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the shark in the wave … possibly

shark

relax man, no worries ok …

Everytime this pic is re-generated a large group of extremely vociferous nuisances come out of the clouds and scoff at the generally held belief that the black fast moving shape that has appeared in front of this very unlucky fellow for just a bowel loosening instant, is in fact a bloody porpoise  .. !!

This is held in scientific dispute and we submit the following as a reasonable argument:

An articulate and well-funded research project conducted by the Ponds Institute of Australia and headed by a Professor Clividian Elthvers B.S.M,. K.Baa., ACCR., BMW., B.Hons., Ph.D., AART., I.N.Do.,  refutes this unlearned position and amongst his most recent findings (refer Blaxton Press Publ., 2009. The Monogamy of Salt Water Carnivores. P199) he states that to be able to discern the difference between a shark and a porpoise in this particular situation is beyond the cognitive abilities of mortal man - ie; chap in photo, above.

- as is irrefutably evidenced by this table of time-elapsed recognitive mental synapses re-produced below. (refer Chambers Publ., 2007. The Timing Relationship of a Nightmare Scenario and the Immediate Evacuative Bowel Instinct of Homo Sapiens. P683. Dr Sturit Winesome BBFW., SWLNT., B.Hons. Darts.)

~

(a) There is a black shadow in the water. … ! .0001 of a second  

(b) It is one metre away from me. .!.. .00015 of a second

(c) It has a FIN … ! .000155 of a second

(d) I’m going to die now. … . .000000111 of a second

~

We follow with a complete renunciation of their base premise:

The porpoise believers <sic> argue that although the silhouette of the dorsal fin is not itself a clear indication of the nature of the beast, they do righteously point out the disposition of the Big Black Beasts’s tail fluke.

‘ It’s HORIZONTAL, ‘ they crow belligerently from their dusty ledgers, their book-lined studies, their academic pulpits –  those fat-arsed bastards (not you dear reader, never you).

What these idle scoffers have never taken the time to study is the relationship between (1)  the speed at which the Beasty is moving, and (2) the state of the nearsight-ocular capacity of the very unlucky fellow confonted with the vision (chap in photo, as before) and (3) The Snell-Descartes Law of Refraction.

Demonstrated in simple form here for those whose knowledge of the subject matter is incomplete:

Snell’s law states that the ratio of the sines of the angles of incidence and refraction is equivalent to the ratio of phase velocities in the two media, or equivalent to the opposite ratio of the indices of refraction:

This is an axiom, and we all probably have it memorised. My apologies if I’m seen to be going over old ground.

\frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1}
~
Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.
Now how could it be simpler eh?  This is a basic human understanding, almost instinctive. I’m surprised you asked.
~
- and P is the Poor Fellow, Q is Quite a Big Underwater Beasty and in other words, what you are seeing is not precisely what is there. Ipso facto.
~
Now that all that has been made clear we consider the entire matter finalised and closed. Nec ista sollicitudine (Lat)
~
However ….
The Vector Form – Just in case what has been explained ^ has left you unconvinced.
Given a normalized light vector l (pointing from the light source toward the surface) and a normalized plane normal vector n, one can simply work out the normalized reflected and refracted rays: Quite a basic mathematical equation really.
\cos\theta_1=\mathbf{n}\cdot (-\mathbf{l})
\cos\theta_2=\sqrt{1-\left(\frac{n_1}{n_2}\right)^2\left(1-\left(\cos\theta_1\right)^2\right)}
\mathbf{v}_{\mathrm{reflect}}=\mathbf{l}+\left(2\cos\theta_1\right)\mathbf{n}
- still with us ? Perhaps if you sat up a little straighter.
\mathbf{v}_{\mathrm{refract}}=\left(\frac{n_1}{n_2}\right)\mathbf{l} + \left( \frac{n_1}{n_2}\cos\theta_1 - \cos\theta_2\right)\mathbf{n}
- couldn’t be more obvious could it. A simple and basic parameter.

Note: \mathbf{n}\cdot(-\mathbf{l}) must be positive. Otherwise, use the following combination of refractive indices, these cosines may of course be used in the Fresnel Equations (carefully now)

\mathbf{v}_{\mathrm{refract}}=\left(\frac{n_1}{n_2}\right)\mathbf{l} + \left(-\frac{n_1}{n_2}\cos\theta_1 + \cos\theta_2\right)\mathbf{n}.
- nearly there. 

Simple Example, laughable in fact:

\mathbf{l}=\{0.707107, -0.707107\},~\mathbf{n}=\{0,1\},~\frac{n_1}{n_2}=0.9
- problem? 
\mathbf{~}\cos\theta_1=0.707107,~\cos\theta_2=0.771362
\mathbf{v}_{\mathrm{reflect}}=\{0.707107, 0.707107\} ,~\mathbf{v}_{\mathrm{refract}}=\{0.636396, -0.771362\}
- done.
~ ~ ~ ~ ~ ~ ~ ~ ~ 
Succinctly put in our view, and as demonstrated just as clearly below for the less mathematically inclined.
We suggest that you do not stare at this graphic for longer than 30 seconds at a time – you may go blind.
2 Comments Post a comment
  1. This blog never fails to make me smile – or like today – laugh.

    July 1, 2011
  2. Rusty Steele #

    Hehehehe…. BMW… Heheh

    July 1, 2011

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