Numerical Study of Focusing Effects of Microwaves inside Wood Due to Timber Ring Structure

: The aim of this study is the detailed calculation of microwave propagation inside raw timber in cylindrical conﬁgurations. Two different approaches have been used. The ﬁrst one uses an exact formulation and analytical approximations in order to explore the electromagnetic ﬁeld distribution inside dry wood. The introduction of conductivity in the exact model makes it so complex that the equations are unsuitable for analytical manipulation. In order to further explore the effect of moisture in cylindrical wood structures, a full scale numerical simulation using commercial software has been performed. The results show that for microwave frequencies in the 3 GHz range and for typical wood parameters, a cylindrical log behaves as a kind of Fresnel lens. This work has important applications in microwave treatment and sensing of wood. and C.A.-C. analyzed the data and results. R.S.-M., P.-L.L.-E., J.A.M.-R., J.A.-H. and C.A.-C.


Introduction
Microwave treatment and sensing of wood is increasingly important. A review of different microwave wood testing techniques is given in [1] and these include propagation modelling, measurement techniques, hardware implementation, and determination of wood properties. According to [2,3], microwave heating, for example, offers several advantages over other drying methods and chemical procedures because its technique is eco-friendlier and causes less damage in the timber sample compared to employing toxic chemicals. Besides, microwave heating method is faster and gentler in comparison to live steam. As a contactless and nondestructive testing technique, microwave scanning is able to estimate several important wood parameters including density, moisture content, and grain angle [4][5][6][7][8][9][10][11][12][13]. Following Bucur [14], other non-destructive techniques use X-rays, gamma rays, thermal imaging, ultrasound, nuclear magnetic resonance, and neutron imaging. Microwave based nondestructive techniques have been applied to a variety of applications, including in situ microwave tomography, moisture content determination, anisotropy measurement, online imaging, internal inspection of logs, mechanical grading, and defect detection. Microwaves and ultrasound have similar resolution, but microwaves are not limited by the contact nature of ultrasound transducers.
As far as we know, most models of microwave propagation inside wood are based on rectangular configurations, although some studies in more general geometries are published [15,16]. In [17], the effect of annual growth rings on nondestructive measurements is explicitly studied. This limits the application of microwave techniques in areas such as measurement of living trees in situ.
Important efforts have been made to develop generalized models for water content prediction in wood using its dielectric properties, but detailed models of electromagnetic propagation inside wood

Semianalytical Model
Raw cylindrical timber can be considered a stack of concentrically dielectric layers, so a model with arbitrary periodic variation of dielectric permittivity is the most accurate. Considerable symbolic manipulation of the Maxwell equations in cylindrical coordinates is necessary for arbitrary permittivity functions which can depend only on the radius of the cylinder. The equations for the exact model are derived from our previous works [24,25]. They are reproduced here for completeness.
