- Email: [email protected]
Unified derivation of reflectivity for all waves
Journal Pre-proofs Microarticle Unified Derivation of Reflectivity for All Waves Peng Li, Xue-Ping Cheng PII: DOI: Reference:
S2211-3797(19)32459-3 https://doi.org/10.1016/j.rinp.2019.102710 RINP 102710
To appear in:
Results in Physics
Received Date: Accepted Date:
13 August 2019 25 September 2019
Please cite this article as: Li, P., Cheng, X-P., Unified Derivation of Reflectivity for All Waves, Results in Physics (2019), doi: https://doi.org/10.1016/j.rinp.2019.102710
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
Unified Derivation of Reflectivity for All Waves Peng Li1, ∗ and Xue-Ping Cheng1 1
School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan 316022, China (Dated: September 15, 2019)
We present a unified method to derive the reflectivity of a plane wave normally incident on an interface. The key idea is to use the flux continuity and the proposed principle of least reflection to acquire the needed generality and conciseness. The result provides a special understanding on the origin of reflection and the half-wave loss. PACS numbers: 01.55.+b, 43.20.+g, 42.25.Gy, 03.65.-w
Reflection is a universal phenomenon which can be observed in many physically distinct waves [1–7]. The amount of reflection is conveniently quantified by the reflectivity [4], the ratio of wave flux between the reflected and the incident waves. In most textbooks reflectivity is derived by invoking the continuity of two physical quantities across the medium interface. For example, displacement and stress are used in mechanical wave [1], electric and magnetic field intensities are used in electromagnetic wave [2], while the quantum wave function and its derivative are used in matter wave [3]. In this point of view, one needs to find two independent continuity equations to calculate the reflectivity. Usually, one of them is obvious because it is kinematic, e.g., the displacement continuity. However, the other is less obvious due to its dynamical nature, e.g., the stress continuity. Therefore, conventional derivations lack the generality and conciseness to help understand the universality of reflection. Here we propose a new method to calculate the reflectivity of all waves in a unified framework. The key idea is to replace the dynamical continuity by a general flux continuity. Moreover, we carefully investigated the phases of the participant waves and found that continuity equations do not fully interpret the observed reflectivity. The calculated reflectivity is a function of the relative phase between the incident and the reflected waves. Interestingly, the real reflectivity is found to be the minimal value of the virtual reflectivity. Therefore we are encouraged to propose the principle of least reflection as a premise in our theory. Next we show the method in brief. We assume the normally incident, the transmitted and the reflected waves are described by the following wave functions: y1 = A1 cos(ωt − k1 x + φ1 ) y2 = A2 cos(ωt − k2 x + φ2 ) , (1) y = A cos(ωt + k x + φ ) 3 3 3 3 where Ai ’s, ki ’s, φi ’s and ω are the amplitudes, wave numbers, initial phases, and angular frequency of the respective waves. Because the incident and the reflected waves are in the same medium their wave numbers
∗ Electronic
address: [email protected]
are equal in magnitude, i.e., k1 = k3 . Moreover, we assume there exists a time-averaged flux density which is a quadratic function of the wave amplitude I¯ = Z · A2 ,
(2)
where the wave impedance Z depends on the medium. Now we write two continuity equations on the interface: one for the wave function itself and the other for the flux density: y1 + y3 = y2 , (3) I¯1 − I¯3 = I¯2 . At the interface (i.e., x = 0) eqs. (3) can be simplified as A1 cos(ωt + φ1 ) + A3 cos(ωt + φ3 ) = A2 cos(ωt + φ2 ), (4) and A1 2 − A3 2 = Z r · A2 2 ,
(5)
where Zr = Z2 /Z1 is the relative wave impedance. Eqs.(4) and (5) pave the theoretical ground for this work. Let us temporarily pretend there is no reflection (i.e., A3 = 0). Then we obtain A1 = A2 (as well as φ1 = φ2 ) from eq. (4) and A21 = Zr A22 from eq. (5). Obviously they contradict each other unless Zr = 1, which corresponds to the disappearance of the interface. This provides a special understanding on the origin of reflection: nature needs it to save both continuities (4) and (5). When reflection is present (i.e., A3 6= 0), the two continuities can hold simultaneously, and our goal is to find the expression for the flux reflectivity (abbreviated as reflectivity) 2 defined by R ≡ I¯3 /I¯1 = (A3 /A1 ) . Now we look carefully into the phases in eq. (4). At a first glance, all φi ’s are arbitrary. However, one can eliminate one of the phases without loss of generality, because it is equivalent to choose a preferred global phase. Here we make it by assuming φ2 = 0. Then eq. (4) can be regarded as a summation of rotating wave vectors as shown in Fig.1. The rule of vector summation asserts that A1 2 + A3 2 + 2A1 A3 cos β = A2 2 ,
(6)
2
FIG. 1: The vector summation of wave amplitudes.
