# Question on TEM representation used in Griffiths book

• yungman
In summary, the book "Introduction to Electrodynamics" 3rd edition by Dave Griffiths discusses the representation of traveling waves and their time-harmonic forms. The book uses a convention where the spatial phase dependence is always positive and the sign of the time dependence changes for forward and reflected waves. This results in a different complex wave function for the reflected wave compared to other books that use a different convention. This difference in time convention does not affect the real part of the wave function.

#### yungman

I am confused with this part of "Introduction to Electrodynamics" 3rd edition by Dave Griffiths.

On page 367, the traveling wave is represented by:

$$f(z,t)\;=\; A cos[k(z-vt) \;+\; \delta]$$ 9.7

Where v is velocity and $kv=\omega$. This give:

$$f(z,t)\;=\; A cos[kz \;-\;\omega t \;+\; \delta] \;\hbox { for forward moving wave and }$$ 9.12

$$f(z,t)\;=\; A cos[kz \;+\;\omega t \;-\; \delta] \;\hbox { for backward moving wave and }$$ 9.13

$$f(z,t) \;=\; Re[A e^{i(kz-\omega t + \delta)} ] \;\hbox { and }$$ 9.16

$$\tilde{f}(z,t) \;=\; \tilde{A}e^{i(kz-\omega t )}$$ 9.17

I have no problem with 9.16 because the real part is cosine and it is an even function. The convension way is always written like:

$$f(z,t) \;=\; Re[A e^{i(\omega t \;-\;kz \;+\; \delta)}]$$

I have issue with 9.17 because this is not just the real part.

$$e^{i (kz-\omega t) } = cos (kz\;-\;\omega t) \;+\; i sin (kz\;-\;\omega t) \;\hbox { which is not the same as } \;e^{i (\omega t \;-\; kz) } \;=\; cos (\omega t \;-\; kz) \;+\; i sin (\omega t \;-\;kz)$$

The sine portion has opposite sign.

Then on P384 section 9.3.2 , the book give the equation of the incident electric wave and reflected electric wave :

$$\tilde { E} _I(z,t) \;=\; \hat x \tilde {E}_{0_I} e^{i(kz-\omega t)} \;\hbox { and }$$ 9.75

$$\tilde { E} _R(z,t) \;=\; \hat x \tilde {E}_{0_R} e^{i(-kz-\omega t)}$$ 9.76

I have issue with 9.76 because this is not just the real part.

$$e^{i (-kz-\omega t) } = cos (kz\;+\;\omega t) \;-\; i sin (kz\;+\;\omega t) \;\hbox { which is not the same as } \;e^{i (kz\;+\;\omega t) } \;=\; cos (kz\;+\;\omega t) \;+\; i sin (kz\;+\;\omega t)$$

Something is wrong on the equation of the reflected wave. But This book have no error that I can find so far, so what did I do wrong? Please help.

Thanks

Alan

The time domain signal is the real part of the complex time-harmonic form.

EDIT: Barring the above, I still do not understand the problem.

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Born2bwire said:
The time domain signal is the real part of the complex time-harmonic form.

EDIT: Barring the above, I still do not understand the problem.

My question is:

1) Why the book use $-\omega t \;\hbox { instead of } \;\omega t \;$ for the forward wave?:

$$f(z,t)\;=\; A cos[kz \;-\;\omega t \;+\; \delta] \;\hbox { for forward moving wave and }$$ 9.12

It only use +ve $\omega t$ for the reflected wave 9.13?

2) In the book, the complex wave function:

$$\tilde { E} _I(z,t) \;=\; \hat x \tilde {E}_{0_I} e^{i(kz-\omega t)} \;\hbox { and }$$ 9.75

$$\tilde { E} _R(z,t) \;=\; \hat x \tilde {E}_{0_R} e^{i(-kz-\omega t)}$$ 9.76

Should be

$$\tilde { E} _I(z,t) \;=\; \hat x \; \tilde {E}_{0_I} e^{i(\omega t - kz)} \;\hbox { and }$$

$$\tilde { E} _R(z,t) \;=\; \hat x \; \tilde {E}_{0_R} e^{i(\omega t +kz)}$$

Notice the Reflected wave is not the same? I understand the real part is the same, but why the book use different convension from all the other books I read?

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Every book needs to properly define their time-harmonic convention. I have seen all four possibilities in books. Normally, most texts use:

$$\mathbf{A}(\mathbf{r},t) = \Re \left[ \mathbf{A}(\mathbf{r}) e^{-i\omega t} \right]$$

Or

$$\mathbf{A}(\mathbf{r},t) = \Re \left[ \mathbf{A}(\mathbf{r}) e^{j\omega t} \right]$$

But I have seen, far less often both exp(i\omega t) and exp(-j\omega t) time conventions. So differences in expressions between texts can sometimes be attributed to different time conventions.