The equations were obtained as follows. First, we write the complete set of Maxwell equations in cylindrical coordinates 1 r ∂(rD r ) ∂r Performing some algebraic manipulations, the full set of Maxwell equations can be reduced to a pair of coupled ordinary differential equations. The following ansatz is inserted into the cylindrical Maxwell Equations (1)- (8), where r is the radius, z is the length of the wood log, and phi is the angle measured from the radius. E is the electric field strength, H is the magnetic field, D is the electric displacement field, B is the magnetic flux density, omega is the angular frequency, t is the time, and l is the angular moment number. E r,φ,z = E 0[r,φ,z] e r,φ,z (r)e (ilφ) × e (iβz) × e −(iωt) (9) H r,φ,z = H 0[r,φ,z] h r,φ,z (r)e (ilφ) × e (iβz) × e −(iωt) (10) This produces a set of separable equations, assuming ρ = 0 (charge density) and J = 0 (current density), which implies that conductivity is not considered. Equations whose terms are in the form βE or βH are selected. These are Equations (1), (2), (4), (6) and (7). Now, the Equations (4), (6) and (7) can be algebraically solved with respect to the radial components e r (r), e φ (r), h r (r), and h φ (r), Substituting this solution into Equations (1) and (2), another pair of ordinary differential equations is obtained. In order to analyze these equations, it is convenient to decoupled them. Using e z (r) = e zR (r) + ie zI (r); h z (r) = h zR (r) + ih zI (r) and taking both real and imaginary parts in the resulting equations, the following expressions are finally obtained e zR (r) + A 11 e zR (r) + A 12 e zI (r) + A 13  where A ij (i = 1, 2, 3 and j = 1, ..., 9) coefficients try to simplify the writing of the final equations, because the complete formulae are rather long. These coefficients multiply the radial components, both real and imaginary, of the axial electric and magnetic field inside the cylindrical wood sample. The value of coefficients A ij defined in the Equation (15) are detailed in the following equations (from Equations (19)-(22)) and where A ij (i = 2 and j = 1, ..., 9) coefficients employed in the Equation (9) are defined in the Equations (23)-(26) and where, In the following equations from Equations (27)-(30) the values of the A ij (where i = 3 and j = 1, ..., 9) used in the Equation (17) are detailed. and, In the Equations (31)-(34) the value of the coefficients A ij (where i = 4 and j = 1, ..., 9) which have been defined in the Equation (18), are detailed. and ε re f f is the effective permittivity parameter. We can observe that this system of equations is not completely coupled. Equations (15) and (18) form a pair of mutually coupled equations in e zR (r) and h zI (r). The same can be said for Equations (16) and (17). Each pair of mutually coupled equations have a similar mathematical structure. Then, it is possible to reduce the problem of microwave propagation inside cylindrical wood samples with arbitrary radial relative permittivities to a pair of ordinary differential equations. We shall choose Equations (15) and (18) for convenience. The results were calculated and simplified by means of the Maxima program [26].
In order to solve the final equations, an implicit high order Runge-Kutta algorithm was used. Implicit RK algorithms are more stable than their explicit counterparts [27] and this fact eliminates some problems related with the periodic nature of the dielectric structure of wood and the extreme sensibility of the solutions to the value of the effective permittivity. Although the core equations of this problem are the same as the equations for a Bragg fiber in our previous work [24], there are important conceptual and computational differences between both problems. First of all, the relative permittivities of adjacent layers in the wood model are larger than in a Bragg fiber and conductivity is not present in optical fibers. More important even is the consideration of the grain angle in the wood model which introduces new complexities and new propagation modes. This grain angle would be similar to a Bragg fiber with torsion, which was not considered in our previous work. Also, the ratio between the wavelength and the dimensions of the rings is different, producing subtle effects that have forced us to develop a new numerical approach to solve the final equations. While in our previous calculations, collocation methods were successfully used, in this case an implicit, unconditionally stable, RK algorithm has been developed in order to avoid large numerical instabilities produced by the mode splitting introduced by the grain angle dependence on the relative permittivity.

Numerical Simulation
Due to mathematical constraints, conductivity cannot be easily introduced in the exact model, because the Maxwell equations cannot be decoupled. Therefore, a detailed numerical simulation of an idealized cylindrical sample of wood has been performed in order to study the effect of moisture on microwave propagation. The commercial software CST Microwave Studio ® [22], which implements a finite-integral technique (FIT) approach, was used.
The sample was formed by concentrically layers of two different permittivities corresponding to typical epsilon values of poplar [27] using the dielectric Cartesian tensor facility of CST (Computer Simulation Technology). The exact values are shown in Figure 1. Summer and spring rings are independently considered. Experimental dielectric and conductivity data for both dry and wet wood were obtained from [28][29][30][31]. The thickness of growth rings inside poplar has been taken from pictures of several samples from the Madrid area and a rounded mean value has been used for the simulations. Thus, this study must be considered as a general description of microwave propagation inside an actual ring structured wood but not a detailed analysis of a particular wood sample.
