Theory for Effective Advection Effect: Spatiotemporal Modulation

Abstract

In this chapter, we introduce spatiotemporal modulation to realize thermal wave nonreciprocity. The major mechanism is the effective advection effect of spatiotemporal modulation in an open thermal system. We further analyze the phase difference between two spatiotemporally modulated parameters, which offers a tunable parameter to control nonreciprocity. We further define a rectification ratio based on the reciprocal of spatial decay rates and discuss the nonreciprocity conditions accordingly. Finite-element simulations are performed to confirm theoretical predictions, and experimental suggestions are provided to ensure the feasibility of spatiotemporal modulation. These results have potential applications in realizing thermal detection and thermal stabilization simultaneously.

1 Opening Remarks

Ever since the concept of spatiotemporal modulation was proposed [1], intensive studies have been conducted not only in wave systems [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] including photonics [2,3,4,5], acoustics [6,7,8,9], and metasurfaces [10,11,12] but also in diffusion systems [17,18,19]. A direct application of spatiotemporal modulation is to realize nonreciprocity which refers to asymmetric propagation in opposite directions. Although many different kinds of waves have been studied to achieve nonreciprocity based on spatiotemporal modulation, thermal waves have received little attention despite being an important phenomenon. In terms of mechanism, thermal waves are a special kind of wave that is dominated by a diffusion equation (i.e., the Fourier equation), thus also called diffusion waves [20]. In terms of application, thermal waves can realize nondestructive detection (i.e., thermal wave imaging), widely applied in aerospace, machinery, and electricity [21,22,23]. Some recent studies also focused on diffusion waves to realize anti-parity-time symmetry [24,25,26,27], negative thermal transport [28], cloaks [20, 29,30,31], and crystals [32,33,34].

However, a mechanism to achieve thermal wave nonreciprocity is still lacking. Thermal waves can be treated as periodic temperature fluctuations, usually a double-edged sword. On the one hand, they are desirable for thermal detection. On the other hand, they are unwanted for thermal stabilization. Therefore, it is crucially important to realize thermal wave nonreciprocity. For this purpose, we explore spatiotemporal modulation to achieve thermal wave nonreciprocity, inspired by pioneering studies on nonreciprocal thermal materials [18]. It has been revealed that an advection term appears in the conduction equation at quasi-steady states if thermal conductivity and mass density are spatiotemporally modulated, thus achieving nonreciprocity. However, the applicability of thermal waves was not discussed. On the one hand, thermal waves feature completely transient states where the Willis term should be considered. On the other hand, the phase difference between two spatiotemporally modulated parameters remains explored.

Here, we thoroughly discuss thermal wave nonreciprocity based on spatiotemporal modulation. Since there is a phase difference between two spatiotemporally modulated parameters, we construct two backward cases (Fig. 14.1) with different nonreciprocity conditions. The results demonstrate that the phase difference offers a flexible and tunable parameter to control nonreciprocity. We also discuss the heat flux to reveal the feature of spatiotemporal modulation.

2 Theoretical Foundation

We consider a passive thermal conduction process in one dimension, dominated by

$$\begin{aligned} \rho \left( x-u t\right) \frac{\partial T}{\partial t}+\frac{\partial }{\partial x}\left[ -\sigma \left( x-u t\right) \frac{\partial T}{\partial x}\right] =0, \end{aligned}$$

(14.1)

where $\sigma \left( x-u t\right)$ is thermal conductivity and $\rho \left( x-u t\right)$ is the product of mass density and heat capacity. The spatiotemporally modulated parameters in Fig. 14.1a take the form of

$$\begin{aligned} \sigma \left( x-u t\right)&=\sigma _A+\sigma _B\cos \left[ K\left( x-u t\right) \right] ,\end{aligned}$$

(14.2a)

$$\begin{aligned} \rho \left( x-u t\right)&=\rho _A+\rho _B\cos \left[ K\left( x-u t\right) +\alpha \right] , \end{aligned}$$

(14.2b)

where $\sigma _A$, $\sigma _B$, $\rho _A$, and $\rho _B$ are four constants. $K=2\pi /\gamma$ is wave number, $\gamma$ is wavelength, u is modulation speed, and $\alpha$ is phase difference. Since $\sigma \left( x-u t\right)$ and $\rho \left( x-u t\right)$ are periodic functions, the Bloch theorem is applicable and the temperature solution can be expressed as

$$\begin{aligned} T=\phi \left( x-u t\right) \textrm{e}^{\textrm{i}\left( kx-\omega t\right) }, \end{aligned}$$

(14.3)

where k and $\omega$ are, respectively, the wave number and circular frequency of a thermal wave. $\phi \left( x-u t\right)$ is an amplitude modulation function that has the same periodicity as $\sigma \left( x-u t\right)$ and $\rho \left( x-u t\right)$. Equation (14.1) can then be homogenized with the approximations of $k\ll K$ and $\omega \ll uK$ [18],

$$\begin{aligned} \tilde{\rho }\frac{\partial \tilde{T}}{\partial t}+C\frac{\partial \tilde{T}}{\partial x}-\tilde{\sigma }\frac{\partial ^2 \tilde{T}}{\partial x^2}-S\frac{\partial ^2 \tilde{T}}{\partial x \partial t}=0, \end{aligned}$$

(14.4)

where the homogenized parameters can be expressed as

$$\begin{aligned} \tilde{\sigma }&\approx \sigma _A\left( 1-\frac{\sigma _B^2}{2\sigma _A^2}\frac{1}{1+\Gamma ^2}\right) ,\end{aligned}$$

(14.5a)

$$\begin{aligned} \tilde{\rho }&\approx \rho _A\left( 1-\frac{\rho _B^2}{2\rho _A^2}\frac{\Gamma ^2}{1+\Gamma ^2}\right) ,\end{aligned}$$

