The wave equation: BASIC SEISMOLOGY
نویسنده
چکیده
T he foundation of seismology is the theory of wave motion, a complicated concept that is still after centuries of experiments and speculations by many of the very greatest scientists an area of active research in many disciplines. Even simple forms of wave motion are difficult to describe verbally; but, ironically, the simplest type of wave is remarkably easy to describe (and subsequently analyze) mathematically. This is one of those areas where, in the words of Nobel Prize physicist Steven Weinberg, mathematics has a “spooky” correlation to the physical world. Although some naturally occurring crystals have perfect geometric shapes, right triangles are a purely mathematical concept. They exist outside our ordinary experience of the physical world. Have you ever found a perfectly righttriangular rock, or blade of grass or leaf in your back yard or on a field trip? Yet we remember from elementary trigonometry (the mathematical analysis of the properties of triangles) that the graph of the sine function nothing more than the ratio of two sides of a right triangle perfectly represents certain periodic motions, such as the (small) oscillations of a pendulum. This type of sinusoidal motion is called simple harmonic motion. The pure sine curve, u = sin x, is quite restricted. The value of u can never be greater than 1 or less than -1 and x must traverse a distance of 2t radians before one cycle of motion is completed. These limitations are, however, not serious. The sine function is easily tailored to represent any regularly repeating motion no matter what its height/depth (or amplitude), its frequency of oscillation, or its value when it crosses a “starting” point (often the x = 0 line). Such an all-purpose sine function can be written, supposing u to be the disturbance caused by the motion. as
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