SECOND ORDER HARMONICS OF SURFACE WAVES IN ISOTROPIC SOLIDS

Journal of Sound and Vibration (1995) 187(3), 369–379

SECOND ORDER HARMONICS OF SURFACE WAVES IN ISOTROPIC SOLIDS Z. W. Q Institute of Acoustics, Academia Sinica, Beijing 100080, People’s Republic of China (Received 17 December 1993, and in final form 16 August 1994) Based on the second order surface wave equations proposed by the author, the second order harmonics of the surface wave generated in an isotropic solid by a primary bulk wave or a primary surface wave are investigated. Only the latter (not the former) ones can generate directly a second order surface wave which has a contribution to the displacements. They have an accumulation solution which increases with distance only along the normal direction to the boundary surface. It is shown by computation that the second order harmonics of the Rayleigh wave are trapped in a layer near the boundary surface which is thinner than that of the primary ones. 7 1995 Academic Press Limited

1. INTRODUCTION

Non-linear surface acoustic waves (SAWs) are of interest in both engineering and physics. For example, elastic convolvers use the non-linear effect (cf., the paper by Grasslet et al. in reference [1]). Also in reference [1], one finds that Harvey et al. investigated nonlinear elastic and piezoelectric surface waves; Abo-El-Nour Abd-Alla and Maugin dealt with non-linear SAWs on magnetostrictive crystals, finite amplitude Love waves on an incompressible elastic layered half space was discussed by Teymur, Kalyanasundaram considered non-linear mode coupling between Rayleigh and Love waves on an isotropic layered half-space, and David et al. considered the problems of non-linear evolution of piezoelectric SAWs and surface solitons. Since the second order potentical theory is convenient for dealing with boundary problems [2, 3], it is used in this paper to investigate non-linear surface waves in isotropic solids. In order to find the solutions of the inhomogeneous wave equations, Lagrange’s method of variation of parameters will be used to obtain the accumulation solutions (i.e., the solutions proportional to the distances from the boundary surface).

2. BASIC EQUATIONS

In order to investigate the boundary value problems of non-linear acoustics in cases of oblique incidence conveniently, the author proposed the theory of second order potentials [2] and subsequently the wave equations of the second order bulk and surface waves were obtained [3] by means of a Green function method as (v) q2fb−T  (2) ¯ D ), 11v=−Lv (r

(v) q2fbs−T(2) ¯ D ), 11v=−Ls (r

q2fs+L (s) (q2fs )=0,

(1–3)

369 0022–460X/95/430369+11 $12.00/0

7 1995 Academic Press Limited

. . 

370 for the longitudinal wave and

q2s c2b+T (2) 13v=−[(l+2m)/m](1/1x)

g

Lv(v) (r¯ D ) dz,

(4)

q2s c2bs+T  (2) 13s=−[(l+2m)/m](1/1x)

g

Ls(v) (r¯ D ) dz,

(5)

q2s c2s=[(l+2m)/m](1/1x)

g

q2fs dz.

(6)

for the transverse wave, respectively, where the meanings of the mathematical symbols are as defined in reference [3]. Obviously, equations (1) and (4) are for the bulk waves and are those used in references [4–6] to investigate the reflection and refraction due to a plane boundary, in which results describe the relationships between the second-order bulk waves and the elastic constants (SOE and TOE) were presented. Equation (2), (3), (5) and (6) are for the second order surface waves, in which fs and c2s are independent of the primary waves. These were used in reference [7] to deal with the second order surface waves. In this paper a further analysis of them is given. 3. SECOND ORDER SURFACE WAVES GENERATED BY A PRIMARY BULK WAVE

As pointed out in reference [3], if a primary wave is a bulk one one has Ls(v) (r¯ D )=−[F/(4K 02 )] exp{−2kx (=z=+jx)}. Substituting this into equations (2) and (5) gives fbs=−[F/(16k 2K 02 )] exp{2kx (=z=+jx)},

c2bs=jfbs ,

K 02=kx2+K2,

(7)

where K  is the z direction wavenumber of the bulk wave. Now one can proceed to solve equation (3) as well as equation (6). Obviously, fs0=F00 exp{−2kx (=z=+jx)}, is a solution of equation (3). In order to find the accumulation solutions, one assumes them to be of the form fs=f(x, z) exp{−2kx (=z=+jx)}. Consider the expansion N

f(x, z)=s s FN,N−n x nz N−n , N n=0

where Nq0 to avoid having negative powers of z. At first, suppose N=1. One then has fs=(F00+F10 x+F01 z) exp{−2kx (=z=+jx)}.

