ANALI PAZU
 
Vol. 13, No. 1, pp. 15–32, June 2023 
 
 
 
https://doi.org/10.18690/analipazu.13.1.15-32.2023 
Besedilo © Pušenjak, 2023 
 
 
 
 
 
COMPUTATION OF DISPERSION 
CURVES OF RAYLEIGH-LAMB WAVES 
BY NEWTON-RAPHSON METHOD 
  
Accepted  
3. 8. 2022 
 
Revised 
2. 10. 2022 
 
Published 
9. 6. 2023 
RUDOLF PUŠENJAK 
Faculty of Industrial Engineering Novo mesto, Novo mesto, Slovenia, 
rudolf.pusenjak@fini-unm.si. 
 
CORRESPONDING AUTHOR 
rudolf.pusenjak@fini-unm.si 
 
Keywords: 
symmetric wave modes, 
asymmetric wave 
modes,  
Helmholtz decoupling, 
homotopy,  
arc-length continuation 
The paper treats the numerical construction of dispersion 
curves which belong to the propagation of Rayleigh-Lamb 
waves in isotropic elastic media. The Newton-Raphson 
method.is applied for computation of real, imaginary and 
complex wavenumbers in dependence on the frequency of the 
wave. Due to multivaluedness, unexpected jumps between 
modes of dispersion curves occur in branch tracing, which 
prevents successful direct application of the method. This 
shortcoming is eliminated in the article by introducing the arc-
length continuation method. The efficiency of this supplement 
to the Newton-Raphson method is shown on the example of 
an aluminum plate with constructions of dispersion curves for 
real, imaginary and complex wave numbers. 

ANALI PAZU
 
Let. 13, Št. 1, pp. 15–32, junij 2023 
 
 
 
https://doi.org/10.18690/analipazu.13.1.15-32.2023 
Text © Pušenjak, 2023 
 
 
 
RAČUNANJE DISPERZIJSKIH KRIVULJ 
RAYLEIGH-LAMBOVIH VALOV Z 
NEWTON-RAPHSONOVO METODO 
 
  
RUDOLF PUŠENJAK 
Sprejeto 
3. 8. 2022 
 
Recenzirano 
2. 10. 2022 
 
Izdano 
9. 6. 2023 
Fakulteta za industrijski inženiring Novo mesto, Novo mesto, Slovenija, 
e-pošta: rudolf.pusenjak@fini-unm.si. 
 
DOPISNI AVTOR 
rudolf.pusenjak@fini-unm.si 
 
Članek obravnava numerično konstrukcijo disperzijskih krivulj, 
ki pripadajo širjenju Rayleigh-Lambovih valov v izotropnih 
elastičnih sredstvih. Za izračun realnih, imaginarnih in 
kompleksnih valovnih števil v odvisnosti od frekvence 
valovanja je uporabljena Newton-Raphsonova metoda. Zaradi 
večličnosti se pri konstrukciji ob sledenju vej pojavljajo 
nepričakovani preskoki med načini valovanja disperzijskih 
krivulj, kar onemogoča uspešno neposredno uporabo metode. 
Ta pomanjkljivost je v članku odpravljena z uvedbo metode 
nadaljevanja po ločni dolžini. Učinkovitost te dopolnitve 
Newton-Raphsonove metode je prikazana na primeru 
aluminijaste plošče s konstrukcijami disperzijskih krivulj za 
realna, imaginarna in kompleksna valovna števila. 
Ključne besede: 
simetrični načini 
valovanja,  
asimetrični načini 
valovanja, 
Helmholtzova ločitev, 
homotopija, 
nadaljevanje po ločni 
dolžini 

