory Nkemzi [2] derived a formula for the speed of Rayleigh waves expressed as a continuous function of γ for any range of values. It is rather cumbersome [3] and the final result as printed in his paper [2] is incorrect [4]. Recently, Malischewsky [4] obtained a formula for the speed of Rayleigh waves for any range of values of γ by using Cardan’s formula together with trigonometric formulas for the roots of a cubic equation and MATHEMATICA. It is expressed as a continuous function of γ and contains a signum function, specifically sign(−γ + 1/6). Malischewsky considered his formula as probably the simplest representation for the Rayleigh wave speed in compressible isotropic materials. In Malischewsky’s paper [4] it is not shown, however, how Cardan’s formula together with the trigonometric formulas for the roots of the cubic equation are used with MATHEMATICA to obtain the formula. The aim of the present paper is twofold: first, to present a technique, based solely on the theory of cubic equations, with which to obtain a formula for the Rayleigh wave speed; second, to explain Malischewsky’s formula, in particular to clarify the role of the function sign(−γ + 1/6) in his formula. It is interesting that this function does not appear in our formula.
∗ Corresponding author. Tel.: +44-1413304550; fax: +44-1413304111. E-mail address: rwo@maths.gla.ac.uk (R.W. Ogden).

0165-2125/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2003.08.004

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2. Secular equation It is well-known that for compressible isotropic elastic solids the original secular equation for Rayleigh waves has the form √ (2 − x)2 = 4 1 − x 1 − γx, (1) (see [5]) where x= c2 , 2 c2
2 c2 =

µ , ρ

0 < x < 1, 0 < γ ≡

µ < 1, λ + 2µ

(2)

where ρ is the mass density of the material, c the Rayleigh wave speed and µ and λ are the classical Lamé moduli. Note that strictly it suffices to restrict attention to 0 < γ < 3/4, the upper bound corresponding to the usual inequality 3λ + 2µ > 0, but the analysis applies equally for the extended range of admissible values of γ in (2), which was also used by some authors. It is not difficult to verify that Eq. (1) is equivalent to the equation √ (1 − x)[4(1 − γ) − x] − x 1 − x 1 − γx = 0. (3) It is noted that Eq. (3) can be obtained directly from the secular equation for Rayleigh waves in pre-stressed compressible elastic materials [6] by specializing to the case with no pre-stress. In terms of the variable η defined by η= 1 − γx , 1−x x= 1 − η2 , γ − η2 η ∈ (1, ∞), (4)

Eq. (3) becomes f(η) ≡ η3 + a2 η2 − η + a0 = 0, a0 = −(1 − 2γ)2 , (5)

where the coefficients a0 and a2 are given by a2 = 4γ − 3. (6) Since f(1) = −4(1 − γ)2 < 0 and f(η) → ∞ as η → ∞, Eq. (5) has at least one root in the interval (1, ∞). From (5) we have f (η) = 3η2 + 2a2 η − 1. (7)

2 The discriminant of the equation f (η) = 0 is 4(a2 + 3) > 0; hence the equation has two distinct real roots, denoted by ηmin and ηmax , corresponding, respectively, to the minimum and maximum values of the function f . It follows from (7) that ηmin ηmax = −1/3 < 0 and hence

ηmax < 0 < ηmin .

(8)

Uniqueness of solution of Eq. (5) in the interval (1, ∞) is therefore assured. Note that if Eq. (5) has two or three distinct real roots, the largest one corresponds to the Rayleigh wave.

3. A formula for the wave speed As mentioned in Section 2, in order to find the Rayleigh wave speed we have to find the largest (real) root of Eq. (5), which we denote by η0 . On introducing the variable z defined by z = η + 1 a2 3 (9)

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193

Eq. (5) becomes z3 − 3q2 z + r = 0, where q=
1 3 1 2 a2 + 3 ≡ 2 (ηmin − ηmax ),

(10)

r=

3 1 27 (2a2

+ 9a2 + 27a0 ).

(11)

Our task is now to find the largest (real) root z0 of Eq. (10). According to the theory of cubic equations, the three roots of Eq. (10) are given by Cardan’s formula (see [7]) in the form √ √ 1 1 1 1 z1 = S + T, z2 = − 2 (S + T) + 2 i 3(S − T), z3 = − 2 (S + T) − 2 i 3(S − T), (12) where i2 = −1 and √ 3 S = R + D, √

T =

3

R−

D,

1 D = R2 + Q3 , R = − 2 r, Q = −q2 .

(13)

Remark. (a) The cube root of a negative real number is taken as the real negative root; (b) if, in the expression for √ S, R + D is complex, the phase angle in T is taken as the negative of the phase angle in S, so that T = S ∗ , where S ∗ is the complex conjugate of S. The nature of the roots (12) of Eq. (10) depends on the sign of its discriminant D. In particular, • if D > 0, (10) has one real root and two complex conjugate roots; • if D = 0, (10) has three real roots, at least two of which are equal; • if D < 0, (10) has three distinct real roots. Now we show that in each case the largest real root z0 of (10) is given by z0 = z1 =
3