But the time convention does not matter because regardless of the time convention the real part always comes out the same as you have shown above.

The distinguishing characteristic of the forward or reverse traveling wave is the spatial phase dependence relative to the time dependence. Most of the time we do not even explicitly include the time convention when we work in time-harmonic signals (but of course it is still there).

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yungman said:
I am confused with this part of "Introduction to Electrodynamics" 3rd edition by Dave Griffiths.

On page 367, the traveling wave is represented by:

$$f(z,t)\;=\; A cos[k(z-vt) \;+\; \delta]$$ 9.7

Where v is velocity and $kv=\omega$. This give:

$$f(z,t)\;=\; A cos[kz \;-\;\omega t \;+\; \delta] \;\hbox { for forward moving wave and }$$ 9.12

$$f(z,t)\;=\; A cos[kz \;+\;\omega t \;-\; \delta] \;\hbox { for backward moving wave and }$$ 9.13
These two equations show that Griffiths kept $kz$ always +ve and change the sign of $\omega t$ from -ve for forward wave to +ve for reflected wave. so the complex wave is:

$$\tilde E_I (z,t) \;=\; A e^{(kz-\omega t)} \;\hbox {for forward wave and }$$

$$\tilde E_R (z,t) \;=\; A e^{(kz+\omega t)}$$

for reflected wave. But if you look at 9.75 and 9.76.

$$\tilde { E} _I(z,t) \;=\; \hat x \tilde {E}_{0_I} e^{i(kz-\omega t)} \;\hbox { and }$$ 9.75

$$\tilde { E} _R(z,t) \;=\; \hat x \tilde {E}_{0_R} e^{i(-kz-\omega t)}$$ 9.76

Notice here he kept $\omega t$ -ve and flipping $kz$ from +ve in forward wave to -ve for reflected wave. This is big inconsistency.

$$f(z,t) \;=\; Re[A e^{i(kz-\omega t + \delta)} ] \;\hbox { and }$$ 9.16

$$\tilde{f}(z,t) \;=\; \tilde{A}e^{i(kz-\omega t )}$$ 9.17

I have no problem with 9.16 because the real part is cosine and it is an even function. The convension way is always written like:

$$f(z,t) \;=\; Re[A e^{i(\omega t \;-\;kz \;+\; \delta)}]$$

I have issue with 9.17 because this is not just the real part.

$$e^{i (kz-\omega t) } = cos (kz\;-\;\omega t) \;+\; i sin (kz\;-\;\omega t) \;\hbox { which is not the same as } \;e^{i (\omega t \;-\; kz) } \;=\; cos (\omega t \;-\; kz) \;+\; i sin (\omega t \;-\;kz)$$

The sine portion has opposite sign.

Then on P384 section 9.3.2 , the book give the equation of the incident electric wave and reflected electric wave :

$$\tilde { E} _I(z,t) \;=\; \hat x \tilde {E}_{0_I} e^{i(kz-\omega t)} \;\hbox { and }$$ 9.75

$$\tilde { E} _R(z,t) \;=\; \hat x \tilde {E}_{0_R} e^{i(-kz-\omega t)}$$ 9.76

I have issue with 9.76 because this is not just the real part.

$$e^{i (-kz-\omega t) } = cos (kz\;+\;\omega t) \;-\; i sin (kz\;+\;\omega t) \;\hbox { which is not the same as } \;e^{i (kz\;+\;\omega t) } \;=\; cos (kz\;+\;\omega t) \;+\; i sin (kz\;+\;\omega t)$$

Something is wrong on the equation of the reflected wave. But This book have no error that I can find so far, so what did I do wrong? Please help.

Thanks

Alan

I am using my original post to point out some inconsistency, please read the blue print above. I double check the equations I have here to make sure I am not miss typing.

Born2bwire said:
Every book needs to properly define their time-harmonic convention. I have seen all four possibilities in books. Normally, most texts use:

$$\mathbf{A}(\mathbf{r},t) = \Re \left[ \mathbf{A}(\mathbf{r}) e^{-i\omega t} \right]$$

Or

$$\mathbf{A}(\mathbf{r},t) = \Re \left[ \mathbf{A}(\mathbf{r}) e^{j\omega t} \right]$$

But I have seen, far less often both exp(i\omega t) and exp(-j\omega t) time conventions. So differences in expressions between texts can sometimes be attributed to different time conventions.

But the time convention does not matter because regardless of the time convention the real part always comes out the same as you have shown above.