(8)

Substituting equation (8) into equation (3) and using the residue theorem give F01=3jF10 .

(9)

-    

371

fs=[F00+F10 (x3jz)+F20 x 2+F11 xz+F02 z 2 ) exp{−2kx (=z=+jx)}.

(10)

When N=2, one has

Obviously, the first two terms in the brackets are the solutions corresponding to the cases N=0, 1, respectively. Now consider the case N=2 and write fs as fs(2) , so that one has q2fs(2)={−4k 2(F20 x 2+F11 xz+F02 z 2 )−4kx [(2jF202F11 )x +(jF112F02 )z]+2(F20+F02 )} exp{−2kx (=z=+jx)}, L (s) (q2fs )=3

1 2p

g

a

{−4k 2F20 (j 2+2xj+x 2 )−4kx [(2jF202F11 )(x+j)]

−a

+2(F20+F02 )}[2z/(j 2+z 2 )] exp{−2jkx (j+x)} dj.

(11)

Consider the following term on the right side of the second of equation (11): I'2 =3

2k 2F20 p

g

a

2zj 2 exp{−2jkx (x+j) dj. j 2+z 2 −a

Note that 2zj 2/(j 2+z 2 ) exp(−2jkx j)=2z[1−z 2/(j 2+z 2 )] exp(−2jkx j), and the term −8k 2F20 z exp(−2jkx x)

1 2p

g

a

exp(−2jkx j) dj=−8k 2F20 zd(2kx ) exp(−2jkx x),

−a

become infinite as long as kx=0, which corresponds to the situation of normal incidence on to the boundary surface. Hence, F20=0 must be required to avoid divergence of the solution (10). Thus, in L (s) (q2fs ) the terms xz and z 2 disappear and from equation (3) these terms in equation (11) also disappear, and one has F20=0, F11=0 and F02=0: i.e., the accumulation solution cannot include quadratic terms in x and z. Similarly, the accumulation terms of Nq2 will disappear in the solutions. Thus, an accumulation solution of equations (3) can be written as fs (x, z)=[F00+F10 (x3jz)] exp[−2kx (=z=+jx),

(12)

Substituting equation (12) into equation (6) gives c2s=2jfs .

(13)

From the relationship between the potentials and the displacements, us=1fs /1x−1cs /1z,

ws=1fs /1z+1cs /1x,

(14)

one has us=0, ws=0, which means that the primary bulk waves do not generate any second order surface waves. 4. SECOND ORDER SURFACE WAVES GENERATED BY PRIMARY SURFACE WAVE

When the primary waves are the surface ones, they can be expressed as [8] fs(1)=A exp(−qz−jkx x),

c2s(1)=B exp(−sz−jkx x),

(15)

. . 

372

where the time factors have been omitted and q 2=kx2−k 2,

s 2=kx2−K 2,

(16)

in which k and K are the longitudinal and the transverse wavenumbers, respectively, and kx can be determined by a characteristic equation [8]. For the sake of convenience, suppose zq0 (a situation for zQ0 can be dealt with similarly). Substituting equations (15) into equations (14) and the results into equation (3) of reference [2] gives the second order stress of the primary waves as 2 −2qz T(2) +b'12 B 2 e−2sz+b'13 AB e−(q+s)z] exp(−2jkx x), 11 A e 11 =[B'

(17)

2 −2qz T(2) +b'23 AB e−(q+s)z] exp(−2jkx x), 21 A e 13 =[b'

(18)

2 2 2 2 −2qz T(2) +b'12 B 2 e−2sz 11−{(l+3m+2m)/(l+2m)}k (q +kx )A e 33 ={[b'

+[b'13−{(l+3m+2m)/(l+2m)}2jkx s]AB e−(q+s)z} exp(−2jkx x),

(19)

where b'11=12 k 4+kx2 k 2+

m+2m 2 2 l+2m 4 qk+ k, l+2m l+2m b'21=−j

b'12=

l+2m+m 4 K, 2(l+2m)

l+3m+2m kx qk 2, m

b'13=jkx [(l+3m+2m)/(l+2m)][2skx2−q(kx2+s 2 )], b'23={[(m+m)/m]K 2+[(l+3m+2m)/m]s 2}k 2.