R. Pušenjak: Computation of Dispersion Curves of Rayleigh-Lamb Waves by Newton-
Raphson Method 
17.
 
 
1 Introduction 
 
Guided waves, especially Lamb waves are ideally suited for nondestructive testing of 
large structures such as metal plates, sheets or composites [3,5]. Although the 
governing equations of propagation of Rayleigh-Lamb waves in isotropic elastic 
media have been known for a long time, the construction of dispersion curves for 
real, imaginary and complex wavenumbers, which these equations produce, is still a 
challenging problem and different approaches by many researchers were undertaken 
in the past [2,4]. Newton-Raphson iterative procedure is a classical numerical 
method having quadratic convergence, which works equally well for real, imaginary 
and complex wavenumbers, respectively. Due to this property, the method is a 
natural choice for solving dispersion relations. However, dispersion relations for 
Rayleigh-Lamb waves have transcendental character, which is the cause of 
multivaluedness. Thus, in general, many different dispersion curves determining the 
course of wavenumbers versus frequency will arise in the investigated frequency 
domain. The direct use of the Newton-Raphson method, which is usually followed 
by branch tracing through a homotopy procedure, for example by incrementing of 
the frequency parameter after the iteration procedure in some point of the frequency 
domain was successfully accomplished, seldom produces the desired result. This is 
due to sudden and unexpected jumps from one wave mode to another. To prevent 
such an undesired behaviour and to provide a feasible and powerful method of 
construction of dispersion curves in the case of transcendental dispersion relations, 
the Newton-Raphson method must be supplemented by the some kind of the 
predictor. The role of the predictor is the computation of the starting point, lying 
inside the actual basin of attraction for performing the next Newton-Raphson 
iteration step. In this paper, the arc-length continuation as predictor is proposed, 
which is proven successful for construction of dispersion curves in all 
aforementioned cases of real, imaginary and complex wavenumbers. 
 
The paper is organized as follows. In the second section, the dispersion relation for 
propagation of Rayleigh-Lamb waves in an infinitely long homogeneous isotropic 
elastic plate is derived and the physical meaning of possible appearance of real, 
imaginary and complex wave numbers, respectively is explained. The third section 
contains the general formulation of the Newton-Raphson method, which can be 
applied for computation of the complex wavenumbers as roots of dispersion 
equation, as well as to computation of real and imaginary roots, respectively, wieved 

18  ANALI PAZU.  
 
 
in numerical sense as special cases. The fourth section is devoted to solving the 
problem of uncontrolled jumping between wave modes during branch tracing by 
introducing an auxiliary procedure, called the arc-length continuation. In the fifth 
section, the construction of dispersion curves for propagation of Rayleigh-Lamb 
waves in the infinitely long, homogeneous elastic aluminium plate is presented and 
analyzed. The sixth section summarizes the conclusions and provides directions for 
further work. 
 
2 The Rayleigh-Lamb dispersion relation 
 
For the reader's convenience, a derivation of the Rayleigh-Lamb dispersion relation 
is presented in which all detailed algebraic manipulations are omitted for brevity. 
The derivation starts with governing equations for 3D linear elasticity. 
 
2.1 Equation of motion in 3D elastic solid 
 
The governing equations of 3D linear homogeneous isotropic elastic solid comprise 
a) the equations of motion of 3D elasticity 
 
,
,
i j j i i
f u       
       (1) 
(b) the stress-strain relations, representing the Hooke's law 
 
2 ,
i j k k i j i j
       
      (2) 
(c) the strain-displacement (or Cauchy's) relations 
 
 
1
, ,
2
,
i j i j j i
u u   
       (3) 
where u i  are the displacement components, σ ij  are components of the stress tensor, 
ε ij  are components of the deformation tensor, ε kk  denotes the trace of deformation 
tensor, f i  are components of volume force, 𝜌 is the density, λ and μ are Lamé 
coefficients. The Einstein's summation convention is assumed for i=1,2,3. 
Substituting first the strain-displacement relations (3) into Eq. (2) and then the 
resulting stress-displacement relations into Eq. (1), the Navier's equation of motion 
is obtained 

R. Pušenjak: Computation of Dispersion Curves of Rayleigh-Lamb Waves by Newton-
Raphson Method 
19.
 
 
 
, ,
.
j j i i j j i i
u u f u           
     (4) 
 
The Navier's equation of motion can be expressed in the compact vector form 
 
   
2
, , , , u v w               u u f u u  
   (5) 
where ∇
 is the Laplace operator. In the derivation of the Rayleigh-Lamb dispersion 
relation, the body forces are assumed to be zero, giving the reduced form of the 
equation of motion 
 
 
2
,            u u u  
       (6) 
 
which is a coupled system of equations in the three displacement components, 
denoted conventionally by u, v and w. This system can be uncoupled by using the 
Helmholtz decomposition [1] in which the displacement vector u is expressed as the 
sum of the gradient of the scalar potential function φ and curl of the vector potential 
function 𝚿  
 
, 0.         u ψ ψ
      (7) 
 
Substituting Eq. (7) into equation of motion (6), taking into account the rules 
∇ ∙ ∇𝜑 = ∇
 𝜑 and ∇ ∙ ∇ × 𝚿 = 0, we obtain the equation 
 
 
2 2
2 .       
   
