Solve the wave equation on the half line with the Utt – cʻuxx = 0, x> 0, t > 0 и, (0, 1) %3D h(), 1>0 и(х,0) %3D и,(х,0) %3D 0, х>0. х> 0, 1> 0 .
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- Show that the function Z = sin(wct)sin(wx) satisfies the wave equation5C. Under suitable assumptions derive one dimensional wave equation.4. Consider a wave equation on an infinite line, J²u J²u 9 Ət² əx² = 0. = Find the characteristics though the point (1,3). Draw the domains of depen- dence and influence of the point (1,3).
- 3. Solve the radial wave in R3 with initral data gler) =4-r², Yer)=0 equation = Au U, =ii. Find parametric equations for the Line through (7, 5) and (-5, 7) 7. Calculate dy/dx at the point indicated: f(0) = (7tan 0, cos O), 0=a/4) Solve the inhomogeneous wave equation on the real lineUtt − c2Uxx = sin x, x ∈ RU(x, 0) = 0, Ut(x, 0) = 0.Explain what theory you are using and show your full computations.
- 8) Find the position vector r(t) for a particle with acceleration a(t) = (5t, 5 sin t, cos 6t), initial velocity (0) = (3, -3, 1) and initial position (0) = (5, 0, -2).The vector parametric equation for the line through the points (1, 2, 4) and (5, 1,-1) is L(t)=Show whether the following functions are wave functions or not. 1. У(х, t) еxp(ikx) = A- exp (i(ot-Ф)) кЗх3-0313-3kоxt(kx-ot)-iф)) 2 У(х, t) 3. y(x, t) Aexp(i(k³x³-w³t³-3kwxt(kx-wt)-ip)) Аехр (i(-kx? + оt))
- for wave equation, seperation of vairables u(x,t)=X=(x)T(t)What did you write for the wave equation at the beginning? is that the same as Schrodinger's Eq.?2. The position vector of a particle is given by r(t)= (2 cos t sin t)i +(cos^2 t - sin^2 t)j + (3t)k If the particle begins its motion at t = 0 and ends at t = pi, find the difference between the length of the path traveled and the distance between start position and end position