Steps in GRAPHICAL METHOD
To find this,from we note that I_n+1/(I_n); graphically, to get (I_n),we start at on the horizontal axis and move vertically until we hit the graph y = f(x). Now this current value of y must be transferred from the vertical axis back to the horizontal axis so that it can be used as the next seed for the quadratic map. The simplest way to transfer the y axis to the x axis is by folding the x-y plane through the diagonal line y = x since points on the vertical axis are identical to points on the horizontal axis on this line. This insight suggests the following graphical recipe for finding the orbit for some initial condition :
1.Start at 0 on the horizontal axis.
2.Move vertically up until you hit the graph f(x).
3.Move horizontally until you hit the diagonal line y = x.
4.Move vertically--up or down--until you hit the graph f(x).
5.Repeat steps 3 and 4 to generate new points.
1.Start at 0 on the horizontal axis.
2.Move vertically up until you hit the graph f(x).
3.Move horizontally until you hit the diagonal line y = x.
4.Move vertically--up or down--until you hit the graph f(x).
5.Repeat steps 3 and 4 to generate new points.
What is INTERFERENCE?
Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). Absolute value snapshots of the (real-valued, scalar) wave field. As time progresses, the wave fronts would move outwards from the two centers, but the dark regions (destructive interference) stay fixed.
Interference is the addition of two or more waves that results in a new wave pattern.
As most commonly used, the term interference usually refers to the interaction of waves which are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency.
Two non-monochromatic waves are only fully coherent with each other if they both have exactly the same range of wavelengths and the same phase differences at each of the constituent wavelengths.
The total phase difference is derived from the sum of both the path difference and the initial phase difference (if the waves are generated from 2 or more different sources). It can then be concluded whether the waves reaching a point are in phase(constructive interference) or out of phase (destructive interference).
Interference is the addition of two or more waves that results in a new wave pattern.
As most commonly used, the term interference usually refers to the interaction of waves which are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency.
Two non-monochromatic waves are only fully coherent with each other if they both have exactly the same range of wavelengths and the same phase differences at each of the constituent wavelengths.
The total phase difference is derived from the sum of both the path difference and the initial phase difference (if the waves are generated from 2 or more different sources). It can then be concluded whether the waves reaching a point are in phase(constructive interference) or out of phase (destructive interference).
Reflections in mirrors
what images look like in concave and convex mirrors
Mirrors
Mirrors
When you focus on an object, a single point, your eyes are receiving light waves diverging from that point. This must be true for an object, or an image of an object, to be visible. To put it simply, if our eyes detect light waves diverging from a point, that point will be visible. As you will see, this is very important in terms of how mirrors work.
For plane (flat) mirrors, light is reflected according to the law of reflection. When the eyes receive these light waves, it looks as if the waves are diverging from behind the mirror, making it appear as if the object is behind the mirror as well. This type of image is called a virtual image, because light waves do not actually pass through that point, it only appears so. The distance between the object and the mirror is called the object distance and the distance between the virtual image and the mirror is the image distance. Notice that on plane mirrors, the object distance is equal to the image distance.
Curved mirrors are slightly more complicated. There are basically two types of curved mirrors: concave and convex. A concave mirror curves toward the incoming light while a convex mirror curves away from the incoming light. For now, we will assume that light waves striking the lens are from an object infinitely far away, therefore, the light waves will be parallel with the principal axis.
When light strikes a concave mirror of curvature radius R, the light waves will reflect and converge at a point on the principal axis that is 1/2 * R in front of the mirror. This point is called the focal point. Since light is converging at the focal point, it is also diverging from that point on the other side. Therefore, the image of the object is created at the focal point, appearing as if the object is actually there. Notice that this image is not like the image of the plane mirror, light actually pass through where the image is. This type of image is called a real image.
When light strikes a convex mirror of curvature radius of R, the light waves will reflect and appear to diverge from a point on the optic axis that is 1/2 * R behind the mirror. Just like that of the concave mirror, this point is also called the focal point. The image of the object, even if the object is infinitely far away, will appear as if it is 1/2 * R behind the mirror.
Notice that the focal point of both the concave and convex mirrors are 1/2 * R away from the mirror. This distance between the mirror and the focal point, 1/2 * R, is called the focal length. The focal length of a concave mirror is always positive while that of the convex mirror is always negative.
Now, obviously, objects can not be infinitely far away, so we can not have it so easy as to have all the light waves always being parallel to the principal axis. If the light waves are not parallel to the principal axis, what then? No sweat! We can still locate the image, where the light waves converge then diverge off, by using three principal rays and finding where they converge.
Notice that the first and second set of principal rays are essentially the same, so any one of the first two principal rays along with the third is all that is required to determine the image location.
The image of an object from a concave mirror is a smaller, inversed version of the object. From a concave mirror, the image is a smaller, upright version of the object.
The object distance, image distance, and focal length are all related by the image equation: 1/do + 1/di = 1/f
When you focus on an object, a single point, your eyes are receiving light waves diverging from that point. This must be true for an object, or an image of an object, to be visible. To put it simply, if our eyes detect light waves diverging from a point, that point will be visible. As you will see, this is very important in terms of how mirrors work.
For plane (flat) mirrors, light is reflected according to the law of reflection. When the eyes receive these light waves, it looks as if the waves are diverging from behind the mirror, making it appear as if the object is behind the mirror as well. This type of image is called a virtual image, because light waves do not actually pass through that point, it only appears so. The distance between the object and the mirror is called the object distance and the distance between the virtual image and the mirror is the image distance. Notice that on plane mirrors, the object distance is equal to the image distance.
Curved mirrors are slightly more complicated. There are basically two types of curved mirrors: concave and convex. A concave mirror curves toward the incoming light while a convex mirror curves away from the incoming light. For now, we will assume that light waves striking the lens are from an object infinitely far away, therefore, the light waves will be parallel with the principal axis.
When light strikes a concave mirror of curvature radius R, the light waves will reflect and converge at a point on the principal axis that is 1/2 * R in front of the mirror. This point is called the focal point. Since light is converging at the focal point, it is also diverging from that point on the other side. Therefore, the image of the object is created at the focal point, appearing as if the object is actually there. Notice that this image is not like the image of the plane mirror, light actually pass through where the image is. This type of image is called a real image.
When light strikes a convex mirror of curvature radius of R, the light waves will reflect and appear to diverge from a point on the optic axis that is 1/2 * R behind the mirror. Just like that of the concave mirror, this point is also called the focal point. The image of the object, even if the object is infinitely far away, will appear as if it is 1/2 * R behind the mirror.
Notice that the focal point of both the concave and convex mirrors are 1/2 * R away from the mirror. This distance between the mirror and the focal point, 1/2 * R, is called the focal length. The focal length of a concave mirror is always positive while that of the convex mirror is always negative.
Now, obviously, objects can not be infinitely far away, so we can not have it so easy as to have all the light waves always being parallel to the principal axis. If the light waves are not parallel to the principal axis, what then? No sweat! We can still locate the image, where the light waves converge then diverge off, by using three principal rays and finding where they converge.
Notice that the first and second set of principal rays are essentially the same, so any one of the first two principal rays along with the third is all that is required to determine the image location.
The image of an object from a concave mirror is a smaller, inversed version of the object. From a concave mirror, the image is a smaller, upright version of the object.
The object distance, image distance, and focal length are all related by the image equation: 1/do + 1/di = 1/f