where β = φ3 − φ1
(7)
is the relative phase between the incident and the reflected waves. Combining eqs. (5) and (6) we find a concise set of equations 1 − r 2 = Zr t2 , (8) 1 + r2 + 2r cos β = t2 ,
Remarkably, eq. (11) is exactly the observed amplitude reflectivity. This indicates that nature make a special choice [i.e., eq. (11)] from all the virtual solutions [i.e., eq. (10)]. Besides, at β = 0 (i.e., φ1 = φ3 ) eq. (4) reduces to (A1 + A3 ) cos(ωt + φ1 ) = A2 cos(ωt + φ2 ), which holds if only φ1 = φ2 and A1 + A3 = A2 (i.e., 1 + r = t). Therefore the special choice of β = 0 yields that φ1 = φ2 = φ3 and t > 0 (because |r| 6 1). This means that the transmitted wave is always in-phase with the incident wave, while the reflected wave can be either in-phase (as r > 0) or reverted (as r < 0) with respect to the incident. The latter case occurs at Zr > 1 and is conventionally called the half-wave loss. Based on these findings, we are convinced to propose the principle of least reflection as the basis of our theory. It suggests that the freedom in virtual variable β should eventually minimize the virtual flux reflectivity. The form of this principle mimics the principle of least action, while its spirit is consistent with the law of inertia.
where r ≡ A3 /A1 and t ≡ A2 /A1 are the amplitude reflectivity and the amplitude transmittance, respectively. By eliminating t from (8) we arrive at r2 (Zr + 1) + 2rZr cos β + Zr − 1 = 0, and r is shown to have two branches of solutions p −Zr cos β ± 1 − Zr 2 sin2 β r± (β) = . Zr + 1
(9)
(10)
The flux reflectivity is simply the square of r, i.e., R± (β) = |r± (β)|2 . Eq. (10) provides an infinite number of solutions that all satisfy the needed continuities. However, there is only one observed reflectivity for a given Zr . Therefore, they should be regarded as virtual functions of β and more physics is needed to explain which β is chosen by nature. To unveil this subtlety, we draw the virtual functions r± (β) and R± (β) for Zr < 1 and Zr > 1 with representative values in Fig.2. It is not difficult to discover (as well as to analytically verify) that β = 0 is the stationary point for all the virtual functions. Moreover, at β = 0 the minimal values of R± (β) are all found on the “+” branch, where the corresponding amplitude reflectivity can be calculated from eq. (10) as r+ (0) = −
Zr − 1 . Zr + 1
(11)
[1] An-Shen Qi and Chan-Ying Du, Mechanics (Second edition in Chinese, Higher Education Press, Beijing, 2005); [2] J. D. Jackson, Classical Electrodynamics (Third edition, Higher Education Press, Beijing, 2004); [3] D. J. Griffiths, Introduction to Quantum Mechanics
FIG. 2: Virtual functions r± (β) (top) and R± (β) (bottom) with representative values for Zr (0.1, 0.3, 0.6, 0.9 for Zr < 1 and 1.2, 1.8, 3, 6 for Zr > 1), where solid and dashed lines correspond to the “+” and “−” branches, respectively.
In summary, we have obtained the wave reflectivity via a universal method aided with the principle of least reflection. The virtual reflectivity functions provide more insights into the nature of reflection. The result can be easily applied to various physical waves with properly defined Zr derived from eq.(2). This work is supported by NSFC. No. 11975204.