         
   
ψ ψ 0    
   (8) 
 
Equation (7) therefore satisfies the equation of motion (6), if it satisfies the following 
uncoupled wave equations 
 
 
2
2 2 2 2
1
2 0 , ,
L
L
c
c
 

      

           
  (9) 
2
2 2 2
1
, ,
T
T
c
c


         ψ ψ 0 ψ ψ    
   (10) 

20  ANALI PAZU.  
 
 
 
where c L is the longitudinal wave velocity and c T is the transverse wave velocity. 
Evidently, the longitudinal wave velocity is greater than the transverse wave velocity, 
what can be explained by taking into account that shear stiffness has the stiffening 
effect on the material. 
 
  

R. Pušenjak: Computation of Dispersion Curves of Rayleigh-Lamb Waves by Newton-
Raphson Method 
21.
 
 
2.2 Raylegh-Lamb Waves in elastic plate with traction-free surfaces 
 
In the following, we limit the topic to the problem of the propagation of Rayleigh-
Lamb waves in a homogeneous isotropic elastic plate. Then it suffices, to solve the 
wave equations (9) and (10) under plane strain condition and by assumption of 
traction-free surfaces. The geometry of the plate can be reduced on twodimensional 
problem in spatial coordinates x and z, while the coordinate y is omitted due to 
invariability of the solution in any plane y=const, see Figure 1. 
 
 
  
Figure 1: Propagation of a) symmetric and b) asymmetric Rayleigh-Lamb wave modes in a 
thin homogeneous isotropic elastic plate. 
Source: own. 
 
The scalar potential function in two variables is φ= φ(x, z,) and the vector potential 
function has only one nonzero component 𝚿 =(0,-ψ y(x, z,),0). If the plane strain 
condition holds in the xz plane, Eq. (7) reduces to 
 
, ,
y y
x z z x
u w
 
 
 
 
   
   
     (11) 
 
 
while from Eqs. (2) and (3) we can derive stresses in the form 
 
 
 
2 ,
2 ,
.
u w u
x x
x z x
u w w
z z
x z z
u w
x z
z x
  
  
 
  
  
  
  
 
 
  
  
 
     (12)  
  

22  ANALI PAZU.  
 
 
Further, instead of the vector wave equation (10) for all three components, only the 
scalar wave equation ∇
 𝜓 (𝑥 , 𝑧 ) = 𝜓 ̈ (𝑥 , 𝑧 )/𝑐 needs to be solved. The free wave 
solution is sought in the form 
 
   
 
   
 
, ,
, ,
i k x t
i k x t
y
x z z e
x z z e


 
 




      (13) 
 
where k represents the wavenumber, which can be real, imaginary or complex 
number and ω denotes the angular frequency of the wave. Solutions φ(x,z) and 
ψ y(x,z) represent traveling waves in the x direction with respect to the function e
i(kx-
ωt)
 and standing waves in the z direction in respect to functions Λ(z), Θ(z). By noting 
that the wavenumber k appears only in the function e
i(kx-ωt)
, we can learn more about 
the nature of wave propagation if we write k as a complex number in the form 
k=k r+i·k i. (i=√-1) with a real part k r and an imaginary part i·k i and examine 
individual possibilities. If k is a pure real number, k= k r, then wave is harmonic, 
traveling in the x direction with constant amplitude. If k is a pure imaginary number, 
k=i k i, the wave is not traveling wave but is evanescent. If k is complex with nonzero 
k r and i·k i parts, the wave travels in the x direction with a decreasing amplitude. To 
determine functions Λ(z), Θ(z) in Eq. (13), potential functions are substituted into 
Eq. (9) and (10), whereby we get 
 
     
 
     
 
2
2 2
2
2 2
sin cos , ,
sin cos , .
L
T
c
c
z A z B z k
z C z D z k


   
   
   
   
  (14) 
 
Determination of constants A, B, C and D depend on boundary conditions. 
Parameters α and β, which are generally complex, influence the course of functions 
Λ(z), Θ(z) in z direction, quotients 
 and 
  denote the wavenumbers of the 
longitudinal (bending) and transversal (shear) wave, respecively and k denotes the 
wavenumber along the direction of the wave propagation. The solving of the 
boundary value problem becomes much more simple, if geometric symmetry around 
axis z=0 is taken into account. The modes of wave propagation in a thin plate may 

R. Pušenjak: Computation of Dispersion Curves of Rayleigh-Lamb Waves by Newton-
Raphson Method 
23.
 
 
be separated into symmetric and asymmetric modes, respectively, if instead of Eq. 
(14) the ansatz 
 
        cos , sin z B z z C z      
   
 (15) 
is used for symmetric modes and ansatz 
 
        sin , cos z A z z D z      
   (16) 
 
for asymmetric modes. In what follows, only derivation of the dispersion relation 
for symmetric modes will be presented, because derivation of dispersion relation for 
asymmetric modes is quite analogous. 
 