The selected frequency for our study has been 3 GHz for numerical convenience in the case of the semianalytical calculations. The permittivity values can be considered almost unaltered from the more common 2.45 GHz according to [27]. However, according to the electric field distribution shown in Figures 2 and 3 for dry and wet poplar sample, the effect of the microwave frequency is noticeable in spite of the constant value of the permittivity at those frequencies. An important consequence of these results is the requirement of several frequencies to achieve a more homogenous electric field distribution inside wood.       A limitation of our study is that real wood has different growth ring thicknesses, heterogeneous zones with defects, moisture gradients, gaps, etc., which could be introduced in future studies.
Forests 2018, 9, x FOR PEER REVIEW 7 of 16 A limitation of our study is that real wood has different growth ring thicknesses, heterogeneous zones with defects, moisture gradients, gaps, etc., which could be introduced in future studies.

Results and Discussion
In both models, analytical and numerical, axial excitation and propagation of microwaves are studied. The timber is excited by an incident plane wave along the z-axis, the electric field is perpendicular to the cylinder axis, and so the magnetic field, as shown in Figure 1. The relative permittivity function used for the semianalytical analysis is shown in Figure 4. The semianalytical model shows a slow modulation of the amplitude of the electric field due to the periodic nature of the dielectric layers of wood, as can be seen in Figures 5 and 6. The behavior of this permittivity function is modelled on [28]. For simplicity and easier convergence of the RK algorithm, uniform growth ring thickness and a lower variation between different wood dielectric values in the radial direction was used. This is clearly a limitation of the semianalytical model and one of the main reasons for using also a full numerical simulation tool. Thus, the results are not exactly the same as the ones calculated using experimental poplar data.
For the previous reasons, the semianalytical method is restricted to only 5 rings, while the full numerical simulations range from 5 growth rings to 20. The thickness of growth rings are taken form [27]. The semianalytical results can be seen as a validation technique for this study. The length of the sample is 1 m in both methods. The semianalytical and numerical simulations have been done at 3 GHz because the wavelength at that frequency is comparable to the diameter of the wood sample.

Results and Discussion
In both models, analytical and numerical, axial excitation and propagation of microwaves are studied. The timber is excited by an incident plane wave along the z-axis, the electric field is perpendicular to the cylinder axis, and so the magnetic field, as shown in Figure 1. The relative permittivity function used for the semianalytical analysis is shown in Figure 4. The semianalytical model shows a slow modulation of the amplitude of the electric field due to the periodic nature of the dielectric layers of wood, as can be seen in Figures 5 and 6. The behavior of this permittivity function is modelled on [28]. For simplicity and easier convergence of the RK algorithm, uniform growth ring thickness and a lower variation between different wood dielectric values in the radial direction was used. This is clearly a limitation of the semianalytical model and one of the main reasons for using also a full numerical simulation tool. Thus, the results are not exactly the same as the ones calculated using experimental poplar data.
For the previous reasons, the semianalytical method is restricted to only 5 rings, while the full numerical simulations range from 5 growth rings to 20. The thickness of growth rings are taken form [27]. The semianalytical results can be seen as a validation technique for this study. The length of the sample is 1 m in both methods. The semianalytical and numerical simulations have been done at 3 GHz because the wavelength at that frequency is comparable to the diameter of the wood sample. Moreover, that frequency is similar to the value which is employed in commercial microwave drying ovens [32,33]. Moreover, that frequency is similar to the value which is employed in commercial microwave drying ovens [32,33].  The periodic structure produces the modification of the values of the relative permittivity, which, in turn, modifies the electromagnetic field modes inside the wood cylinder. In order to apply microwave propagation to the detailed study of internal ring structure of wood, higher frequencies up to tens of GHz (microwaves in the mm range) are necessary. However, at such short wavelengths, the effect of limited skin depth and increased attenuation should be studied carefully. Besides, the computational demands are far higher, so this will be considered in future works. If such accuracy could be practically achieved, a non-invasive determination of tree ages would be possible, for example.