(Third edition, Cambridge University Press, Cambridge, 2018); [4] Information on: https://en.wikipedia.org/wiki/Reflectance [5] G. Norman and I. Saitov, Phys. Rev. E 94, 043202 (2016); [6] M. Bordag, G. L. Klimchitskaya, V. M. Mostepanenko,
3 and V.M. Petrov, Phys. Rev. D 91, 045037 (2015); [7] Chuan He, et al, Phys. Rev. B 95, 184302 (2017)
S2211-3797(19)32459-3 https://doi.org/10.1016/j.rinp.2019.102710 RINP 102710
To appear in:
Results in Physics
Received Date: Accepted Date:
13 August 2019 25 September 2019
Please cite this article as: Li, P., Cheng, X-P., Unified Derivation of Reflectivity for All Waves, Results in Physics (2019), doi: https://doi.org/10.1016/j.rinp.2019.102710
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
Unified Derivation of Reflectivity for All Waves Peng Li1, ∗ and Xue-Ping Cheng1 1
School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan 316022, China (Dated: September 15, 2019)
We present a unified method to derive the reflectivity of a plane wave normally incident on an interface. The key idea is to use the flux continuity and the proposed principle of least reflection to acquire the needed generality and conciseness. The result provides a special understanding on the origin of reflection and the half-wave loss. PACS numbers: 01.55.+b, 43.20.+g, 42.25.Gy, 03.65.-w
Reflection is a universal phenomenon which can be observed in many physically distinct waves [1–7]. The amount of reflection is conveniently quantified by the reflectivity [4], the ratio of wave flux between the reflected and the incident waves. In most textbooks reflectivity is derived by invoking the continuity of two physical quantities across the medium interface. For example, displacement and stress are used in mechanical wave [1], electric and magnetic field intensities are used in electromagnetic wave [2], while the quantum wave function and its derivative are used in matter wave [3]. In this point of view, one needs to find two independent continuity equations to calculate the reflectivity. Usually, one of them is obvious because it is kinematic, e.g., the displacement continuity. However, the other is less obvious due to its dynamical nature, e.g., the stress continuity. Therefore, conventional derivations lack the generality and conciseness to help understand the universality of reflection. Here we propose a new method to calculate the reflectivity of all waves in a unified framework. The key idea is to replace the dynamical continuity by a general flux continuity. Moreover, we carefully investigated the phases of the participant waves and found that continuity equations do not fully interpret the observed reflectivity. The calculated reflectivity is a function of the relative phase between the incident and the reflected waves. Interestingly, the real reflectivity is found to be the minimal value of the virtual reflectivity. Therefore we are encouraged to propose the principle of least reflection as a premise in our theory. Next we show the method in brief. We assume the normally incident, the transmitted and the reflected waves are described by the following wave functions: y1 = A1 cos(ωt − k1 x + φ1 ) y2 = A2 cos(ωt − k2 x + φ2 ) , (1) y = A cos(ωt + k x + φ ) 3 3 3 3 where Ai ’s, ki ’s, φi ’s and ω are the amplitudes, wave numbers, initial phases, and angular frequency of the respective waves. Because the incident and the reflected waves are in the same medium their wave numbers
∗ Electronic
address: [email protected]
are equal in magnitude, i.e., k1 = k3 . Moreover, we assume there exists a time-averaged flux density which is a quadratic function of the wave amplitude I¯ = Z · A2 ,
(2)
where the wave impedance Z depends on the medium. Now we write two continuity equations on the interface: one for the wave function itself and the other for the flux density: y1 + y3 = y2 , (3) I¯1 − I¯3 = I¯2 . At the interface (i.e., x = 0) eqs. (3) can be simplified as A1 cos(ωt + φ1 ) + A3 cos(ωt + φ3 ) = A2 cos(ωt + φ2 ), (4) and A1 2 − A3 2 = Z r · A2 2 ,
(5)
where Zr = Z2 /Z1 is the relative wave impedance. Eqs.(4) and (5) pave the theoretical ground for this work. Let us temporarily pretend there is no reflection (i.e., A3 = 0). Then we obtain A1 = A2 (as well as φ1 = φ2 ) from eq. (4) and A21 = Zr A22 from eq. (5). Obviously they contradict each other unless Zr = 1, which corresponds to the disappearance of the interface. This provides a special understanding on the origin of reflection: nature needs it to save both continuities (4) and (5). When reflection is present (i.e., A3 6= 0), the two continuities can hold simultaneously, and our goal is to find the expression for the flux reflectivity (abbreviated as reflectivity) 2 defined by R ≡ I¯3 /I¯1 = (A3 /A1 ) . Now we look carefully into the phases in eq. (4). At a first glance, all φi ’s are arbitrary. However, one can eliminate one of the phases without loss of generality, because it is equivalent to choose a preferred global phase. Here we make it by assuming φ2 = 0. Then eq. (4) can be regarded as a summation of rotating wave vectors as shown in Fig.1. The rule of vector summation asserts that A1 2 + A3 2 + 2A1 A3 cos β = A2 2 ,
(6)
2
FIG. 1: The vector summation of wave amplitudes.
