This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
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
 MattKeti
 Multivariable Calculus

Click to edit the document details