(20)

r¯ D=[b'31 A 2 e−2qz+b'33 AB e−(s+q)z] exp(−2jkx x),

(21)

From reference [2],

where b'31=4[(l+3m+2m)/(l+2m)]q 2k 4, b'33=2j[(l+3m+2m)/(l+2m)]/kx k 2(s+q)(kx2−sq).

(22)

According to the definitions in references [2, 3], we have L (v) (r¯ D )={b'31 A 2[exp(−2qz)−exp(−2kx z)]/(4k 2 )+b'33 AB[exp(−qz−sz) −exp(−2kx z)]/[4kx2−(s+q)2 ]} exp(−2jkx x).

(23)

Substituting equations (17) and (23) into equation (2) and taking into account that a constant amplitude surface wave A exp[−2kx (=z=+jx)] is not a contribution to the displacements (cf. the previous section), one can write equation (2) as (v) q2fs=T(2) ¯ D )=[b'l1 A 2 exp(−2qz)+b'l2 B 2 exp(−2sz) 11 −L (r

+b'13 AB exp(−sz−qz)] exp(−2jkx x),

(24)

where b'l1=b'11−b'31 /(4k 2 ),

b'l2=b'12 ,

b'l3=b'13−b'33 /[4kx2−(q+s)2 ].

(25)

-    

373

Similarly, substituting equations (18) and (23) into equation (5) gives q2s c2s=−T  (2) 13 −{(l+2m)/m}(1/1x)

g

L (v) (r¯ D ) dz=bs3 AB exp[−(s+q)z−2jkx x], (26)

where bs3=−b'23−2jkx b'33 /{(s+q)[4kx2−(q+s)2 ]}(l+2m)/m.

(27)

Now one can proceed to find the solution of the inhomogeneous equations (24) and (26), which consist of a homogeneous solution, for example, fs(h) and a special solution fs(s) : i.e., fs=fs(h)=fs(s) ,

(28)

where q2fs(h)=0.

(29)

When the right side of equation (24) is zero, the special solution f is zero. From references [9, 10], one can see that only the special solution corresponding to the first term of the right side of equation (24) is an accumulation solution, and this can be obtained by the Lagrangian method of the variation of parameters, i.e., (s) s

fs(s)=fs(s1)+fs(s2) ,

(30)

and q2fs(s1)=A 2b'l1 exp[−2qz+jkx x), q2fs(s2)=[b'l2 B 2 exp(−2sz)+b'ls AB exp(−sz−qz)] exp(−2jkx x), where fs(s2) is a non-accumulation special solution corresponding to the rest of the terms on the right side of equation (24), which can easily be obtained as fs(s2)=

$

%

b'l2 B 2 b'l3 AB e−(q+s)z exp(−2jkx x). e−2sz+ (q+s)2−4q 2 4(s 2−q 2 )

(31)

Now, to find the accumulation solution fs(s1) , suppose that fs(s1)=f1 (x) exp(−2qz)+f2 (x) exp(−2jkx x),

(32)

f01 (x)+4kx2 f1 (x)=a exp(−2jkx x).

(33)

f02 (z)−4q 2f2 (z)=(b'l1 A 2−a) exp(−2qz),

(34)

and

One then has

where a is a constant to be determined. By means of the parameter variation method [9, 10], the solutions of (33) and (34) can be expressed as f1 (x)=a[−x/(4jkx )+1/(16kx2 )] exp(−2jkx x),

(35)

f2 (z)=(b'l1 A 2−a)[−z/(4q)−1/(16q 2 )] exp(−2qz),

(36)

On the other hand, if one supposes that f02 (z)−4q 2f2 (z)=a' exp(−2qz),

(37)

. . 

374

then instead of equation (33), after substitution of equation (33) into equation (24), one will have f01 (x)+4kx2 f1 (x)=(b'l1 A 2−a') exp(−2jkx x),

(38)

where a' is another constant to be determined. Thus, the solutions of equations (37) and (38) become f1 (x)=(b'l1 A 2−a')[−x/(4jkx )+1/(16kx2 )] exp(−2jkx x),

(39)

f2 (z)=a'[−z/(4q)−1/(16q 2 )] exp(−2qz).

(40)

Note that f as well as f is a special solution so that it must be zero when the inhomogeneous term b'l1 A 2 on the right side of equation (24) equals zero. However, from equations (35) and (36), or (39) and (40), one can see that f1 (x) and f2 (z) not equal to zero even if b'l1 A 2=0, which violates the definition of a special solution unless a=0 or a'=0. Thus, one obtains the special solution fs(s1) as fs(s1)=fs(s11) , or fs(s12) , where (s1) s

(s2) s

fs(s11)=b'l1 A 2[−z/(4q)−1/(16q 2 )] exp(−2qz−2jkx x),

(41)

fs(s12)=b'l1 A 2[−x/(4jkx )+1/(16kx2 )] exp(−2qz−2jkx x).