Putting Eqs. (15) into Eq. (13) and forming required partial derivatives, which appear 
in Eqs. (11) and (12), we get the system of equations: 
 
   
 
   
 
 
   
 
 
 
   
 
 
 
   
 
2 2 2
2 2 2
2 2
cos cos ,
cos sin ,
2 cos 2 cos ,
2 cos 2 cos ,
sin 2 sin .
i k x t
i k x t
i k x t
x x
i k x t
z z
i k x t
x z
u i k B z C z e
w A z i k C z e
k k B z i k C z e
k B z i k C z e
k C z i k B z e





  
  
       
        
     





   
 
   
 
 
    
 
 
    
 
 
  
 
(17) 
 
By applying the traction free boundary conditions σ zz=σ xz=0 for z=±h, we obtain 
from last two equations of the system (17) the following system 
 
 
   
 
 
 
2 2 2
2 2
2 cos 2 cos
0
0
2 sin sin
k h i k h
B
C
i k h k h
       
     
   
   
   
   

   
 
   
 
 
 
. (18) 
 

24  ANALI PAZU.  
 
 
Since the system of Equation (18) is homogeneous, nontrivial solution for constants 
B and C is possible, if determinant vanishes, which results in the desired dispersion 
relation. For this purpose, let's first prove the following identity 
 
   
2 2 2 2 2
2 k k          
    
 (19) 
and the Rayleigh-Lamb dispersion relation is 
 
 
 
 
2
2
2 2
tan
4
tan
0,
k
h
k
h


 


 
      (20) 
 
which is evidently transcendental. It is fully clear, that the Rayleigh-Lamb dispersion 
relation for propagation of asymmetric modes can be derived in an analogous way. 
The dispersion relation is similar to Eq. (20) with the only difference that the second 
quotient on the left side of the equation is reciprocal. Thus, the dispersion relation 
for symmetric and asymmetric modes can be written with one equation 
 
 
 
 
2
2
2 2
1
tan
4
tan
0,
k
h
k
h


 



 
 
 
 
 
 
      (21) 
 
where the value of the exponent +1 belongs to the symmetric modes, and the value 
of the exponent -1 to the asymmetric modes. 
 
3 The application of the Newton-Raphson method for construction of the 
dispersion curves 
 
The Newton-Raphson method is an iterative algorithm for finding the roots of an 
equation f(x)=0, where the argument x can be real, imaginary or complex. The 
method has been in past widely used for solving non-linear equations with single or 
multiple variables. It should be noted, that the dispersion relation (21) is of the form 
f(ω,k)=0, that is, solutions are pairs of numbers if the wavenumber k is real or 

R. Pušenjak: Computation of Dispersion Curves of Rayleigh-Lamb Waves by Newton-
Raphson Method 
25.
 
 
imaginary, and are even triples of numbers if the wavenumber is complex. We shall 
present this method in complex and vector space for complex arguments, while the 
functions of the real valued or imaginary valued arguments, respectively can be 
treated as special cases. For a function f(ω,k): C→C with a complex argument 
k=k r+i·k i,

C, (i=√-1), where 𝑘 , 𝑘 𝜖𝑅 and where first two derivatives 𝑓 (𝜔 , 𝑘 ) =
 ( , )
 , 𝑓 (𝜔 , 𝑘 ) =
 ( , )
 are assumed to be continuous functions in the vicinity 
of the roots, we create a sequence of iterations 
 
 
 
,
1
,
n
n
f k
n n
f k
k k




 
      (22) 
 
that by selecting a good enough initial guess k 0 leads to the approximate solution of 
equation f(ω,k)=0. Newton-Raphson method in the complex space is characterized 
by basins of attraction with accompanied fractal structure for many functions and 
by divergent iterations for some starting points. Typically for transcendental 
functions as they appear in dispersion relation (21), which contains expressions 𝛼 =
 (𝜔 /𝑐 )
 − 𝑘 and  𝛽 = (𝜔 /𝑐 )
 − 𝑘  are discontinuities in f(ω,k) and 
𝑓 (𝜔 , 𝑘 ) where arg(ω,k)=(2j+1) /2 and 𝑗𝜖𝑍 . 
 