The solutions are valid only for certain values of the effective relative permittivity (permittivity eigenvalues) due to the existence of discrete propagation modes inside the cylindrical material. These eigenvalues are calculated by means of an iterative procedure which runs the RK algorithm searching for the correct behavior of the solutions. In this case, the azimuthal number of the selected propagation mode is l = 1. Therefore, every mode is characterized by a definite value of the effective  Moreover, that frequency is similar to the value which is employed in commercial microwave drying ovens [32,33].  The periodic structure produces the modification of the values of the relative permittivity, which, in turn, modifies the electromagnetic field modes inside the wood cylinder. In order to apply microwave propagation to the detailed study of internal ring structure of wood, higher frequencies up to tens of GHz (microwaves in the mm range) are necessary. However, at such short wavelengths, the effect of limited skin depth and increased attenuation should be studied carefully. Besides, the computational demands are far higher, so this will be considered in future works. If such accuracy could be practically achieved, a non-invasive determination of tree ages would be possible, for example.
The solutions are valid only for certain values of the effective relative permittivity (permittivity eigenvalues) due to the existence of discrete propagation modes inside the cylindrical material. These eigenvalues are calculated by means of an iterative procedure which runs the RK algorithm searching for the correct behavior of the solutions. In this case, the azimuthal number of the selected propagation mode is l = 1. Therefore, every mode is characterized by a definite value of the effective The periodic structure produces the modification of the values of the relative permittivity, which, in turn, modifies the electromagnetic field modes inside the wood cylinder. In order to apply microwave propagation to the detailed study of internal ring structure of wood, higher frequencies up to tens of GHz (microwaves in the mm range) are necessary. However, at such short wavelengths, the effect of limited skin depth and increased attenuation should be studied carefully. Besides, the computational demands are far higher, so this will be considered in future works. If such accuracy could be practically achieved, a non-invasive determination of tree ages would be possible, for example.
The solutions are valid only for certain values of the effective relative permittivity (permittivity eigenvalues) due to the existence of discrete propagation modes inside the cylindrical material. These eigenvalues are calculated by means of an iterative procedure which runs the RK algorithm searching for the correct behavior of the solutions. In this case, the azimuthal number of the selected propagation mode is l = 1. Therefore, every mode is characterized by a definite value of the effective relative permittivity, which can be expressed as a correction factor multiplying the mean effective relative permittivity. relative permittivity, which can be expressed as a correction factor multiplying the mean effective relative permittivity. The exact model allows the variation of the grain angle, α. The grain angle is defined as the angle between the timber axis and the wood logs. It is the longitudinal arrangement of wood timbers or the pattern resulting from this. From an analytical perspective, if the grain angle had been inserted directly in the cylindrical Maxwell's equations, they would not be separable. In fact, helicoidal coordinates should be used, but the model was too complex for the present study. Instead of a full helicoidal model, we have made the hypothesis that the effect of the grain angle can be considered using only its variation inside the dielectric permittivity profile.
The numerical simulation of the grain angle would imply a helicoidally arrangement of different wood elements, so that parametric functions should be used inside the Cartesian dielectric tensor of the CST software. In this regard, the analytical model is more general, although conductivity effects are hard to introduce, if additional hypotheses are not made. Due to the limitations of both approaches, we have opted for a combined study, comparing the results for a neutral grain angle and dry wood. Then, the effect of moisture is studied using numerical analysis and the grain angle variation uses the analytical model.
In Figures 5 and 6, we can compare the results for α = 45° and 30°, respectively. It must be noted that the correction factor for the effective relative permittivity changes with the value of the grain angle, so that the corrections are 0.70 for α = 45° and 0.66 α = 45°. The ability to calculate the grain angle is important, because grain angle is related with the mechanical strength of wood. The effective value of the relative permittivity due to the grain angle correction is calculated by means of an iterative bisection algorithm which finds the correct asymptotic behavior of the oscillating fields inside wood after a number of rings. These calculations are time consuming and numerically unstable, even using the implicit Runge-Kutta algorithm for the propagation step. Due to this, only two decimal figures and five rings are considered in Figures 5 and 6, while the numerical modelwhich discards the grain angle influence-can be calculated for a larger number of rings.