where β = φ3 − φ1
(7)
is the relative phase between the incident and the reflected waves. Combining eqs. (5) and (6) we find a concise set of equations 1 − r 2 = Zr t2 , (8) 1 + r2 + 2r cos β = t2 ,
Remarkably, eq. (11) is exactly the observed amplitude reflectivity. This indicates that nature make a special choice [i.e., eq. (11)] from all the virtual solutions [i.e., eq. (10)]. Besides, at β = 0 (i.e., φ1 = φ3 ) eq. (4) reduces to (A1 + A3 ) cos(ωt + φ1 ) = A2 cos(ωt + φ2 ), which holds if only φ1 = φ2 and A1 + A3 = A2 (i.e., 1 + r = t). Therefore the special choice of β = 0 yields that φ1 = φ2 = φ3 and t > 0 (because |r| 6 1). This means that the transmitted wave is always in-phase with the incident wave, while the reflected wave can be either in-phase (as r > 0) or reverted (as r < 0) with respect to the incident. The latter case occurs at Zr > 1 and is conventionally called the half-wave loss. Based on these findings, we are convinced to propose the principle of least reflection as the basis of our theory. It suggests that the freedom in virtual variable β should eventually minimize the virtual flux reflectivity. The form of this principle mimics the principle of least action, while its spirit is consistent with the law of inertia.
where r ≡ A3 /A1 and t ≡ A2 /A1 are the amplitude reflectivity and the amplitude transmittance, respectively. By eliminating t from (8) we arrive at r2 (Zr + 1) + 2rZr cos β + Zr − 1 = 0, and r is shown to have two branches of solutions p −Zr cos β ± 1 − Zr 2 sin2 β r± (β) = . Zr + 1
(9)
(10)
The flux reflectivity is simply the square of r, i.e., R± (β) = |r± (β)|2 . Eq. (10) provides an infinite number of solutions that all satisfy the needed continuities. However, there is only one observed reflectivity for a given Zr . Therefore, they should be regarded as virtual functions of β and more physics is needed to explain which β is chosen by nature. To unveil this subtlety, we draw the virtual functions r± (β) and R± (β) for Zr < 1 and Zr > 1 with representative values in Fig.2. It is not difficult to discover (as well as to analytically verify) that β = 0 is the stationary point for all the virtual functions. Moreover, at β = 0 the minimal values of R± (β) are all found on the “+” branch, where the corresponding amplitude reflectivity can be calculated from eq. (10) as r+ (0) = −
Zr − 1 . Zr + 1
(11)
[1] An-Shen Qi and Chan-Ying Du, Mechanics (Second edition in Chinese, Higher Education Press, Beijing, 2005); [2] J. D. Jackson, Classical Electrodynamics (Third edition, Higher Education Press, Beijing, 2004); [3] D. J. Griffiths, Introduction to Quantum Mechanics
FIG. 2: Virtual functions r± (β) (top) and R± (β) (bottom) with representative values for Zr (0.1, 0.3, 0.6, 0.9 for Zr < 1 and 1.2, 1.8, 3, 6 for Zr > 1), where solid and dashed lines correspond to the “+” and “−” branches, respectively.
In summary, we have obtained the wave reflectivity via a universal method aided with the principle of least reflection. The virtual reflectivity functions provide more insights into the nature of reflection. The result can be easily applied to various physical waves with properly defined Zr derived from eq.(2). This work is supported by NSFC. No. 11975204.
(Third edition, Cambridge University Press, Cambridge, 2018); [4] Information on: https://en.wikipedia.org/wiki/Reflectance [5] G. Norman and I. Saitov, Phys. Rev. E 94, 043202 (2016); [6] M. Bordag, G. L. Klimchitskaya, V. M. Mostepanenko,
3 and V.M. Petrov, Phys. Rev. D 91, 045037 (2015); [7] Chuan He, et al, Phys. Rev. B 95, 184302 (2017)