Phasor Addition Explained - Wave Physics

[MrsTesla]

Hello,

In my lecture notes for Wave Physics, I have that phasor addition can be represented as a geometric progression. This is what was said in the lecture (see attachment).
Can anyone explain me why are the mathematical contributions like that? (aka 5.22 in the attachment )
I've been trying to understand but I really don't get it.

Thanks in advance.

 

[sophiecentaur]

Hello,

In my lecture notes for Wave Physics, I have that phasor addition can be represented as a geometric progression. This is what was said in the lecture (see attachment).
Can anyone explain me why are the mathematical contributions like that? (aka 5.22 in the attachment )
I've been trying to understand but I really don't get it.

View attachment 234631

Thanks in advance.

The equation just expresses the contributions of each of the 'rays', using complex exponential notation. You can do the calculation using just sin or cos but the mechanics are not as elegant and don't deliver that smart answer. The terms can be expressed in terms of a geometrical progression because they contain powers of a common term. One needs to get used to the way Mathematicians often re-write expressions with different variables to reveal the patterns involved. The sum of a series like that is basic algebra.

 

[Tom.G]

Or in finer detail;

• a₀ is the energy in the incoming light ray
• α is the proportion of the light reflected back to the top of the oil film
• Therefore αa₀ is the energy in the first reflected ray
• The second ray has already been reflected once, so it starts with only the power from the first from the first reflection, or α⋅α⋅a₀. Which is α²a₀
• And this sequence continues for the subsequent rays

The e^iφ, e^2iφ,... indicate a phase shift at each reflection. eⁱ is an alternate way of representing an angle using imaginary numbers. If drawing a graph using x-y co-ordinates, the 'y' axis is replaced with the 'i' axis, with 'i' being √-1.

Hope this helps.

Cheers,
Tom