(42)

Substituting these results into equation (28) gives the complete solutions of equation (24) as fs=fs(h)+fs(s2)+{fs(s11)

or

fs(s12) },

(43a, b)

which means that one can select only f or (not and) f as the accumulation solution. Since equation (25) has no accumulation, one can easily obtain the solution as (s11) s

(s12) s

c2s=c2s(h)+c2s(s) ,

(44)

where c2s(h) is the homogeneous solution which satisfies q2c2s(h)=0,

(45)

and c is a special solution, which can be expressed as (s) 2s

c2s(s)={AB/[(q+s)2−4s 2 ]}bs3 exp[−(q+s)z−2jkx x].

(46)

Now consider the homogeneous solutions of equations (29) and (45). Note that the only boundary surface here is z=0, so that one can write the homogeneous solution as fs(h)=H1 exp(−2qz−2jkx x),

c2s(h)=H2 exp(−2sz−2jkx x).

(47)

Substituting these results into equations (28) and (45) gives second-order potentials fs and c2s . By using equations (14) the second order displacements us(2) and ws(2) can be obtained as (some errors in equations (33)–(35) of reference [7] are corrected here) us(2)=

6

0 1

jkx b'l1 A 2 1 jk b' z+ e−2qz− 2x l2 2 B 2 e−2sz 2q 4q 2(q −s )

−

$

%

2jkx b'l3 (q+s)bs3 − AB e−(s+q)z (q+s)2−4q 2 (s+q)2−4s 2

−2jkx H1 e−2qz+2sH2 e−2sz} exp[2j(vt−kx x)],

(48)

-    

375

6 0 1

vs(2)=

b'l1 A 2 z−

1 2

$

−

1 s e−2qz+ 2 2 b'l2 B 2 e−2sz 4q 2(q −s )

%

q+s 2jkx b' − b AB e−(q+s)z (q+s)2−4q 2 ls (q+s)2−4s 2 s3

−2qH1 e−2qz−2jkx H2 e−2sz} exp[2j(vt−kx x)].

(49)

Note that it is preferable to select f rather than f as an accumulation solution because the latter does not satisfy the boundary conditions when one takes equation (47) for the homogeneous solutions (a further investigation of this problem will be given in the next section). From reference [3], the second order stresses can be written as (s11) s

(2) s33 =(l+2m)

(s12) s

1ws(2) 1u (2) (2) , +l s +T33 1z 1x

(2) s13 =m

0

1

1us(2) 1ws(2) (2) +T13 , + 1z 1x

(50, 51)

which will be zero at a free surface z=0. Substituting equations (48) and (49) into equations (50) and (51) at z=0 gives H1 and H2 as H1=[2kx sDi−(s 2+kx2 )Er ]/{4[−qskx2 m−(s 2+kx2 )(lk 2−2q 2m)]}, H2={−j[2kx qEr+(lk 2−2mq 2 )Di /m}/{4[−qskx2 m−(s 2+kx2 )(lk 2−2q 2m)]},

(52)

where Di=m{−2qD1+D0+2sD3+(s+q)(D4−D5 )−2kx (−E1+E3−E4+E5 ) −b21+b'23 =B/A=}A 2, Er=l{2kx (D1−D3−D4+D5 )}A 2+(l+2m){2qE1+E0−2sE3+(s+q)(E4−E5 ) +(b'11+b'12 =B/A=2}A 2−(l+3m+2m){(q 2+kx2 )−2kx s =B/A=}k 2A 2, D0=kx bl1 /2q,

Ds=−{kx bl2 /[2(q 2−s 2 )]}=B/A=2,

D1=D0 /4q,

D4=−{2kx bl3 /[(q+s)2−4q 2 ]}=B/A =, E0=bl1 /2,

E1=E0 /4q,

D5={(s+q)bs3 /[(s+q)2−4s 2 ]}=B/A =, E3=−{sbl2 /[2(q 2−s 2 )]}=B/A=2,

E4=−{(s+q)bl3 /[(s+q)2−4q 2}=B/A =, =B/A==2qkx /[s 2+kx2 ],

b21=jb'21 ,

E5=2kx bs3 /[(q+s)2−4s 2 ]=B/A=,

b13=−jb'13 ,

b33=−jb'33 ,

bl3=−jb'l3 .