The distinguishing characteristic of the forward or reverse traveling wave is the spatial phase dependence relative to the time dependence. Most of the time we do not even explicitly include the time convention when we work in time-harmonic signals (but of course it is still there).

So you mean people can have different way of expressing the wave because we only care about the real part of the equation which is the cosine part of it. The cosine being even function so we come up with the same answer even when we switch polarity of the content inside the cosine? But the complex form of the instanteneous waves are different in different representation!

But I think the other books like Cheng's make a lot more sense to have +ve $\omega t$ all the time because you don't have negative time in real life. Then using $cos ( \omega t \;-\;kz ) \;=\; 0$ to keep track of the peak of the wave and differentiate $\omega t -kz$ respect to t to give you:

$$\frac {dz}{dt}= \frac{\omega}{k} = velocity$$

With this, for the reflected wave would have +ve$kz$ and the velocity would be

$$\frac {dz}{dt}= -\frac{\omega}{k}$$

This make a lot more physical sense!

So, the real part of 9.17 is:

$$\\ \Re(\tilde{f}) = \hat{x}\left[\Re(\tilde{A})\cos(\omega t - kx) + \Im(\tilde{A})\sin(\omega t - kx)\right]$$.

The above is general solution to the homogeneous wave equation for a single wave traveling in the +z-direction. The sign in front of the sin term doesn't matter because the boundary and initial conditions on the wave will give you whatever is appropriate for the imaginary part of A. You're worried about sign conventions for equations that have undetermined constants before these conditions are in place.

The reality is that for a given set of boundary conditions, the waves described by

$$\\ \tilde{f}(z,t)=\tilde{A}e^{-i\omega t+ikz} \text{ and }\\ \tilde{g}(z,t)=\tilde{B}e^{-ikz+i\omega t}\\$$

are identical. You will find that,

$$\\ \Re(\tilde{A})=\Re(\tilde{B}) \text{ and }\\ \Im(\tilde{A})=-\Im(\tilde{B}) \\$$

necessary to make the equations for complex f and g describe the exact same wave. This is an example of the beauty of differential equations with complex numbers. The only sign that mattered here was the sign of the time component relative to the space component, which gives rise to the physical interpretation of a wave going in a specified direction.

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ednobj has pretty much stated what I have been trying to say above. But you (yungman) are mistaken about the equations for velocity.

The phase velocity is

$$\nu_p = \frac{\omega}{k}$$

The group velocity is

$$\nu_g = \frac{d \omega}{d k}$$

Taken in this respect one does not find the same discrepancy since the time convention does not affect your dispersion relations.

Personally, I much prefer the exp{-i\omega t} convention. Notice that the direction of the wave is in the \hat{k} direction where k is the wave vector. The resulting plane wave is thus,

$$\mathbf{E}(\mathbf{r}, t) = \mathbf{E}_0(\mathbf{r}) e^{i\mathbf{k}\cdot\mathbf{r}} e^{-i\omega t}$$

Of course if you keep in mind the proper convention then for the j\omega t convention the same plane wave becomes,

$$\mathbf{E}(\mathbf{r}, t) = \mathbf{E}_0(\mathbf{r}) e^{-j\mathbf{k}\cdot\mathbf{r}} e^{j\omega t}$$

So as long as you mentally keep in mind the proper form of +/- i or +/- j you can apply the appropriate sign and separate the imaginary number from the dot product of the wave vector and the position vector. Thinking of electromagnetics in terms of the time-domain is fairly cumbersome and it is much easier to keep it in the time-harmonic complex form.

## 1. What is TEM representation?

TEM representation, also known as the Transfer Matrix Method, is a mathematical technique used to calculate the transmission and reflection properties of layered materials. It is commonly used in the field of material science and engineering to study the behavior of materials at the nanoscale.

## 2. How is TEM representation used in Griffiths book?

Griffiths book, "Introduction to Electrodynamics", uses TEM representation to analyze the propagation of electromagnetic waves in layered media. It presents a detailed explanation of the mathematical principles and applications of TEM representation in the context of electromagnetism.

## 3. What are the benefits of using TEM representation?

TEM representation offers several benefits, including its ability to model complex layered structures accurately, its ease of use in solving difficult electromagnetic problems, and its efficiency in calculating the transmission and reflection properties of materials.

## 4. Are there any limitations to using TEM representation?

Like any mathematical technique, TEM representation has its limitations. It is most suitable for analyzing planar layered structures and may not be applicable to other types of structures. It also assumes that the layers are infinitely thin, which may not always be the case in real-world materials.

## 5. Can TEM representation be used in other scientific fields?

Yes, TEM representation has applications in various scientific fields, including optics, acoustics, and quantum mechanics. It is a versatile mathematical tool that can be adapted to study the behavior of waves in different types of media.