Instead of solving the original problem (22), the two-dimensional vector space 
Newton-Raphson method is presented in this research work for simplicity and ease 
of use. The equation f(ω,k)=0 implies that equations Re[f(ω,k)]=0, Im[f(ω,k)]=0 hold 
simultaneously and variables 𝑘 , 𝑘 𝜖𝑅 are independent. Then the vector space 
Newton-Raphson method can be formulated in R
2 
by introduction of the vector 
argument k=(k r,k i) and vector function f=(f r,f i), where the notation f r= Re[f(ω,k)], f i= 
Im[f(ω,k)] is used. Two equations of the vector space problem then are f r(ω,k n)=0, 
f i(ω,k n)=0, which are iteratively solved by means of the equation 
 
   
1
1
, , ,
n n n n
 


   k k J k f k
     (23) 
 
where J(ω, k) is the Jacobean matrix, defined as 

26  ANALI PAZU.  
 
 
  , .
r r
r i
i i
r i
f f
k k
f f
k k

 
 
 
 
 
 

 
 
 
J k
       (24) 
The vector method fails if J(ω,k n) is a singular matrix. In respect to the dispersion 
relation (21) the vector equation (23) is reduced to 
 
 
 
 
 
 
 
, ,
, ,
, ,
, 1 , , 1 ,
, ,
,
r n i n
r n i n
f k f k
r n r n i n i n
f k f k
k k k k
 
 
 
 
   
 
 (25) 
for real or imaginary wavenumbers, respectively. Similarly as in the vector problem 
for k=(k r,k i), the iterations (25) fail, if either 𝑓 𝜔 , 𝑘 , = 0 or 𝑓 𝜔 , 𝑘 , = 0. 
 
4 The arc-length continuation 
 
The construction of the dispersion curves can be in principle performed by using 
homotopy. This principle is depicted in Fig. 2, where the attempt to construct the 
curve of the first asymmetric mode for real wavenumbers is tried. The homotopy 
procedure is flexible enough to start at the top or bottom of the curve. In Fig. 2, for 
example, it starts at the top of the first asymmetric mode, where the first (highest 
lying) point is computed by Newton-Raphson iterative procedure. The homotopy 
procedure then decrements the real wave number k r for a prescribed value (of 
course, the incrementing the real wavenumber k r would be applied, when starting 
from the bottom) and uses the computed value of frequency ω as the starting value 
for the next iterative step. In Fig. 2, the homotopy procedure is successful in 10 
subsequent points, but then it follows an unexpected jump to the curve of the next 
asymmetric mode. After two points on the right curve are computed, the Newton-
Raphson iterative process no longer converges and the construction must be 
terminated. 
 

R. Pušenjak: Computation of Dispersion Curves of Rayleigh-Lamb Waves by Newton-
Raphson Method 
27.
 
 
 
 
Figure 2: The fail of the direct use of the Newton-Raphson method for construction of 
dispersion curve for real wavenumbers by homotopy. The jump from first asymmetric mode 
curve to the next asymmetric mode curve is shown. Bellow the second point of the right 
curve, the convergence cannot be reached.  
Source: own. 
 
In order to prevent such sudden jumps and in the same time to reduce the number 
of iterations required that the solution converges, an arc-length continuation 
combined with cubic extrapolation to predict the next point of the dispersion curve 
is applied. For the sake of brevity, let p 0=(ω 0,k 0), p 1=(ω 1,k 1), p 2=(ω 2,k 2) and 
p 3=(ω 3,k 3) be known points and predict the next point p 4=(ω 4,k 4) by 
extrapolation.of the prescribed arc length Δs. To achieve this, the arc length l is used 
as parameter so that l i corresponds to p i, i=0,1,…,3 and corresponding arc lenghts 
are l 0, l 1=s 1, l 2= l 1+ s 2, l 3= l 2+ s 3, l 4= l 3+Δs. The extrapolated point p 4 is then 
calculated by using the formula 
 
4
3 3
4
0 0,
,
j
i j
l l
i
l l
i j
j i


 

 
 
  
 
 
 
p p
       (26) 
  

28  ANALI PAZU.  
 
 
whereby it is 
 
   
1
2
1 1
1
1
f
N
i i i i i
j
s j j

 

       
 
p p p p
     (27) 
 
and N f denotes the number of components of the wavenumber vector k. It is 
obvious that N f=1 in the case of real or imaginary wavenumbers and N f=2 for 
complex wavenumbers. The choice of the incremental length Δs is determined by 
the principal curvature of the hyperspace curve. 
 