(14.5b)

$$\begin{aligned} C&\approx u\frac{\sigma _B\rho _B}{2\sigma _A}\frac{1}{1+\Gamma ^2}P\left( \alpha \right) ,\end{aligned}$$

(14.5c)

$$\begin{aligned} S&\approx \frac{1}{u}\frac{\sigma _B\rho _B}{2\rho _A}\frac{\Gamma ^2}{1+\Gamma ^2}Q\left( \alpha \right) , \end{aligned}$$

(14.5d)

with $\Gamma =\rho _Au\gamma /\left( 2\pi \sigma _A\right)$, $P\left( \alpha \right) =\cos \alpha +\Gamma \sin \alpha$, and $Q\left( \alpha \right) =\cos \alpha +\Gamma ^{-1}\sin \alpha$. $\tilde{T}$ can be treated as the envelope line of the actual temperature T. Here, we extend the results reported in Ref. [18] by additionally considering a phase difference of $\alpha$. $\tilde{\sigma }$ and $\tilde{\rho }$ are irrelevant to $\alpha$, but C and S are dependent on $\alpha$, offering a tunable parameter.

Fig. 14.1

Adapted from Ref. [35]

Thermal wave nonreciprocity. a Forward case. b Backward-1 case by changing the source position. c Backward-2 case by changing the modulation direction.

We then qualitatively discuss the nonreciprocity induced by spatiotemporal modulation. In what follows, the subscripts of f, b1, and b2 denote the parameters related to the forward case in Fig. 14.1a, the backward-1 case in Fig. 14.1b, and the backward-2 case in Fig. 14.1c, respectively. The two backward cases are equivalent only when $\alpha =0$. Since $\tilde{\sigma }$ and $\tilde{\rho }$ do not contribute to nonreciprocity, we mainly discuss C and S in detail.

Fig. 14.2

Adapted from Ref. [35]

C and S as functions of $\alpha$. Parameters: $\sigma _A=300$ W m$^{-1}$ K$^{-1}$, $\sigma _B=100$ W m$^{-1}$ K$^{-1}$, $\rho _A=3\times 10^6$ J m$^{-3}$ K$^{-1}$, $\rho _B=5\times 10^5$ J m$^{-3}$ K$^{-1}$, and $u=0.05$ m/s.

For the forward case, we know $C_f=C$ and $S_f=S$. For the backward-1 case, we can derive $C_{b1}=-C$ and $S_{b1}=-S$. Nonreciprocity requires $C_f\ne C_{b1}$ (or $S_f\ne S_{b1}$). Therefore, as long as $C\ne 0$ (or $S\ne 0$), nonreciprocity will occur and a larger C (or S) yields larger nonreciprocity. For clarity, we plot the functions of $C\left( \alpha \right)$ and $S\left( \alpha \right)$ in Fig. 14.2 with $\Gamma =0.5,\,1,\,2$. The maximum and minimum values of C appear at $\alpha =-\textrm{arccot}\Gamma +\pi /2$ and $\alpha =-\textrm{arccot}\Gamma -\pi /2$, respectively; and the zero value occurs at $\alpha =\arctan \Gamma \pm \pi /2$. The maximum and minimum values of S appear at $\alpha =-\arctan \Gamma +\pi /2$ and $\alpha =-\arctan \Gamma -\pi /2$, respectively; and the zero value occurs at $\alpha =\textrm{arccot}\Gamma \pm \pi /2$. For the backward-2 case, we can obtain $C_{b2}=C\left( -u\right)$ and $S_{b2}=S\left( -u\right)$. Nonreciprocity requires $C_f\ne C_{b2}$ (or $S_f\ne S_{b2}$). Therefore, as long as $C\left( u\right) \ne C\left( -u\right)$ [or $S\left( u\right) \ne S\left( -u\right)$], nonreciprocity will occur. We can also observe that $\alpha =\pm \pi /2$ makes $P\left( \alpha \right)$ and $Q\left( \alpha \right)$ two odd functions of u. C and S then become two even functions of u, so nonreciprocity disappears. In one word, the nonreciprocity condition for the backward-1 case is $C\ne 0$ (or $S\ne 0$), and that for the backward-2 case is $C\left( u\right) \ne C\left( -u\right)$ [or $S\left( u\right) \ne S\left( -u\right)$]. Especially when $\alpha =0$, $C\left( -u\right) =-C\left( u\right)$ [or $S\left( -u\right) =-S\left( u\right)$], the nonreciprocity condition for the backward-2 case can then be reduced to $C\ne 0$ (or $S\ne 0$), which is the same as that for the backward-1 case.

We then consider a transient case that can support thermal waves’ propagation. Qualitative analysis is insufficient since both C and S can contribute to nonreciprocity. Therefore, we quantitatively discuss a rectification ratio. For this purpose, we apply a periodic temperature at the left side of the structure in Fig. 14.1a to generate a forward thermal wave described by Eq. (14.3). The periodic temperature has a form of $T_p=\phi _0\textrm{e}^{-\textrm{i}\omega t}+T_0$ where $\phi _0$ denote the temperature amplitude. We set the reference temperature $T_0=0$ K in theoretical discussions for brevity. The envelope line of the actual temperature T can then be expressed as

$$\begin{aligned} \tilde{T}=\phi _0\textrm{e}^{\textrm{i}\left( k x-\omega t\right) }. \end{aligned}$$

(14.6)

The real part of Eq. (14.6) makes sense, which has been experimentally realized by periodically heating a material  [24, 25]. The substitution of Eq. (14.6) into Eq. (14.4) yields

$$\begin{aligned} -\textrm{i}\omega \tilde{\rho }+\textrm{i}kC+k^2\tilde{\sigma }-\omega kS=0. \end{aligned}$$

