The parabola

A parabolic mirror with a lamp at its focal point creates a parallel beam of light, this is called a search-light, because it travels a long distance with minimum dispersion.. Likewise, a sound source placed at the focus of a parabolic reflector will transmit a concentrated sound-beam over a great distance. Conversely, a long-distance microphone has a parabolic reflector behind it, so it can focus on a sound source  a great distance away and pick it up.

All these reflectors work in this way because a sound wave emanating from the focus of a parabola, and hitting the surface of the reflector, will be reflected parallel to the axis of symmetry of the parabola.

The standard equation of a parabola is y2 = 4ax   where a is the distance from the focus to the curve along the axis of symmetry. In the demonstration curve, a = 4, so the coefficient c at the top of the screen = 4 times 4 = 16. All the other coefficients are zero. The type of equation is set to square root of polynomial, to account for the fact that the left hand side is y2 

The simplest equation for a parabola is y = x2   or for more flexibility, y = bx2, where decreasing coefficient b to less than 1 will make the parabola wider, spreading out more rapidly. This creates a parabola facing upwards, with all y values positive, and x ranging from negative to positive side of the Y-axis. You can try this by selecting a polynomial from the drop-down box (not square root of a polynomial), and typing 0.1 for coefficient b, all the rest are zero. Set the range of x from -50 to +50, and the range of y from 0 to 100.


Parabolic.htm     02 November 2006         MENU  Theory -> Maths and Physics -> Mathematical equations for music