Question 3 Consider the one-dimensional homogeneous wave equation: 22u at2 a2u c2 ar2 subject to the initial conditions

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Question 3 Consider The One Dimensional Homogeneous Wave Equation 22u At2 A2u C2 Ar2 Subject To The Initial Conditions 1 (99.62 KiB) Viewed 20 times

Question 3 Consider the one-dimensional homogeneous wave equation: 22u at2 a2u c2 ar2 subject to the initial conditions u(x,0) = f(x), and ди (x,0) = g(x). at = a. Using the transformation & = X – ct and n = x + ct, show that the cannonical form of the one-dimensional wave equation is given by 22u agan 0, and find the general solution of this cannonical form. b. Show that the solution to the wave equation is 1 1 u(x, t) $(f (x + ct) + f(x – ct)) + x ) 1 $"969).ds 2c c. Using the specific initial conditions to the wave equation u(x,0) = 0, and au (x,0) = sin(x), at show that the solution to the wave equation is given by u(x, t) 1 sin(x) sin(ct) 2c