Solve Envelope Amplitudes Homework

[nordmoon]

Homework Statement

This isn't homework, but rather a task I have to solve at work. I am a bit lost and don't know where to start and would like some hints on how to make this an automated calculation in MATLAB.

I have an area, let say a square that is rotated 45 degree (this can change in other iterations) and is of certain size, let say 5 a.u.

I have two lines, one is vertical and one is oriented also 45 degrees. Line 2 have a length of let's say 20 a.u (line 1 is same length but not rotated). The two lines is placed 2 a.u. from each other (the minimal distance at the top).

These two lines are scanned past the area of the square as function time. That is, when time t = 0, first line in the scanning sequence (rotated in the same direction as the square) enters the square. At the time t = 2.a.u./speed, the second line (vertical) enters the square,.. and continue like this until the both have past the square.

In the first iteration, the top of the lines are placed at the centre line of the square (rotated 45 degrees). I need to calculate the amplitude envelope of each line as function of time and add their amplitude vs. time curves to get some kind of combined "amplitude" envelope of the two.

The things is, I like to do this automated in MATLAB because I need to move the square up or down in space to find the highest "amplitude" or the highest "contribution" from the two scanned lines in this square area. And I don't know how to do this.

I also need to as a next step, change the angle of the square (scanned area) to for example vertical and maybe even go further in another iteration and change the square to rectangle of arbitrary units.

Homework Equations

See below.

The Attempt at a Solution

I did a calculation of this for one location (the simplest one), when the top of the lines are placed at centre line of the square. I get the following equations.

L1 = length of line 1.
L2 = length of line 2 = 20 a.u.
L = is the length of the square.
v = is the scanning speed of the two lines

Line 1 (rotated 45 degrees) should give an amplitude that is the largest at the beginning t = 0 with an maximal amplitude of L. It then goes down to zero as it leaves the square. (Since in this location we are scanning a triangle area of the square).

A1 = L/L1*( 1 - v*t/(L*sqrt(2)) )

Line 2 (vertical) has an amplitude that is zero at t=0s and t =L*sqrt(2)/v. It has a maximum in the half the time, t = L/sqrt(2)/v and a maximal amplitude that is L/sqrt(2).

A2 = t*v/L2 - 2.a.u/L2 t <= L/sqrt(2)/v
A2 = (2a.u.+5L*sqrt(2)-t*v)/L2 t > L/sqrt(2)/v

Now the thing is I don't know how to do this automated, and how to move the position of the lines relative the area and get the new amplitude equations. Let say I move the centre line of the rectangle 1 a.u. down ward and repeat the calculations. How can I do this? Is there a way,... ?

 

[Nidum]

Moving the square up and down just changes the time interval between the arrivals of the two lines at any particular point and the actual geometry of traverse of the lines stays the same ?

 

[Nidum]

The geometry of traverse of the lines is going to require a piecewise solution .

For any line angle and any angle and size of rectangle first identify the positions where the geometry changes as the line passes over the rectangle . Between these positions there will be zones where there is a simple relationship between length of line bounded by the edges of the rectangle and how far the line has traversed across the zone .

For a completely general case a test will be needed to ensure that the line does not stop short inside the rectangle .

A similar procedure can be used for shapes other than rectangles .