Question on TEM representation used in Griffiths book

Main Question or Discussion Point

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

On page 367, the travelling 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

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Born2bwire
Gold Member
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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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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Born2bwire
Gold Member
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 travelling 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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I am confused with this part of "Introduction to Electrodynamics" 3rd edition by Dave Griffiths.

On page 367, the travelling 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.

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 travelling 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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Born2bwire
Gold Member
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.