(14.7)

Since thermal conduction features dissipation, the wave number k should be complex, i.e., $k=\mu +\textrm{i}\xi$ with $\mu$ and $\xi$ being two real numbers. Equation (14.6) can then be rewritten as $\tilde{T}=\phi _0\textrm{e}^{-\xi x}\textrm{e}^{\textrm{i}\left( \mu x-\omega t\right) }$. Therefore, the physical meaning of $\mu$ is the wave number and that of $\xi$ is the spatial decay rate. With the complex k, Eq. (14.7) can be further reduced to

$$\begin{aligned} -\textrm{i}\omega \tilde{\rho }+\textrm{i}\left( \mu +\textrm{i}\xi \right) C+\left( \mu +\textrm{i}\xi \right) ^2\tilde{\sigma }-\omega \left( \mu +\textrm{i}\xi \right) S=0. \end{aligned}$$

(14.8)

By independently considering the real and imaginary parts of Eq. (14.8), we can derive two equations,

$$\begin{aligned} -\xi C+\left( \mu ^2-\xi ^2\right) \tilde{\sigma }-\omega \mu S&=0,\end{aligned}$$

(14.9a)

$$\begin{aligned} \omega \tilde{\rho }-\mu C-2\mu \xi \tilde{\sigma }+\omega \xi S&=0. \end{aligned}$$

(14.9b)

The solution to Eq. (14.9) is

$$\begin{aligned} \mu&=\frac{2S\omega +\sqrt{2}\varepsilon }{4\tilde{\sigma }},\end{aligned}$$

(14.10a)

$$\begin{aligned} \xi&=\frac{-4C\omega \left( 2\tilde{\sigma }\tilde{\rho }-CS\right) +2\sqrt{2}\left( C^2-S^2\omega ^2\right) \varepsilon +\sqrt{2}\varepsilon ^3}{8\tilde{\sigma }\left( 2\tilde{\sigma }\tilde{\rho }-CS\right) \omega }, \end{aligned}$$

(14.10b)

with $\varepsilon =\sqrt{-C^2+S^2\omega ^2+\sqrt{\left( C^2+S^2\omega ^2\right) ^2+16\omega ^2\tilde{\sigma }\tilde{\rho }\left( \tilde{\sigma }\tilde{\rho }-CS\right) }}$. Although Eq. (14.10) is complicated, we can discuss some special conditions to have a rough idea. For the forward case, we can know $\mu _f=\mu$ and $\xi _f=\xi$. For the backward-1 case, we can derive $\mu _{b1}=\mu \left( -C,\,-S\right)$ and $\xi _{b1}=\xi \left( -C,\,-S\right)$. Due to $\varepsilon \left( C,\,S\right) =\varepsilon \left( -C,\,-S\right)$, it does contribute to nonreciprocity, so the nonreciprocity origins of $\mu$ and $\xi$ lie in S and C, respectively (Eq. (14.10)). We can then conclude that nonreciprocal $\mu$ requires $S\ne 0$ (i.e., $\alpha \ne \textrm{arccot}\Gamma \pm \pi /2$) and nonreciprocal $\xi$ requires $C\ne 0$ (i.e., $\alpha \ne \arctan \Gamma \pm \pi /2$). For the backward-2 case, we can derive $\mu _{b2}=\mu \left[ C\left( -u\right) ,\,S\left( -u\right) \right]$ and $\xi _{b2}=\xi \left[ C\left( -u\right) ,\,S\left( -u\right) \right]$. When $\alpha =\pm \pi /2$, C, S, and $\varepsilon$ are all even functions of u, so nonreciprocity will disappear. Therefore, nonreciprocal $\mu$ (or $\xi$) requires $\alpha \ne \pm \pi /2$.

In general, it makes little sense to define a rectification ratio based on wave numbers. However, it is meaningful to define a rectification ratio $\left( R_T\right)$ based on the temperature amplitude $\left( \phi _0\textrm{e}^{-\xi x}\right)$ or the reciprocal of spatial decay rate $\left( 1/\xi \right)$,

$$\begin{aligned} R_{T1}=\frac{1/\xi _f-1/\xi _{b1}}{1/\xi _f+1/\xi _{b1}}=\frac{\xi _{b1}-\xi _f}{\xi _{b1}+\xi _f},\end{aligned}$$

(14.11a)

$$\begin{aligned} R_{T2}=\frac{1/\xi _f-1/\xi _{b2}}{1/\xi _f+1/\xi _{b2}}=\frac{\xi _{b2}-\xi _f}{\xi _{b2}+\xi _f}, \end{aligned}$$

(14.11b)

where $R_{T1}$ and $R_{T2}$ are defined for the backward-1 and backward-2 cases, respectively. We plot $R_{T1}$ and $R_{T2}$ as functions of $\alpha$ in Fig. 14.3. The results demonstrate that a smaller $\Gamma$ or a smaller $\omega$ yields larger nonreciprocity. Therefore, both $R_{T1}$ and $R_{T2}$ can theoretically reach 1, and we can obtain a perfect thermal wave diode. Especially when $\alpha =0$, Eq. (14.11) can be reduce to

$$\begin{aligned} R_{T1}=R_{T2}=\frac{2\sqrt{2}C\omega \left( 2\tilde{\sigma }\tilde{\rho }-CS\right) }{2\left( C^2-S^2\omega ^2\right) \varepsilon +\varepsilon ^3}, \end{aligned}$$

(14.12)

indicating that the two backward cases are equivalent when $\alpha =0$.

Fig. 14.3

Adapted from Ref. [35]

$R_{T1}$ and $R_{T2}$ as functions of $\alpha$. The parameters are the same as those for Fig. 14.2.

