Trying to understand a constant in the phase shift (or difference?) of 2 waves

[fluidistic]

I'm reading through Hecht's book on Optics and I fail to understand something. I think it's the third edition, page 380, chapter 9 (Interference).
So he's talking about spherical waves emitted by 2 sources. He says that the waves can be written under the form $$\vec E _1 (r_1 ,t)=\vec E_{01} (r_1)} e^{i(kr_1 -\omega t + \varepsilon _1)}$$ and $$\vec E _2 (r_2 ,t)=\vec E_{02} (r_1)} e^{i(kr_2 -\omega t + \varepsilon _2)}$$.
First questions: Hecht's was always meticulous writing $$\vec k \cdot \vec x$$ for plane waves, now he dropped the vector notation? I don't understand why. Ok k and r are parallels in this case so $$\vec k \cdot \vec r =kr$$, but he never justified it, I find it very strange. I'm likely missing something. Any help to understand here will be very welcome.
Then he went to say "The terms $$r_1$$ and $$r_2$$ are the radii of the spherical wavefronts overlapping at P; they specify the distances from the sources to P. In this case $$\delta = k(r_1-r_2)+(\varepsilon _1 - \varepsilon _2)$$."
In case you wonder, P is just a considered point over a screen far away from the sources. $$\delta$$ is the phase difference according to Hecht.
I do not understand why $$\delta$$ is worth what it's worth. I realize that the difference in optical path of the waves emitted by both sources is $$\frac{(r_1-r_2)}{n}$$ where n is the refractive index of the medium. How do you reach $$\delta$$ form it?

 

[tiny-tim]

hi fluidistic!

First questions: Hecht's was always meticulous writing $$\vec k \cdot \vec x$$ for plane waves, now he dropped the vector notation? I don't understand why. Ok k and r are parallels in this case so $$\vec k \cdot \vec r =kr$$, but he never justified it, I find it very strange. I'm likely missing something. Any help to understand here will be very welcome.

(it would be k.x = kr, not k.r = kr )

because it would be complicated and confusing …

the point P is at x, say, but r₁ and r₂ are measured from two different points, x₁ and x₂ say …

so the exponent would have a k.(x - x₁) and k.(x - x₂) …

it would look really unhelpful

Then he went to say "The terms $$r_1$$ and $$r_2$$ are the radii of the spherical wavefronts overlapping at P; they specify the distances from the sources to P. In this case $$\delta = k(r_1-r_2)+(\varepsilon _1 - \varepsilon _2)$$."
…
I do not understand why $$\delta$$ is worth what it's worth.

he's looking at a fixed point x and seeing how the two phases differ, as a function of t …

r₁ and r₂ are (generally) measured along different lines, so you're not going to get something simple like (r₁ - r₂)/n

 

[fluidistic]

hi fluidistic!


(it would be k.x = kr, not k.r = kr )

because it would be complicated and confusing …

the point P is at x, say, but r₁ and r₂ are measured from two different points, x₁ and x₂ say …

so the exponent would have a k.(x - x₁) and k.(x - x₂) …

it would look really unhelpful

Thanks for your reply. Ok I understand this, though the $$\vec k$$ aren't parallel I think so I'm guessing that your last equation is an approximation (that is, assuming that the screen and the point P are very far from the sources so that the k vectors can be considered as parallel).

he's looking at a fixed point x and seeing how the two phases differ, as a function of t …

r₁ and r₂ are (generally) measured along different lines, so you're not going to get something simple like (r₁ - r₂)/n

Hmm ok but I'm not able to show it mathematically. Can you help me on that?

 

[tiny-tim]

hi fluidistic!

(just got up :zzz: …)

hmm ok but I'm not able to show it mathematically. Can you help me on that?

you seem determined to use vectors …

it really isn't helpful for spherically symmetric waves like this …

Hecht uses k.x for plane waves, but scalar k for spherical ones because each simplifies the maths for that case

using the scalar k (as in your first post) is completely accurate, and gives you the phase difference immediately