The CST simulations have been done without considering the grain angle. The numerical values for the simulations were: length = 1 m, number of cylindrical layers = from 5 to 20, radius = from 35 to 145 mm. Growth ring thickness are different depending on the season. One type has been set to 10 mm and the other to 5 mm. The values of the dielectric constant of each layer have been taken from [27] and they are shown in Figure 1. The exact model allows the variation of the grain angle, α. The grain angle is defined as the angle between the timber axis and the wood logs. It is the longitudinal arrangement of wood timbers or the pattern resulting from this. From an analytical perspective, if the grain angle had been inserted directly in the cylindrical Maxwell's equations, they would not be separable. In fact, helicoidal coordinates should be used, but the model was too complex for the present study. Instead of a full helicoidal model, we have made the hypothesis that the effect of the grain angle can be considered using only its variation inside the dielectric permittivity profile.
The numerical simulation of the grain angle would imply a helicoidally arrangement of different wood elements, so that parametric functions should be used inside the Cartesian dielectric tensor of the CST software. In this regard, the analytical model is more general, although conductivity effects are hard to introduce, if additional hypotheses are not made. Due to the limitations of both approaches, we have opted for a combined study, comparing the results for a neutral grain angle and dry wood. Then, the effect of moisture is studied using numerical analysis and the grain angle variation uses the analytical model.
In Figures 5 and 6, we can compare the results for α = 45 • and 30 • , respectively. It must be noted that the correction factor for the effective relative permittivity changes with the value of the grain angle, so that the corrections are 0.70 for α = 45 • and 0.66 α = 45 • . The ability to calculate the grain angle is important, because grain angle is related with the mechanical strength of wood. The effective value of the relative permittivity due to the grain angle correction is calculated by means of an iterative bisection algorithm which finds the correct asymptotic behavior of the oscillating fields inside wood after a number of rings. These calculations are time consuming and numerically unstable, even using the implicit Runge-Kutta algorithm for the propagation step. Due to this, only two decimal figures and five rings are considered in Figures 5 and 6, while the numerical model-which discards the grain angle influence-can be calculated for a larger number of rings.
The CST simulations have been done without considering the grain angle. The numerical values for the simulations were: length = 1 m, number of cylindrical layers = from 5 to 20, radius = from 35 to 145 mm. Growth ring thickness are different depending on the season. One type has been set to 10 mm and the other to 5 mm. The values of the dielectric constant of each layer have been taken from [27] and they are shown in Figure 1.
The numerical FIT simulation allows the study of a layer of growth rings. In general, the results in Figures 7-12 prove the critical effect of the number of rings on the electric field distribution inside of poplar wood. Besides, this effect is not trivial according to the radial electric field variation seen in Figures 13-18. The electric field maxima varies with the number of the growth rings both, in the radial and longitudinal directions. A direct consequence of these results for heating applications is that the thermal distribution inside wood would vary critically from one sample to another with different thicknesses. For the calculations, two extreme cases of the permittivity variation between growth rings (50% and 100%), according to the literature [28], have been studied. Comparing Figure 7, Figure 9, and Figure 11 (50% dielectric variation) with Figures 8, 10 and 12 (100% dielectric variation) it can be concluded that the effect of this variable is very important. For example, in Figures 7 and 8 it can be observed that the most remarkable changes in the electric field distribution, are produced between the 5-ring samples and the 15-ring samples, in both, the radial and the axial direction. This implies that, even if two poplar samples have the same dimensions, their electric and thermal behavior can be completed different.
In Figures 9-12 the influence of moisture on the distribution of the electric field inside poplar wood is explored. According to these results, there are two main conclusions of this: the electric field distribution is completely altered by the presence of moisture and the number of rings changes this behavior in a non-predictable way. For example, in the five rings case, moisture produces a noticeable central focusing effect making the wood sample similar to a metallic waveguide. On the contrary, the 10-ring case shows a completely opposite behavior, from a central distribution in the dry case to a much attenuated radial distribution. The axial distribution is also noticeably altered. The implications for heating applications are clear.