Another possibility to define a rectification ratio $\left( R_J\right)$ lies in nonreciprocal heat fluxes J. For this purpose, we define the dynamic heat flux J according to Eq. (14.1),

$$\begin{aligned} \nonumber J&=-\sigma \left( x-u t\right) \frac{\partial T}{\partial x}=-\sigma \left( x-u t\right) \frac{\partial }{\partial x}\left[ \phi \left( x-u t\right) \textrm{e}^{-\xi x}\textrm{e}^{\textrm{i}\left( \mu x-\omega t\right) }\right] \\&=-\sigma \left( x-u t\right) \left[ \phi '\left( x-u t\right) +\left( -\xi +\textrm{i}\mu \right) \phi \left( x-u t\right) \right] \textrm{e}^{-\xi x}\textrm{e}^{\textrm{i}\left( \mu x-\omega t\right) }, \end{aligned}$$

(14.13)

where $\phi '\left( x-u t\right) =\partial \phi \left( x-u t\right) /\partial x$. Since $\sigma \left( x-u t\right)$, $\phi \left( x-u t\right)$, and $\phi '\left( x-u t\right)$ are all periodic functions, the dynamic heat flux described by Eq. (14.13) varies with temporal periodicity, but the heat flux amplitude decays along the x axis due to the term of $\textrm{e}^{-\xi x}$. Therefore, we can also define $R_J$ based on the reciprocal of spatial decay rate $\left( 1/\xi \right)$ which should have the same form as Eq. (14.11), indicating that the whole theoretical framework is self-consistent.

We can then draw a brief conclusion. Spatiotemporal modulation can generate two additional terms: the convective term associated with C and the Willis term related to S. C and S can be flexibly tuned by $\alpha$. We also discuss two backward cases: (I) changing the source position and (II) changing the modulation direction, equivalent only when $\alpha =0$. We further discuss their nonreciprocity conditions and define a rectification ratio ($R_T$ or $R_J$) based on the reciprocal of spatial decay rate $\left( 1/\xi \right)$.

3 Finite-Element Simulation

We then perform simulations with COMSOL Multiphysics to confirm the theoretical analyses. For this purpose, we study the thermal conduction in a one-dimensional structure whose parameters are spatiotemporally modulated as described by Eq. (14.2) with $\Gamma =1$. For accuracy, the mesh size is one-tenth of the modulation wavelength ($\gamma$), and the time tolerance is $10^{-6}$.

Fig. 14.4

Adapted from Ref. [35]

Simulations of the backward-1 case. The parameters are the same as those for Fig. 14.2 with $\Gamma =1$ and $L=0.2$ m. The periodic temperature is set at $T_p=40\cos \left( -2\pi t/50\right) +323$ K. The detected position locates at the center of the structure. a and b Temperature evolution. c and d Heat flux evolution.

We firstly discuss the backward-1 case, which requires changing the source position but keeping the modulation direction (Fig. 14.1b). As theoretically predicted (Eq. (14.11)), $R_{T1}=0$ occurs when $\alpha =\pi /4\pm \pi /2$. For brevity, we set $\alpha =-\pi /4$ to perform simulations. The temperature and heat flux evolutions are presented in Fig. 14.4a, c, respectively. We can observe that the forward and backward-1 propagations are the same, indicating reciprocal propagations. Moreover, $R_{T1}$ reaches the maximum value when $\alpha =\pi /4$ as predicted. We also perform simulations with $\alpha =\pi /4$, and the results are presented in Fig. 14.4b, d. The temperature amplitudes are different, indicating nonreciprocal propagations. The theoretical prediction of the forward and backward-1 temperature amplitudes are 5.13 and 1.87 K, respectively. The simulations show that the forward and backward-1 temperature amplitudes are 5.23 and 1.92 K, respectively. Therefore, the simulations agree well with the theoretical predictions.

Fig. 14.5

Adapted from Ref. [35]

Simulations of the backward-2 case. The parameters are the same as those for Fig. 14.4. The difference from Fig. 14.4 is that here we change the modulation speed instead of changing the source position.

We then discuss the backward-2 case, which requires changing the modulation direction but keeping the source position (Fig. 14.1c). Equation (14.11) tells that $R_{T2}=0$ appears when $\alpha =\pm \pi /2$, and we set $\alpha =-\pi /2$ to perform simulations (Fig. 14.5a, c). The forward and backward-2 propagations have the same temperature (or heat flux) amplitudes, indicating reciprocal thermal waves. In addition, $R_{T2}$ reaches the maximum value when $\alpha =0$ as predicted, and the simulation results are presented in Fig. 14.5b, d. The theoretical prediction of the forward and backward-2 temperature amplitudes are 4.48 and 2.19 K, respectively. The simulations demonstrate that the forward and backward-1 temperature amplitudes are 4.53 and 2.24 K, respectively. Again, the simulations and theories have good agreement.

We finally provide some experimental suggestions to ensure the feasibility of practical implementations. The most crucial is to realize spatiotemporal modulations of $\sigma$ (thermal conductivity) and $\rho$ (the product of mass density and heat capacity). We firstly discuss the spatiotemporal modulation of $\sigma$. Many studies have shown that thermal conductivities can be flexibly controlled by external fields like electric fields [36, 37] and light fields [38]. The in-plane thermal conductivity can change two orders of magnitude with an out-of-plane electric field [36]. We then discuss the spatiotemporal modulation of $\rho$ by considering heat capacity. Many materials have a phase change [39] in the presence of an electric field, so heat capacities change with the phase change. Therefore, spatiotemporal modulations of $\sigma$ and $\rho$ can be realized with an electric field. Moreover, Ref. [19] also provides an insight into practical implementations, although the experiments were conducted in electrics. Since thermotics and electrics follow similar equations (thermal conductivity corresponds to electric conductivity and heat capacity corresponds to electric capacity), spatiotemporal modulations of $\sigma$ and $\rho$ might also be realized by rotating disks, as presented in Ref. [19]. A periodic temperature can be obtained by directly using a pulse heat source or alternately using a ceramic heater and a semiconductor cooler. Therefore, these results should be possible to be experimentally validated. Here, thermal waves are based on the Fourier law, and many other kinds of thermal waves remain further explored, i.e., those considering thermal relaxation [40,41,42,43].

