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Sine waves, angles, and circles - a useful visualization
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<blockquote data-quote="Phil Graham" data-source="post: 10173" data-attributes="member: 430"><p>Hey all,</p><p></p><p>Over time I've experienced a conceptual divide when guiding people through some of the basics behind wave physics, usually in the context of measurement. Some people quickly start dealing with wave length, group delay, and phase angles in their heads, and others are missing a speedy jump between these concepts. Being able to work through them quickly is very handy. I came across the following image, and thought it would be useful to help people tie these concepts together:</p><p><img src="http://i.stack.imgur.com/p8O4P.gif" alt="" class="fr-fic fr-dii fr-draggable " data-size="" style="" /></p><p></p><p></p><ul> <li data-xf-list-type="ul">In the image above, a triangle is inscribed in a circle. The vertex of the triangle is rotating anti-clockwise around the circle, and the value of the vertex is driving the behavior of the oscillating waves. The hypotenuse is always the same length, namely the radius of the circle.</li> <li data-xf-list-type="ul">The trig ratios between the sides of the inscribed triangle form the familiar sine and cosine</li> <li data-xf-list-type="ul">We represent the waves of sound schematically with these trigonometric ratios because they are some of the simplest periodically repeating structures in mathematics.</li> <li data-xf-list-type="ul">If we say that a circle is 360 degrees of total rotation about a fixed radius, then the vertex of the triangle can be seen to represent some value of rotation between 0 degrees and 360 degrees.</li> <li data-xf-list-type="ul">The values of the trigonometric ratios take the vertex's point of rotation and convert it to a number between 0 and 1. These values rise and fall smoothly and give us a convenient way to describe wave behavior at a specific frequency.</li> <li data-xf-list-type="ul">When we talk about phase, or phase rotations, there is no physical rotation taking place, only that the input values to the trig ratios follow the points on a circle. These values are usually represented in degrees or radians but there is nothing magical about the unit of angle</li> </ul></blockquote><p></p>
[QUOTE="Phil Graham, post: 10173, member: 430"] Hey all, Over time I've experienced a conceptual divide when guiding people through some of the basics behind wave physics, usually in the context of measurement. Some people quickly start dealing with wave length, group delay, and phase angles in their heads, and others are missing a speedy jump between these concepts. Being able to work through them quickly is very handy. I came across the following image, and thought it would be useful to help people tie these concepts together: [IMG]http://i.stack.imgur.com/p8O4P.gif[/IMG] [LIST] [*]In the image above, a triangle is inscribed in a circle. The vertex of the triangle is rotating anti-clockwise around the circle, and the value of the vertex is driving the behavior of the oscillating waves. The hypotenuse is always the same length, namely the radius of the circle. [*]The trig ratios between the sides of the inscribed triangle form the familiar sine and cosine [*]We represent the waves of sound schematically with these trigonometric ratios because they are some of the simplest periodically repeating structures in mathematics. [*]If we say that a circle is 360 degrees of total rotation about a fixed radius, then the vertex of the triangle can be seen to represent some value of rotation between 0 degrees and 360 degrees. [*]The values of the trigonometric ratios take the vertex's point of rotation and convert it to a number between 0 and 1. These values rise and fall smoothly and give us a convenient way to describe wave behavior at a specific frequency. [*]When we talk about phase, or phase rotations, there is no physical rotation taking place, only that the input values to the trig ratios follow the points on a circle. These values are usually represented in degrees or radians but there is nothing magical about the unit of angle [/LIST] [/QUOTE]
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