R+

√

D+

3

R−

√

D

(14)

in which each square and cube radical is understood as the complex root taking its principal value. If D > 0, it is clear that Eq. (10) has only one real root, namely z0 given by z1 in (12), in which the radicals are understood as real roots. Since 3λ + 2µ > 0 it follows from (6) that a2 < 0. This, together a0 < 0 and (11)2 implies r < 0 and hence, by (13)4 , R > 0. Because the value of a real cube root of a positive real number coincides with √ the principal value of its correspondent complex root and R > 0, R > D, it is clear that z0 is given by (14). If D = 0 then r = −2q3 and hence Eq. (10) reduces to (z + q)2 (z − 2q) = 0, which has two distinct real roots, namely z = −q (double root) and z = 2q. Hence, in this case z0 = 2q. Also, R = q3 and the formula (14) is therefore valid. If D < 0 then Eq. (10) has three distinct real roots given by (12) and (13), in which complex cube (square) roots can take one of three (two) possible values such that T = S ∗ . In our case we take their principal values and indicate that z1 , as expressed by (12)1 , is the largest real root of (10), so that again (14) is valid. Through the rest of this section, for simplicity, we take the complex roots to have their principal values. From (13) we have S=
3

R + i −R2 − Q3 ,

T = S∗. √

(15)

Since R > 0, the phase angle of the complex number R + i −D is contained in the interval (0, π/2), and hence the phase angle θ of S is in the interval (0, π/6). From (15) this implies that |S| = q, and hence S and T can be expressed as π S = q eiθ , T = q e−iθ , 0 < θ < , (16) 6

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where θ ∈ (0, π/6) satisfies the equation r cos 3θ = − 3 , 2q which is derived by substituting z = S + T = 2q cos θ in Eq. (10). Note that, for D < 0, | − (0, π/6). From (12) and (16) it is easy to verify that z1 = 2q cos θ, z2 = 2q cos θ+ r/2q3 |

(17)

(18) < 1, which ensures Eq. (17) has a unique solution in the interval

2π , 3

z3 = 2q cos

θ+

4π . 3

(19)

Then, from (19), since θ ∈ (0, π/6), it is clear that z1 > z3 > z2 , i.e. z1 is the largest real root of (10) and (14) is once more valid. After some manipulation we obtain, by use of (6), (11) and (13), R = 2(27 − 90γ + 99γ 2 − 32γ 3 )/27, 4 ρc2 = 4(1 − γ) 2 − γ + µ 3 √ D = 4(1 − γ)2 (11 − 62γ + 107γ 2 − 64γ 3 )/27. √
−1

(20)

Using (3), (4), (9) and (14) we obtain finally a formula for the wave speed:
3

R+

D+

3

R−

D

.

(21)

In (21), R and D are given by (20), and we emphasize that the roots are understood as their principal values. 4. Connection with Malischewsky’s formula This section is devoted to an explanation of Malischewsky’s formula for the Rayleigh wave speed [4] based on the analysis presented in Section 3. The role of the function sign(−γ + 1/6) in his formula is clarified. First, we recall that Malischewsky started from the well-known form of the secular equation obtained by squaring (1) and rearranging (see [5]). This may be written as F(x) ≡ x3 + a2 x2 + a1 x + a0 = 0, where a2 = −8, a1 = 8(3 − 2γ), a0 = −16(1 − γ) (23) and x is defined in (2). Note that a0 and a2 differ from those in (5). Malischewsky used Cardan’s formula together with trigonometric formulas for the roots of a cubic equation and MATHEMATICA and he extracted (but without showing how) a formula for x, and hence for the Rayleigh wave speed c, namely x = 2 [4 − 3
3

(22)

h3 (γ) + sign[h4 (γ)] 3 sign[h4 (γ)]h2 (γ)],

(24)

where the functions hi (γ), i = 1, 2, 3, 4, are defined by √ h1 (γ) = 3 3 11 − 62γ + 107γ 2 − 64γ 3 , h3 (γ) = 17 − 45γ + h1 (γ), h2 (γ) = 45γ − 17 + h1 (γ), (25)

h4 (γ) = −γ + 1 . 6

We note that, from three possible values of the cube roots in (24), Malischewsky used those located in the first and fourth quadrants depending the sign of the imaginary part of the radicand. This means that the complex cube roots in (24) are understood to take their principal values.

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From (22) it is clear that F(0) < 0, F(1) > 0 and hence Eq. (22) has at least one root in the interval (0, 1). From (22) we have F (x) = 3x2 − 16x + 8(3 − 2γ). (26)

Thus, F (x) ≥ 0 for all x if γ ≤ 1/6, while, if γ > 1/6, F (x) has two distinct real zeros, denoted as xmin and xmax and satisfying xmin xmax = 8 (3 − 2γ) > 3 and hence 0 < xmax < 1 < xmin or 1 ≤ xmax < xmin . (28)
8 3

(27)

It is easy to verify that uniqueness of solution of Eq. (22) in the interval (0, 1) is ensured. In the case that Eq. (22) has two or three distinct real roots it is the smallest root (see also [1]). We now define the new variable z by z = x + 1 a2 3 and the parameters q and r by q2 = 1 (8(6γ − 1)), 9 z3 − 3q2 z + r = 0. r=
1 27 (16(17 − 45γ))