4 Conclusion

We propose the mechanism of tunable thermal wave nonreciprocity with spatiotemporal modulation. The tunability lies in the phase difference ($\alpha$) between two spatiotemporally modulated parameters. We reveal that the homogenized thermal conductivity ($\tilde{\sigma }$) and the homogenized product of mass density and heat capacity ($\tilde{\rho }$) are independent of the phase difference ($\alpha$). Still, the convective term (C) and the Willis term (S) are crucially dependent on the phase difference ($\alpha$). We also discuss two backward cases: (I) changing the source position and (II) changing the modulation direction. The two cases are equivalent only when $\alpha =0$. We further define a rectification ratio ($R_{T1}$ or $R_{T2}$) based on the reciprocal of spatial decay rate ($1/\xi$) and discuss nonreciprocity conditions. These theoretical analyses are all confirmed by finite-element simulations, and experimental suggestions are also given to ensure feasibility. These results could provide distinct opportunities for nonreciprocal heat transfer.

5 Exercise and Solution

Exercise

• 1. Derive Eq. (14.5) by homogenizing spatiotemporal modulation.

Solution

• 1. We consider two variable substitutions of $n=x-u t$ and $\tau =t$, yielding $\partial /\partial x=\partial /\partial n$ and $\partial /\partial t=\partial /\partial \tau -u \partial /\partial n$. Equation (14.1) can then be reduced to

  $$\begin{aligned} \rho \left( n\right) \frac{\partial T}{\partial \tau }-u\rho \left( n\right) \frac{\partial T}{\partial n}+\frac{\partial }{\partial n}\left[ -\sigma \left( n\right) \frac{\partial T}{\partial n}\right] =0. \end{aligned}$$

  (14.14)

  Similarly, Eq. (14.2) can also be simplified as

  $$\begin{aligned} \sigma \left( n\right)&=\sigma _A+\sigma _B\cos \left( Kn\right) ,\end{aligned}$$

  (14.15a)
  $$\begin{aligned} \rho \left( n\right)&=\rho _A+\rho _B\cos \left( Kn+\alpha \right) . \end{aligned}$$

  (14.15b)

  We rewrite Eq. (14.15) with the Fourier expansion,

  $$\begin{aligned} \sigma \left( n\right)&=\sum _{s=0,\,\pm 1}\sigma _s\textrm{e}^{\textrm{i}K_s n}=\sigma _0\textrm{e}^{\textrm{i}K_0 n}+\sigma _{+1}\textrm{e}^{\textrm{i}K_{+1} n}+\sigma _{-1}\textrm{e}^{\textrm{i}K_{-1} n},\end{aligned}$$

  (14.16a)
  $$\begin{aligned} \rho \left( n\right)&=\sum _{s=0,\,\pm 1}\rho _s\textrm{e}^{\textrm{i}K_s n}=\rho _0\textrm{e}^{\textrm{i}K_0 n}+\rho _{+1}\textrm{e}^{\textrm{i}K_{+1} n}+\rho _{-1}\textrm{e}^{\textrm{i}K_{-1} n}, \end{aligned}$$

  (14.16b)

  with $K_0=0$, $K_{\pm 1}=\pm K$, $\sigma _0=\sigma _A$, $\sigma _{\pm 1}=\sigma _B/2$, $\rho _0=\rho _A$, and $\rho _{\pm 1}=\textrm{e}^{\pm \textrm{i}\alpha }\rho _B/2$. With the Bloch theorem, we can express the temperature solution as

  $$\begin{aligned} \nonumber T\left( n,\,\tau \right)&=\phi \left( n\right) \textrm{e}^{\textrm{i}\left( G n-W \tau \right) }=\left( \sum _{s=0,\,\pm 1}\phi _s\textrm{e}^{\textrm{i}K_s n}\right) \textrm{e}^{\textrm{i}\left( G n-W \tau \right) }\\&=\left( \phi _0\textrm{e}^{\textrm{i}K_0 n}+\phi _{+1}\textrm{e}^{\textrm{i}K_{+1} n}+\phi _{-1}\textrm{e}^{\textrm{i}K_{-1} n}\right) \textrm{e}^{\textrm{i}\left( G n-W \tau \right) }, \end{aligned}$$

  (14.17)

  where G and W are the wave number and circular frequency in the $n-\tau$ frame. $\phi \left( n\right)$ is the amplitude modulation function. We can then express $\partial T/\partial \tau$ and $\partial T/\partial n$ as

  $$\begin{aligned} \frac{\partial T}{\partial \tau }=-\textrm{i}W\left( \phi _0\textrm{e}^{\textrm{i}K_0 n}+\phi _{+1}\textrm{e}^{\textrm{i}K_{+1} n}+\phi _{-1}\textrm{e}^{\textrm{i}K_{-1} n}\right) \textrm{e}^{\textrm{i}\left( G n-W \tau \right) }, \end{aligned}$$

  (14.18)
  $$\begin{aligned} \frac{\partial T}{\partial n}=\textrm{i}\left[ \left( G+K_0\right) \phi _0\textrm{e}^{\textrm{i}K_0 n}+\left( G+K_{+1}\right) \phi _{+1}\textrm{e}^{\textrm{i}K_{+1} n}+\left( G+K_{-1}\right) \phi _{-1}\textrm{e}^{\textrm{i}K_{-1} n}\right] \textrm{e}^{\textrm{i}\left( G n-W \tau \right) }. \end{aligned}$$

