1. Let u be a solution of the wave equation ue20. Let the points A. B, C. D be the vertices of th...
1. Let u be a solution of the wave equation u0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines z--d-q,z-ct-c2, z + ct = di, z + ct d2. Show that u(A) u (C) u(B) +u(D) Use this to find u satisfying till- tLzz =0, u( s,s)-s, u(8,8)-82 for s > 0. For which (r, t) can you determine u (x, t) uniquely this way? 1. Let...
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines r-ctC1,x- ct-2,+ ct- di,r +ct- d2 Show that u (A)+u (C)-u (B) + u (D Use this to find u satisfying For which (x, t) can you determine u (x, t) uniquely this way? 2. Suppose u satisfies the wave equation utt -curr0 in the strip 0...
d1=8 d2=9 lu for Find the solution u(x,t) for the l-D wave equation-=- Qx2 25 at2 (a) oo < x < oo with initial conditions u(x,0)-A(x) , where A(x) Is presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and t, somewhat similar to f(x) on page 85 of the Notes Part 2. d2+5 di+10 di+15dı+20 (b) Check for the wave equation in (a) that if (x...
d1= 3 & d2= 2 Question 2 Find the solution 11(x, 1) for the 1-D wave equation aT = (a) 25-for -o <x < oo with initial conditions it (x,0) = A (x) , where A(x) is presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and 1, somewhat similar to fex) on page 8s of the Notes Part 2. 2 d2+5 r-0 di+10 di+15 di+20 3...
Question 2 ul lu (a) Find the solution u(x,t) for the 1-D wave equationfor -oo < x < oo with initial conditions u (x,0)-A(x) , where A(x) s presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and t, somewhat similar to f(x) on page 85 of the Notes Part 2. di+10 dı+15di+20 (b) Check for the wave equation in (a) that if f(xtct) (use appropriate value...
Consider the inlínitm ar ls sby subject to the initial conditions u(x,0)-f() and u(,0)-9(x) where 0, otherwise (a) What is the wave speed? c (b) What is the domain of dependence for the point (r, t) (2,3)? (c) Using only characteristic lines and your knowledge of d'Alembert's solution, find the following values of u(r,t) c) u9,9)- (b) u2,5) Consider the inlínitm ar ls sby subject to the initial conditions u(x,0)-f() and u(,0)-9(x) where 0, otherwise (a) What is the wave...
I need help with d) please help thank you Question 1 Wave motion appears in all branches of physics. In the lectures we considered the solution of the advection equation, a first-order hyperbolic PDE. Here we consider the solution of the wave equation: c2 where c >0 is constant. , We assume all variables have been non-dimensionalised. (a) Eq. (1) has the general solution (d'Alembert, 1747): u(x,t) F(x -ct) +G(x ct), where F and G are arbitrary functions. Consider the...
Fourier Wave Equation Question 2 The function u(x,t) is governed by the wave equation 82u 182u 8x2 c2 8t2 Subject to the following conditions having c2 3 i. At x 0 and x-1, u 0 for all t 2 0. i. Whent 0, for sx s 1 Use the method of separation of variables to establish that the solution for u(x, t). From the solution established, given a condition t0, usinx(1 + cosx) for 0 s x s1. Find the...
2. Find the travelling wave solution of the regularized long wave equation U + uz + huur - Uunt = 0, in the form uz,t) = w(z), with z = I-ct, with c a constant, where u and its derivatives vanish as too for allt and c is a constant.
(1 point) Consider the wave equation 1(1)utt = uzz for-oo < z < oo, t>0 with initial conditions ut (z,0-0 and u(z,0) = /(z), where (2) f(z) = 1 for 0 < z < 1, (3) f(z) =-1 for-1 < z < 0, and (4) f(z) = 0 for all other. The slanting lines in the figure below show the characteristics for this PDE that originate on the z-axis at the points of discontinuity of the initial data f f(x)...