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

 

[PJC01]

I can understand your confusion and I will do my best to explain the concept of phase difference in interference. First, let's define what we mean by phase difference. Phase difference is the difference in the position of two waves in their respective cycles at a given point in time. This can also be described as the difference in the phase angles of two waves at a specific point.

Now, let's look at Hecht's equations. The first thing to note is that he is using complex notation, where the real part represents the amplitude of the wave and the imaginary part represents the phase. So, \vec E_{01} (r_1) e^{i(kr_1 -\omega t + \varepsilon _1)} can be written as E_{01} (r_1) cos(kr_1 -\omega t + \varepsilon _1) + i E_{01} (r_1) sin(kr_1 -\omega t + \varepsilon _1). Similarly for \vec E_{02} (r_2) e^{i(kr_2 -\omega t + \varepsilon _2)}.

Now, let's look at the phase difference at point P. The waves from the two sources have to travel different distances to reach P, as they are emitted from different sources and have different paths. This difference in distance is represented by (r_1-r_2). However, this is not the only factor that contributes to the phase difference. The waves also have a phase angle, \varepsilon _1 and \varepsilon _2, respectively. This phase angle can be thought of as the starting point of the wave cycle. So, when the two waves reach P, they will have a different phase angle or starting point, which will result in a phase difference.

To calculate the total phase difference, we need to add the difference in distance traveled, (r_1-r_2), and the difference in phase angle, (\varepsilon _1 - \varepsilon _2). This is why Hecht writes \delta = k(r_1-r_2)+(\varepsilon _1 - \varepsilon _2). The term \frac{(r_1-r_2)}{n} represents the difference in optical path, but we also need to consider the phase angle difference to calculate the total phase difference.

As