5. ON SELECTION OF ACCUMULATION AND HOMOGENEOUS SOLUTIONS

In the previous section fs(s11) was selected as an accumulation solution, which has an increase along the normal direction to the surface. According to equation (42), however, there is another accumulation solution which has an increase along the propagation direction x of the surface waves. Furthermore, since the Rayleigh wave is a non-dispersive one, which would make the second order harmonics grow as x increases, so that fs(s12) could be selected as an accumulation solution. It is easily seen, however, that this selection will be a failure as long as the homogeneous solutions are given by equation (47). In this situation, the integral constants H1 and H2 will depend on the variable x, which is in conflict with the definition of integral constants. This problem has been investigated in references [10–12], but the question has remained open. Here one can prove that only fs(s11) is an accumulation solution, as follows.

. . 

376

T 1 Material constants Materials

No.

l(10 )

m(1012 )

l(1012 )

m(1012 )

n(1012 )

r

q/kx

s/kx

Steel Steel Aluminium Aluminum

1 2 1 2

1·11 1·11 0·57 0·58

0·821 0·82 0·276 0·26

−4·61 −3·28 −3·11 −2·24

−6·36 −5·95 −4·01 −2·37

−7·08 −6·68 −4·08 −2·76

7·8 7·8 2·7 2·7

0·87 0·87 0·89 0·89

0·38 0·38 0·36 0·36

12

It is not difficult to verify that either of fs(h)=[H3 (x−jkx z/q)+H1 ] exp(−2qz−2jkx x),

(53)

c2s(h)=[H4 (x−jkx z/s)+H2 ] exp(−2sz−jkx x),

(54)

satisfy the homogeneous wave equation, where H1 , H2 , H3 and H4 are the integral constants to be determined by the boundary conditions. This means that equations (53) and (54) can be regarded as homogeneous solutions. Substituting them into equations (43b) and (44), selecting equations (42) and (46) as the special solutions and using equations (14), (50) and (51) at the free surface z=0 one has {4[(l+2m)q 2−lkx2 ]Hs+8jkx mH4=[jbl1 A 2/kx ][(l+2m)q 2−lkx ]}x +L1 H1+L2 H2+L3=0, {2jkx qH3−(kx2+s 2 )H4−2mqbl1 A 2}x+M1 H1+M2 H2+M3=0,

(55)

where L1 , L2 , L3 , M1 , M2 and M3 are the known constants. Note that there are four integral constants H1 , H2 , H3 and H4 to be determined by the two equations (55). Obviously, all of the constants depend on the variable x, which will be in conflict with the definition of them and makes this problem not solvable. Thus, in references [11, 12] it was proposed that the coefficients standing of x in equations (53) and (54) be equal to zero, which gives [(l+2m)q 2−lkx2 ]H3+2jkx msH4=−j[bl1 A 2/4kx ][(l+2m)q 2−lkx2 ] 2jkx qH3−(kx2+s 2 )H4=bl1 A 2q/2,

Figure 1. Relative displacements vs. z/L for primary and second order Rayleigh waves in aluminum (the curves for Nos. 1 and 2 are the same). For the primary Rayleigh wave: curve 1, us0(1) (z)/us0(1) (0); curve 2, ws0(1) (z)/ws0(1) (0). For the second order Rayleigh wave: curve 3, us0(2) (z)/us0(2) (0); curve 4, ws0(2) (z)/ws0(2) (0).

-    

377

Figure 2. Relative displacements vs.z/L for primary and second order Rayleigh waves in steel (the curves for nos. 1 and 2 are the same). Key as Figure 1.

and one obtains easily Hs=bl1 A 2/[4jkx ],

H4=0.

(56)

Substituting equations (56) into equations (53) and (54) and then the results obtained into equation (43b), yields a solution fs which is the same as equation (43a), which is just an earlier selection in the previous section. Furthermore, since H4=0, c2s is the same in both situations. The result thus obtained is the unexpected one that the second order harmonic Rayleigh wave is not an accumulation one along its propagation direction. 6. NUMERICAL COMPUTATION

Substituting the relevant quantities into equations (48) and (49), and taking their real parts gives us(2)=us0(2) (z) sin (2vt−2kx x),

ws(2)=ws0(2) (z) cos (2vt−2kx x),

(57)

where us0(2) (z) and ws0(2) (z) are real. Note that the displacements of the second order Rayleigh wave have the same forms as the primary ones [8] except that the former includes an accumulation wave.