A detailed study of the radial electric field distribution is shown in Figures 13-18. In this set of figures, the combined and complex interaction among the analyzed variables are more clearly seen. The maximum of the electric field is displaced from the center to a different position depending on the number of the growth rings. Also, the shape of the electric field distribution is greatly affected. The numerical FIT simulation allows the study of a layer of growth rings. In general, the results in Figures 7-12 prove the critical effect of the number of rings on the electric field distribution inside of poplar wood. Besides, this effect is not trivial according to the radial electric field variation seen in Figures 13-18. The electric field maxima varies with the number of the growth rings both, in the radial and longitudinal directions. A direct consequence of these results for heating applications is that the thermal distribution inside wood would vary critically from one sample to another with different thicknesses. For the calculations, two extreme cases of the permittivity variation between growth rings (50% and 100%), according to the literature [28], have been studied. Comparing Figures 7, 9, and 11 (50% dielectric variation) with Figures 8, 10 and 12 (100% dielectric variation) it can be concluded that the effect of this variable is very important. For example, in Figures 7 and 8 it can be observed that the most remarkable changes in the electric field distribution, are produced between the 5-ring samples and the 15-ring samples, in both, the radial and the axial direction. This implies that, even if two poplar samples have the same dimensions, their electric and thermal behavior can be completed different.
In Figures 9-12 the influence of moisture on the distribution of the electric field inside poplar wood is explored. According to these results, there are two main conclusions of this: the electric field distribution is completely altered by the presence of moisture and the number of rings changes this behavior in a non-predictable way. For example, in the five rings case, moisture produces a noticeable central focusing effect making the wood sample similar to a metallic waveguide. On the contrary, the 10-ring case shows a completely opposite behavior, from a central distribution in the dry case to a much attenuated radial distribution. The axial distribution is also noticeably altered. The implications for heating applications are clear.
A detailed study of the radial electric field distribution is shown in Figures 13-18. In this set of figures, the combined and complex interaction among the analyzed variables are more clearly seen. The maximum of the electric field is displaced from the center to a different position depending on the number of the growth rings. Also, the shape of the electric field distribution is greatly affected.          . Electric field distribution inside wet poplar wood with different values of the numbers of growth rings with 50% permittivity variation at 3 GHz. Dielectric constant values were used with 28% moisture content according to [27].   . Electric field distribution inside wet poplar wood with different values of the numbers of growth rings with 50% permittivity variation at 3 GHz. Dielectric constant values were used with 28% moisture content according to [27]. Figure 11. Electric field distribution inside wet poplar wood with different values of the numbers of growth rings with 50% permittivity variation at 3 GHz. Dielectric constant values were used with 28% moisture content according to [27].   Dielectric constant values were used with 28% moisture content according to [27].       . Radial profile of the electric field for a wet poplar sample with a 50% dielectric constant variation between growth rings. Dielectric constant values were used with 0% moisture content according to [27]. Figure 16. Radial profile of the electric field for a wet poplar sample with a 100% dielectric constant variation between growth rings. Dielectric constant values were used with 0% moisture content according to [27].

Figure 17.
Radial profile of the electric field for a wet poplar sample with a 50% dielectric constant variation between growth rings. Dielectric constant values were used with 28% moisture content according to [27]. Figure 18. Radial profile of the electric field for a wet poplar sample with a 100% dielectric constant variation between growth rings. Dielectric constant values were used with 28% moisture content according to [27]. Figure 15. Radial profile of the electric field for a wet poplar sample with a 50% dielectric constant variation between growth rings. Dielectric constant values were used with 0% moisture content according to [27].
Forests 2018, 9, x FOR PEER REVIEW 14 of 16 Figure 15. Radial profile of the electric field for a wet poplar sample with a 50% dielectric constant variation between growth rings. Dielectric constant values were used with 0% moisture content according to [27]. Figure 16. Radial profile of the electric field for a wet poplar sample with a 100% dielectric constant variation between growth rings. Dielectric constant values were used with 0% moisture content according to [27].