  (14.19)

  We can further write $\rho \left( n\right) \partial T/\partial \tau$ as

  $$\begin{aligned} \nonumber \rho \left( n\right) \frac{\partial T}{\partial \tau }=&-\textrm{i}W\left( \rho _0\phi _0+\rho _{+1}\phi _{-1}+\rho _{-1}\phi _{+1}\right) \textrm{e}^{\textrm{i}K_0 n}\textrm{e}^{\textrm{i}\left( G n-W \tau \right) }\\ \nonumber&-\textrm{i}W\left( \rho _0\phi _{+1}+\rho _{+1}\phi _0\right) \textrm{e}^{\textrm{i}K_{+1} n}\textrm{e}^{\textrm{i}\left( G n-W \tau \right) }\\ \nonumber&-\textrm{i}W\left( \rho _0\phi _{-1}+\rho _{-1}\phi _0\right) \textrm{e}^{\textrm{i}K_{-1} n}\textrm{e}^{\textrm{i}\left( G n-W \tau \right) }\\&+o\left( \textrm{e}^{\textrm{i}K_{\pm 1} n}\right) . \end{aligned}$$

  (14.20)

  We can also express $-u\rho \left( n\right) \partial T/\partial n$ and $-\sigma \left( n\right) \partial T/\partial n$ as

  $$\begin{aligned} \nonumber -u\rho \left( n\right) \frac{\partial T}{\partial n}=&-\textrm{i}u\left[ \rho _0\left( G+K_0\right) \phi _0+\rho _{+1}\left( G+K_{-1}\right) \phi _{-1}+\rho _{-1}\left( G+K_{+1}\right) \phi _{+1}\right] \textrm{e}^{\textrm{i}K_0 n}\textrm{e}^{\textrm{i}\left( G n-W \tau \right) }\\ \nonumber&-\textrm{i}u\left[ \rho _0\left( G+K_{+1}\right) \phi _{+1}+\rho _{+1}\left( G+K_0\right) \phi _0\right] \textrm{e}^{\textrm{i}K_{+1} n}\textrm{e}^{\textrm{i}\left( G n-W \tau \right) }\\ \nonumber&-\textrm{i}u\left[ \rho _0\left( G+K_{-1}\right) \phi _{-1}+\rho _{-1}\left( G+K_0\right) \phi _0\right] \textrm{e}^{\textrm{i}K_{-1} n}\textrm{e}^{\textrm{i}\left( G n-W \tau \right) }\\&+o\left( \textrm{e}^{\textrm{i}K_{\pm 1} n}\right) , \end{aligned}$$

  (14.21)
  $$\begin{aligned} \nonumber -\sigma \left( n\right) \frac{\partial T}{\partial n}=&-\textrm{i}\left[ \sigma _0\left( G+K_0\right) \phi _0+\sigma _{+1}\left( G+K_{-1}\right) \phi _{-1}+\sigma _{-1}\left( G+K_{+1}\right) \phi _{+1}\right] \textrm{e}^{\textrm{i}K_0 n}\textrm{e}^{\textrm{i}\left( G n-W \tau \right) }\\ \nonumber&-\textrm{i}\left[ \sigma _0\left( G+K_{+1}\right) \phi _{+1}+\sigma _{+1}\left( G+K_0\right) \phi _0\right] \textrm{e}^{\textrm{i}K_{+1} n}\textrm{e}^{\textrm{i}\left( G n-W \tau \right) }\\ \nonumber&-\textrm{i}\left[ \sigma _0\left( G+K_{-1}\right) \phi _{-1}+\sigma _{-1}\left( G+K_0\right) \phi _0\right] \textrm{e}^{\textrm{i}K_{-1} n}\textrm{e}^{\textrm{i}\left( G n-W \tau \right) }\\&+o\left( \textrm{e}^{\textrm{i}K_{\pm 1} n}\right) . \end{aligned}$$

  (14.22)

  With Eq. (14.22), we can further derive

  $$\begin{aligned} \nonumber&\frac{\partial }{\partial n}\left[ -\sigma \left( n\right) \frac{\partial T}{\partial n}\right] \\ \nonumber&=\left( G+K_0\right) \left[ \sigma _0\left( G+K_0\right) \phi _0+\sigma _{+1}\left( G+K_{-1}\right) \phi _{-1}+\sigma _{-1}\left( G+K_{+1}\right) \phi _{+1}\right] \textrm{e}^{\textrm{i}K_0 n}\textrm{e}^{\textrm{i}\left( G n-W \tau \right) }\\ \nonumber&+\left( G+K_{+1}\right) \left[ \sigma _0\left( G+K_{+1}\right) \phi _{+1}+\sigma _{+1}\left( G+K_0\right) \phi _0\right] \textrm{e}^{\textrm{i}K_{+1} n}\textrm{e}^{\textrm{i}\left( G n-W \tau \right) }\\ \nonumber&+\left( G+K_{-1}\right) \left[ \sigma _0\left( G+K_{-1}\right) \phi _{-1}+\sigma _{-1}\left( G+K_0\right) \phi _0\right] \textrm{e}^{\textrm{i}K_{-1} n}\textrm{e}^{\textrm{i}\left( G n-W \tau \right) }\\&+o\left( \textrm{e}^{\textrm{i}K_{\pm 1} n}\right) . \end{aligned}$$

  (14.23)