5 The results 
 
The presented method of construction of dispersion curves for real, imaginary and 
complex wavenumbers is applied for aluminium plate with the longitudinal wave 
velocity c L =6300 m/s, the transverse wave velocity c T=3100 m/s and half- thickness 
of the plate h=4 mm. The construction of dispersion curves is carried out in the 
frequency range 0 - 1 MHz, whereby the circular frequency ω is converted into 
frequency f using equation ω=2 f. In all figures, the symmetric modes of Lamb 
waves are shown by continuous and asymmetric modes by dashed curves, 
respectively. The dispersion curves are computed in Mathematica programming 
environment, using the strong capabilities of symbolic computing. 
 
5.1 Dispersion curves for real wavenumbers 
 
In general, several dispersion curves appear at each selected frequency interval, the 
calculation of which is time consuming. In order to limit the processing time, the 
computer program prescribes the number of symmetric and corresponding 
asymmetric modes of Lamb waves that we want to calculate. In the Fig. 3, first three 
symmetric and corresponding asymmetric modes of real wavenumbers are shown. 
 

R. Pušenjak: Computation of Dispersion Curves of Rayleigh-Lamb Waves by Newton-
Raphson Method 
29.
 
 
 
 
Figure 3: Dispersion curves of Lamb waves for real wavenumbers. Symmetric modes are 
presented by continuous curves, asymmetric modes by dashed curves. 
Source: own. 
 
5.2 Dispersion curves for imaginary wavenumbers 
 
In third section it was shown that dispersion curves for imaginary wavenumbers can 
be constructed as the special case of the vector space method. However, there exists 
an alternative way, which is actually applied in this paper. If wavenumber is pure 
imaginary, it can be written as k=i·k i. Inserting this substitution into Eq. (14) for α 
and β and rename them as α i and β i, we get 
 
   
 
 
2
2
2
2
2
2 2
2 2
1
tan 4
tan
, ,
0.
L T
i i i i
i
k
i
c
T
i i i i
c c
h k
h
k k

 
  

 
 
 
  
 
   
 

   
 
 
 
 
 
 
 
 
    (28) 
 
By using Eq. (28), the construction of dispersion curves proceeds in the same way 
as for real wavenumbers.  
 

30  ANALI PAZU.  
 
 
 
 
Figure 4: Dispersion curves of Lamb waves for imaginary wavenumbers. Symmetric modes 
are presented by continuous curves, asymmetric modes by dashed curves. 
Source: own. 
 
5.3 Dispersion curves for complex wavenumbers 
 
The presented method can easily be used for the construction of dispersion curves 
for complex wavenumbers. Of course, in this case, it is necessary to separately 
present the course of the real and imaginary part of the complex wavenumber as a 
function of frequency, as shown, for example, in Fig. 5. Figures 5.a and 5.b show 
only individual symmetric (depicted by continuous line) and asymmetric wave mode 
(depicted by dashed line). Unlike the finite number of wave modes for real and 
imaginary wavenumbers, there are infinitely many wave modes for complex wave 
numbers [6]. 
 
 
 
Figure 5: Dispersion curves of Lamb waves for complex wavenumbers: a) real part of 
complex wavenumber, b) imaginary part of complex wavenumber. Symmetric modes are 
presented by continuous curves, asymmetric modes by dashed curves. 
Source: own. 

R. Pušenjak: Computation of Dispersion Curves of Rayleigh-Lamb Waves by Newton-
Raphson Method 
31.
 
 
6 The conclusions 
 
In the paper, the necessary prediction capability is added to the classical Newton-
Raphson method to enable successful construction of curves for transcendental 
dispersion relations, which is the characteristic of Rayleigh-Lamb waves. The 
prediction is performed by arc length continuation combined with cubic 
extrapolation to successfully prevent unexpected and undesired jumps between wave 
modes during the branch tracing. It is proved that the proposed supplement to the 
Newton-Raphson method works equally well in the cases of construction of 
dispersion curves for real, imaginary and complex wave numbers. In the future, we 
hope that the proposed method can be extended to be used in multilayer plates. 
 
 
Acknowledgement 
 
Author gratefully appreciates the Slovenian Research Agency (ARRS) for co-financing the research 
within the framework of the fundamental research project J2-9224. 
 
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