Figure 17.
Radial profile of the electric field for a wet poplar sample with a 50% dielectric constant variation between growth rings. Dielectric constant values were used with 28% moisture content according to [27]. Figure 18. Radial profile of the electric field for a wet poplar sample with a 100% dielectric constant variation between growth rings. Dielectric constant values were used with 28% moisture content according to [27]. Figure 16. Radial profile of the electric field for a wet poplar sample with a 100% dielectric constant variation between growth rings. Dielectric constant values were used with 0% moisture content according to [27].
Forests 2018, 9, x FOR PEER REVIEW 14 of 16 Figure 15. Radial profile of the electric field for a wet poplar sample with a 50% dielectric constant variation between growth rings. Dielectric constant values were used with 0% moisture content according to [27]. Figure 16. Radial profile of the electric field for a wet poplar sample with a 100% dielectric constant variation between growth rings. Dielectric constant values were used with 0% moisture content according to [27].

Figure 17.
Radial profile of the electric field for a wet poplar sample with a 50% dielectric constant variation between growth rings. Dielectric constant values were used with 28% moisture content according to [27]. Figure 18. Radial profile of the electric field for a wet poplar sample with a 100% dielectric constant variation between growth rings. Dielectric constant values were used with 28% moisture content according to [27].  Figure 17. Radial profile of the electric field for a wet poplar sample with a 50% dielectric constant variation between growth rings. Dielectric constant values were used with 28% moisture content according to [27].
Forests 2018, 9, x FOR PEER REVIEW 14 of 16 Figure 15. Radial profile of the electric field for a wet poplar sample with a 50% dielectric constant variation between growth rings. Dielectric constant values were used with 0% moisture content according to [27]. Figure 16. Radial profile of the electric field for a wet poplar sample with a 100% dielectric constant variation between growth rings. Dielectric constant values were used with 0% moisture content according to [27].

Figure 17.
Radial profile of the electric field for a wet poplar sample with a 50% dielectric constant variation between growth rings. Dielectric constant values were used with 28% moisture content according to [27]. Figure 18. Radial profile of the electric field for a wet poplar sample with a 100% dielectric constant variation between growth rings. Dielectric constant values were used with 28% moisture content according to [27].  Figure 18. Radial profile of the electric field for a wet poplar sample with a 100% dielectric constant variation between growth rings. Dielectric constant values were used with 28% moisture content according to [27].

Conclusions
Semianalytical and numerical models of microwave propagation in wood have been performed. The electromagnetic field distribution inside cylindrical wood is a complex phenomenon due to its quasiperiodic ring structure. Both the number of growth rings and moisture content modify the microwave behavior in complex and unexpected ways. The effects of grain angle and moisture have also been considered, using two complementary approaches, semianalytical, and numerical. Grain angle affects the value of the effective permittivity of wood and influences the radial distribution of the electric field inside wood, changing the radial position of the electric field maximum slowly. More critically, moisture completely alters both the radial and axial behavior of microwave propagation inside poplar wood. Depending on the numbers of the growth rings, a cylindrical sample of poplar wood performs like a metallic waveguide. The implications of these results for many applications of microwaves for wood treatment and testing are clear. For example, optimization of heating wood in microwave ovens would be a much more complex process than intuitively expected. Moreover, for nondestructive in situ testing purposes, the interpretation of data could be greatly affected by these heterogeneous focusing phenomena.
Our study is limited by several factors. Semianalytical calculations are very complex and numerically demanding in the end, so they do not present many advantages over the numerical ones, except for theoretical insight. The semianalytical method cannot address the effect of moisture, but can study the grain angle. On the contrary, numerical simulations have no problem with water content, but implementing helicoidally variations of the permittivity is complex. It is expected that these results can be qualitatively and tentatively extrapolated to many species of wood, but in this work, the calculated and simulated values correspond to poplar samples.