  By arranging the terms associated with $\textrm{e}^{\textrm{i}K_0 n}$, $\textrm{e}^{\textrm{i}K_{+1} n}$, and $\textrm{e}^{\textrm{i}K_{-1} n}$ in Eqs. (14.20), (14.21) and (14.23) together, we can obtain three equations,

  $$\begin{aligned} \nonumber&-\textrm{i}\left[ \rho _0\left( W+uG+uK_0\right) \phi _0+\rho _{+1}\left( W+uG+uK_{-1}\right) \phi _{-1}+\rho _{-1}\left( W+uG+uK_{+1}\right) \phi _{+1}\right] \\&+\left( G+K_0\right) \left[ \sigma _0\left( G+K_0\right) \phi _0+\sigma _{+1}\left( G+K_{-1}\right) \phi _{-1}+\sigma _{-1}\left( G+K_{+1}\right) \phi _{+1}\right] =0,\end{aligned}$$

  (14.24a)
  $$\begin{aligned} \nonumber&-\textrm{i}\left[ \rho _0\left( W+uG+uK_{+1}\right) \phi _{+1}+\rho _{+1}\left( W+uG+uK_0\right) \phi _0\right] \\&+\left( G+K_{+1}\right) \left[ \sigma _0\left( G+K_{+1}\right) \phi _{+1}+\sigma _{+1}\left( G+K_0\right) \phi _0\right] =0,\end{aligned}$$

  (14.24b)
  $$\begin{aligned} \nonumber&-\textrm{i}\left[ \rho _0\left( W+uG+uK_{-1}\right) \phi _{-1}+\rho _{-1}\left( W+uG+uK_0\right) \phi _0\right] \\&+\left( G+K_{-1}\right) \left[ \sigma _0\left( G+K_{-1}\right) \phi _{-1}+\sigma _{-1}\left( G+K_0\right) \phi _0\right] =0. \end{aligned}$$

  (14.24c)

  Equation (14.24) is written in the $n-\tau$ frame, and we can also express it in the $x-t$ frame by taking $k=G$ and $\omega =W+uG$ where k and $\omega$ are, respectively, the wave vector and circular frequency in the $x-t$ frame,

  $$\begin{aligned} \nonumber&-\textrm{i}\left[ \rho _0\left( \omega +uK_0\right) \phi _0+\rho _{+1}\left( \omega +uK_{-1}\right) \phi _{-1}+\rho _{-1}\left( \omega +uK_{+1}\right) \phi _{+1}\right] \\&+\left( k+K_0\right) \left[ \sigma _0\left( k+K_0\right) \phi _0+\sigma _{+1}\left( k+K_{-1}\right) \phi _{-1}+\sigma _{-1}\left( k+K_{+1}\right) \phi _{+1}\right] =0,\end{aligned}$$

  (14.25a)
  $$\begin{aligned} \nonumber&-\textrm{i}\left[ \rho _0\left( \omega +uK_{+1}\right) \phi _{+1}+\rho _{+1}\left( \omega +uK_0\right) \phi _0\right] \\&+\left( k+K_{+1}\right) \left[ \sigma _0\left( k+K_{+1}\right) \phi _{+1}+\sigma _{+1}\left( k+K_0\right) \phi _0\right] =0,\end{aligned}$$

  (14.25b)
  $$\begin{aligned} \nonumber&-\textrm{i}\left[ \rho _0\left( \omega +uK_{-1}\right) \phi _{-1}+\rho _{-1}\left( \omega +uK_0\right) \phi _0\right] \\&+\left( k+K_{-1}\right) \left[ \sigma _0\left( k+K_{-1}\right) \phi _{-1}+\sigma _{-1}\left( k+K_0\right) \phi _0\right] =0. \end{aligned}$$

  (14.25c)

  With Eqs. (14.25b) and (14.25c) , we can derive the expressions of $\phi _{+1}$ and $\phi _{-1}$,

  $$\begin{aligned} \phi _{+1}&=-\frac{\left( k+K_{+1}\right) \sigma _{+1}\left( k+K_0\right) -\textrm{i}\rho _{+1}\left( \omega +u K_0\right) }{\left( k+K_{+1}\right) \sigma _0\left( k+K_{+1}\right) -\textrm{i}\rho _0\left( \omega +u K_{+1}\right) }\phi _0,\end{aligned}$$

  (14.26a)
  $$\begin{aligned} \phi _{-1}&=-\frac{\left( k+K_{-1}\right) \sigma _{-1}\left( k+K_0\right) -\textrm{i}\rho _{-1}\left( \omega +u K_0\right) }{\left( k+K_{-1}\right) \sigma _0\left( k+K_{-1}\right) -\textrm{i}\rho _0\left( \omega +u K_{-1}\right) }\phi _0. \end{aligned}$$

  (14.26b)

  We then consider two approximations of $k\ll K$ and $\omega \ll uK$, so Eq. (14.26) can be reduced to

  $$\begin{aligned} \phi _{+1}&=-\frac{K_{+1}\sigma _{+1}k-\textrm{i}\rho _{+1}\omega }{K_{+1}\sigma _0K_{+1}-\textrm{i}\rho _0u K_{+1}}\phi _0,\end{aligned}$$

  (14.27a)
  $$\begin{aligned} \phi _{-1}&=-\frac{K_{-1}\sigma _{-1}k-\textrm{i}\rho _{-1}\omega }{K_{-1}\sigma _0K_{-1}-\textrm{i}\rho _0u K_{-1}}\phi _0. \end{aligned}$$

  (14.27b)

  Similarly, Eq. (14.25a) can also be reduced to

  $$\begin{aligned} k^2\sigma _0\phi _0-\textrm{i}\omega \rho _0\phi _0+\left( k\sigma _{-1}K_{+1}-\textrm{i}\rho _{-1}uK_{+1}\right) \phi _{+1}+\left( k\sigma _{+1}K_{-1}-\textrm{i}\rho _{+1}uK_{-1}\right) \phi _{-1}=0.\nonumber \\ \end{aligned}$$

  (14.28)

