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class18lecturenotes-1 - Class 18 Monday Reading page 231 l...

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Class 18, Monday, February 22, 2010 Reading: page 231 l t 1. , Waves in two and three dimensions So far we've only considered linear waves, i.e. waves that only change along one direction in space. In general, however, waves are three-dimensional phenomena. The most common example of a two-dimensional wave are ripples on a pond, excited by a stone dropped in the water. Each circle corresponding to a maximum displacement (height of the ripple above the average level of the water) constitutes a wave front. Each two successive wave fronts are separated by one wavelength, A.. In nature you will notice that the peaks are not evenly distributed. This is because the ripples are actually a pulse, generated by a one-time event (analogously to once shaking one end of a rope): the stone impacting the water surface. A pulse can be described by adding up an infinite number of waves of different frequencies. In some media, the speeds at which the individual components that make up the pulse travel are independent of their wavelengths, and so the pulse can retain its shape as it travels through the medium. This is not necessarily always the case. In other media, the wave speeds depend on the wavelengths, a property called displlrsion. Thus, the components that make up the pulse travel at different speeds. This changes the shape of the pulse, and alters the distance between consecutive maximum displacements. For example, for surface water waves in deep water: 1h ~ fEI w- vZ7T As a result, in dispersive media, one needs to understand how far a pulse can travel before the information it carries is so altered, that it can no longer be "read". For the waves we study presently, we will assume that they are non-dispersive. For a spherical wave, the wave fronts are surfaces of concentric nested spheres, separated by one wavelength. For a spherical or circular wave, one can sometimes make the simplifying approximation that the shape of the wave front is a line rather than a circle, or a plane rather than the curved surface of a sphere. For waves in 3D, this approximation is called the planar wave approximation. An ant sitting on the surface of an inflating balloon, will perceive the balloon as clearly spherical, when it is small in size. As you keep inflating it, however, the ant will loose track of the curvature and will think of its immediate surroundings as a plane. We can make this approximation whenever we are far enough away from the source of the wave. 11.1
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Interference in Two Dimensions. While the one-dimensional picture gives us a simple and clear idea about how to describe interference and what it entails, in nature we generally deal with 2 or 3-dimensional situations. We described above a 2-dimensional wave as a pattern of concentric fronts, equally spaced by a distance A. This pattern is moving away from the source (the center) at a given speed Vw as illustrated in the figure below. Each circle corresponds to a wave peak. Halfway in between there is a trough.
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class18lecturenotes-1 - Class 18 Monday Reading page 231 l...

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