(29)

(30)

so that Eq. (22) may be written in the same form as (10), namely (31)

We emphasize, however, that q and r differ from the values in (11) and, in particular, here q2 can be negative. The expressions for z in terms of x are also different in the two situations. We now examine the distinct cases dependent on the values of γ in order to explain the formula of Malischewsky. For this purpose we use the theory of Section 3, in particular, the notation defined in (13). In respect of (31) the values of Q, R and D are R=
1 27 (8(45γ

− 17)),

Q = 1 (8(1 − 6γ)), 9

D=

1 27 (64(11 − 62γ

+ 107γ 2 − 64γ 3 ))

(32)

and the connections with the functions in (25) should be noted, in particular D = 8h1 (γ)/27. With D(γ) regarded as a function of γ it is easy to check that D (γ) < 0 for all γ, that D(1/6) > 0, D(17/45) < 0 and hence that the solution, γ = γ ∗ say, of D(γ) = 0 is unique and such that 1/6 < γ ∗ < 17/45. The significance of the values 1/6 and 17/45 can be seen by reference to (25). • Case 1: 0 < γ ≤ 1/6. In this case Q ≥ 0, R < 0, D = R2 + Q3 > 0. This implies that Eq. (31) has only one real root, given by the first equations in (12) and (13). Let this be denoted as z0 . Then, z0 is given by √ √ 3 3 z0 = R + D + R − D, (33) in which the roots are to be understood as real roots. Since R < 0 and Q ≥ 0 it follows that √ √ R − D < 0, R + D ≥ 0. The solution z0 can therefore be expressed as √ √ 3 3 z0 = − −R + D + R + D,

√

(34)

(35)

in which the roots are understood as complex roots with their principal values. From (29) and (35), taking into account the fact that h4 (γ) ≥ 0, we deduce that (24) holds.

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• Case 2: 1/6 < γ < γ ∗ The value γ ∗ , given approximately by γ ∗ = 0.3214984 (see [1]), is the unique root of the equation D(γ) = 0. Note that an √ exact formula √ γ ∗ has been given by Malischewsky [4]. In this case, noting that Q < 0, R < 0, for D > 0, R − D < 0, R + D < 0, we see that, analogously to Case 1, Eq. (31) has only one real root, denoted as z0 and given by √ √ 3 3 z0 = − −R + D − −(R + D), (36) in which the roots are complex roots with their principal values. Since −γ + 1/6 < 0, equations (29) and (36) lead again to (24). • Case 3: γ = γ ∗ When γ = γ ∗ , D = 0 and R < 0. Hence R = −q3 , r = 2q3 and Eq. (13) becomes (z − q)2 (z + 2q) = 0. (37) In this case the relevant solution is z0 = −2q and (24) is applicable. • Case 4: γ > γ ∗ For γ > γ ∗ we have D < 0. In this case, Eq. (31) has three distinct real roots. Following arguments presented in Section 3, the smallest real root is 2q cos (θ + 2π/3), where θ ∈ (0, π/3) is defined by (17). To ensure that (24) is valid in this case we now show that √ √ 2π 3 3 − −R + D − −(R + D) = 2q cos θ + , (38) 3 where the roots are complex roots with their principal values. Following the method used in Section 3, Arg(R + √ √ √ D) = 3θ, Arg(R− D) = −3θ, θ ∈ (0, π/3) being the solution of (17), and hence Arg[−(R+ D)] = 3θ −π, √ √ Arg[−(R − D)] = −3θ + π. Since | − (R + D)| = q it follows that √ √ 3 3 −(R + D) = q ei(θ−π/3) , −(R − D) = q ei(−θ+π/3) , (39) where the roots are complex roots taking their principal values. From (39) we deduce that √ √ 2π π 3 3 − −R + D − −(R + D) = −2q cos θ − = 2q cos θ + 3 3 and (38) is established. From the above arguments we see that the essential role of the function sign[h4 (γ)] is to change real roots to complex ones with their principal values. All four cases can therefore be embraced by a single formula that expresses x, and hence the Rayleigh wave speed c, as a continuous function of the parameter γ. In conclusion, we note that the formula (21) does not contain the signum function and the value of γ = 1/6 is not of special significance except that it is the value for which the first and second derivatives of the cubic (22) with respect to x vanish together, as pointed out in [4]. The expressions for the wave speed given by (21) and (24) are two equivalent formulas. The first is the solution of Eq. (3) for x ∈ (0, 1), while the second is the solution of (22) also for x ∈ (0, 1), on which interval the two equations are equivalent. As a check on the equivalence, we have used MATHEMATICA to plot each solution as a function of γ and the plots are indistinguishable. To translate directly from one formula to the other, however, is extremely cumbersome algebraically.

(40)

Acknowledgements The work is partly supported by the Ministry of Education and Training of Vietnam and completed during a visit of the first author to the Department of Mathematics, University of Glasgow, UK.

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