  The substitution of Eq. (14.27) into Eq. (14.28) yields

  $$\begin{aligned} \nonumber k^2\sigma _0\phi _0-\textrm{i}\omega \rho _0\phi _0&-\frac{\left( k\sigma _{-1}K_{+1}-\textrm{i}\rho _{-1}uK_{+1}\right) \left( K_{+1}\sigma _{+1}k-\textrm{i}\rho _{+1}\omega \right) }{K_{+1}\sigma _0K_{+1}-\textrm{i}\rho _0u K_{+1}}\phi _0\\&-\frac{\left( k\sigma _{+1}K_{-1}-\textrm{i}\rho _{+1}uK_{-1}\right) \left( K_{-1}\sigma _{-1}k-\textrm{i}\rho _{-1}\omega \right) }{K_{-1}\sigma _0K_{-1}-\textrm{i}\rho _0u K_{-1}}\phi _0=0. \end{aligned}$$

  (14.29)

  Equation (14.29) can be further arranged in a physical form,

  $$\begin{aligned} \nonumber&-\textrm{i}\omega \left( \rho _0+\frac{\textrm{i}Ku\rho _{+1}\rho _{-1}}{\sigma _0K^2-\textrm{i}\rho _0u K}+\frac{-\textrm{i}Ku\rho _{+1}\rho _{-1}}{\sigma _0K^2+\textrm{i}\rho _0u K}\right) \phi _0 \\ \nonumber&+\textrm{i}k\left( \frac{K^2u\sigma _{+1}\rho _{-1}}{\sigma _0K^2-\textrm{i}\rho _0u K}+\frac{K^2u\sigma _{-1}\rho _{+1}}{\sigma _0K^2+\textrm{i}\rho _0u K}\right) \phi _0\\ \nonumber&+k^2\left( \sigma _0-\frac{K^2\sigma _{+1}\sigma _{-1}}{\sigma _0K^2-\textrm{i}\rho _0u K}-\frac{K^2\sigma _{+1}\sigma _{-1}}{\sigma _0K^2+\textrm{i}\rho _0u K}\right) \phi _0\\&-\omega k\left( \frac{-\textrm{i}K\rho _{+1}\sigma _{-1}}{\sigma _0K^2-\textrm{i}\rho _0u K}+\frac{\textrm{i}K\rho _{-1}\sigma _{+1}}{\sigma _0K^2+\textrm{i}\rho _0u K}\right) \phi _0=0. \end{aligned}$$

  (14.30)

  By taking $\partial /\partial t=-\textrm{i}\omega$, $\partial /\partial x=\textrm{i}k$, and $\tilde{T}=\phi _0\textrm{e}^{\textrm{i}\left( kx-\omega t\right) }$, we can rewrite Eq. (14.30) as

  $$\begin{aligned} \tilde{\rho }\frac{\partial \tilde{T}}{\partial t}+C\frac{\partial \tilde{T}}{\partial x}-\tilde{\sigma }\frac{\partial ^2 \tilde{T}}{\partial x^2}-S\frac{\partial ^2 \tilde{T}}{\partial x \partial t}=0, \end{aligned}$$

  (14.31)

  where the homogenized parameters take the form of

  $$\begin{aligned} \tilde{\sigma }&=\sigma _0-\frac{K^2\sigma _{+1}\sigma _{-1}}{\sigma _0K^2-\textrm{i}\rho _0u K}-\frac{K^2\sigma _{+1}\sigma _{-1}}{\sigma _0K^2+\textrm{i}\rho _0u K},\end{aligned}$$

  (14.32a)
  $$\begin{aligned} \tilde{\rho }&=\rho _0+\frac{\textrm{i}Ku\rho _{+1}\rho _{-1}}{\sigma _0K^2-\textrm{i}\rho _0u K}+\frac{-\textrm{i}Ku\rho _{+1}\rho _{-1}}{\sigma _0K^2+\textrm{i}\rho _0u K},\end{aligned}$$

  (14.32b)
  $$\begin{aligned} C&=\frac{K^2u\sigma _{+1}\rho _{-1}}{\sigma _0K^2-\textrm{i}\rho _0u K}+\frac{K^2u\sigma _{-1}\rho _{+1}}{\sigma _0K^2+\textrm{i}\rho _0u K},\end{aligned}$$

  (14.32c)
  $$\begin{aligned} S&=\frac{-\textrm{i}K\sigma _{-1}\rho _{+1}}{\sigma _0K^2-\textrm{i}\rho _0u K}+\frac{\textrm{i}K\sigma _{+1}\rho _{-1}}{\sigma _0K^2+\textrm{i}\rho _0u K}. \end{aligned}$$

  (14.32d)

  We can further reduce Eq. (14.32) to

  $$\begin{aligned} \tilde{\sigma }&=\sigma _A\left( 1-\frac{\sigma _B^2}{2\sigma _A^2}\frac{1}{1+\Gamma ^2}\right) ,\end{aligned}$$

  (14.33a)
  $$\begin{aligned} \tilde{\rho }&=\rho _A\left( 1-\frac{\rho _B^2}{2\rho _A^2}\frac{\Gamma ^2}{1+\Gamma ^2}\right) ,\end{aligned}$$

  (14.33b)
  $$\begin{aligned} C&=u\frac{\sigma _B\rho _B}{2\sigma _A}\frac{1}{1+\Gamma ^2}\left( \cos \alpha +\Gamma \sin \alpha \right) ,\end{aligned}$$

  (14.33c)
  $$\begin{aligned} S&=\frac{1}{u}\frac{\sigma _B\rho _B}{2\rho _A}\frac{\Gamma ^2}{1+\Gamma ^2}\left( \cos \alpha +\frac{1}{\Gamma }\sin \alpha \right) , \end{aligned}$$

  (14.33d)

  with $\Gamma =\rho _Au\gamma /\left( 2\pi \sigma _A\right)$.