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)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 positionWhat did you write for the wave equation at the beginning? is that the same as Schrodinger's Eq.?