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Unformatted text preview: The Wave Equation In physics, we are often interested in studying the behaviour of light, sound, and fluids. The motion of these phenomena can be modeled by waves. Using a combination of Newtons and Hookes law, we can derive a partial differential equation modeling wave motion: 2 u ( x,t ) t 2 = c 2 2 u ( x,t ) x 2 (1) where u ( x,t ) is the position of the wave at position x and time t , and c is the speed of wave propagation in the proposed medium. By finding solutions to (1), we can gain a better understanding of the nature of waves. DAlemberts Solution The solution to (1) is attributed to Jean le Rond dAlembert, a mathematician from the eighteenth century. It is remarkable in that it takes on a very simple form. To begin, we define = x + ct and = x- ct . Then we consider the equation u ( , ) and see how it affects (1). Computing the first partials, u t = u t + u t = c u - c u u x = u x + u x = u + u And then computing the second partials, 2 u t 2 = c 2 2 u 2- 2 c 2 2 u + c 2 2 u 2 2 u x 2 = 2 u 2 + 2 2 u + 2 u 2 Putting these expressions back into (1) and simplifying yields 2 u = 0 We integrate each side with respect to : Z 2 u d = Z 0 d = u = g ( ) for some function g . We integrate again with respect to....
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This note was uploaded on 11/22/2010 for the course MATH Math 2d taught by Professor Mattketi during the Spring '10 term at UC Irvine.
- Spring '10
- Multivariable Calculus