Figure 3. Displacements [us0(2) /(kx3 A 2 )]10−2 vs. z/L for aluminum. Curve 1 for no. 1, curve 2 for no. 2 (cf., Table 1)..

378

. . 

Figure 4. Displacements [ws0(2) /(kx3 A 2 )]10−2 vs. z/L for aluminum. Curve 1 for no. 1, curve 2 for no. 2 (cf., Table 1).

In Table 1 listed the second and third order elastic constants [13], by means of which the relative displacements us0(2) (z)/us0(2) (0) and ws0(2) (z)/ws0(2) (0) have been calculated from equation (57) as well as equations (48) and (49). q and s were calculated numerically from equations (2.37) of reference [8]. Table 1 lists q/kx and s/kx . In Figures 1 and 2 is shown the relationship between the relative displacements of the primary and secondary order Rayleigh waves and z/L in aluminum and steel, respectively, where L is the Rayleigh wavelength. As the aluminum and steel are considered, the maximum relative displacements of the second order wave occur closer to the boundary surface than the primary ones and thus the former are trapped in a thinner layer. Note that Table 1 lists two kinds of aluminum and the relative displacements are almost the same as each other, even though the third order elastic constants are very different, as are those of the two kinds of steel. On the other hand, when the displacements us0(2) and ws0(2) are available instead of the relative ones, in Figures 3 and 4 are shown the results for two kinds of aluminum. Obviously, they are very different. However, the differences disappear for the two kinds of steel. From Table 1, there are larger differences of third order elastic constants for the former than for the latter. 7. CONCLUSIONS

A primary bulk wave does not generate second order surface waves directly; these can be generated only by a primary surface wave. For a primary Rayleigh wave, its second order harmonic wave is an accumulation one which increases with distance only along the normal direction to the boundary surface (not along the propagation direction). For aluminum and steel, the second order Rayleigh waves are trapped in a thinner layer and their maximum displacements occur closer to the boundary surface than those of the primary waves. ACKNOWLEDGMENT

This is a project supported by NSFC. REFERENCES 1. D. F. P and G. A. M (editors) 1987 Recent Developments in Surface Acoustic Waves; Proceedings of the European Mechanics Colloquium 226. Berlin: Springer-Verlag.

-    

379

2. Z. W. Q 1989 Journal of the Acoustical Society of America 86, 1965–1967. Equations of the second-order potentials of finite-amplitude waves in elastic solids (see also the erratum in 1992 ibid. 91, 3067.) 3. Z. W. Q 1995 Journal of Sound and Vibration 186, 561–566. Equations of second-order potentials for bulk and surface waves in boundary solids. 4. Z. W. Q 1992 Proceedings of the 14th International Congress on Acoustics, A1–A13. Reflection and refraction of finite-amplitude sound wave at a plane interface between two solids. 5. Z. W. Q 1993 Science in China (Series A) 36, 1468–1478. Nonlinear acoustics bounded solid—reflection and refraction of second-order bulk waves (I): P incidence. 6. Z. W. Q 1993 Science in China (Series A) 37, 693–703. Nonlinear acoustics of bounded solid—reflection and refraction of second-order bulk waves (II): SV or SH wave incidence. 7. Z. W. Q 1993 Advances in Nonlinear Acoustics, 13th International Symposium on Nonlinear Acoustics, 490–495. Bergen: World Scientific. Second-order harmonics of a surface wave in nonlinear acoustics. 8. H. K 1953 Stress Waves in Solids. Oxford: Oxford University Press. 9. Z. W. Q 1982 Scientia Sinica (Series A) 25, 492–501. Reflection of finite-amplitude sound wave at a plane boundary of a half space. 10. Z. W. Q 1993 Acta Physica Sinica 42, 949–953. Special solution of second harmonic wave equation in nonlinear acoustics and applications to boundary-value problem (in Chinese). 11. Z. W. Q and XZiaoyu Zheng 1989 C P L 6, 306–308. A   fi   . 12. Z. W. Q and X Z 1990 Chinese Physics Letters 7, 72–74. Further investigation on reflection and refraction of finite amplitude sound wave. 13. R. T. S, R. S and R. W. B. S 1966 Journal of the Acoustical Society of America 40, 1002–1008. Third-order realistic moduli of polycrystalline metals from ultrasonic velocity measurements.