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Show that the function z = cos(4x + 4ct) satisfies the wave equation ∂2 z/ ∂t2 = c2 (∂2 z /∂x2 ).
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- The graph of f (θ) = Acos θ + B sin θ is a sinusoidal wave for any constants A and B. Confirm this for (A,B) = (1, 1), (1, 2), and (3, 4) by plotting f .Show that the function z = ln(x + ct) satisfies the wave equation ∂2 z/ ∂t2 = c2 (∂2 z /∂x2 ).Find the principal unit norma l r(t) = 6 cos ti + 6 sin tj + vector to the curve at the specified value of the parameter t = 3 π /4.
- a2u satisfies the wave equation əx² -n a²u Verify that U(x, t) = e¬Vkt cos\ax %3D k at2What did you write for the wave equation at the beginning? is that the same as Schrodinger's Eq.?Sketch the curve whose vector equation is Solution r(t) = 6 cos(t) i + 6 sin(t) j + 3tk. The parametric equations for this curve are X = I y = 6 sin(t), z = Since x² + y² = + 36. sin²(t) = The point (x, y, z) lies directly above the point (x, y, 0), which moves counterclockwise around the circle x² + y2 = in the xy-plane. (The projection of the curve onto the xy-plane has vector equation r(t) = (6 cos(t), 6 sin(t), 0). See this example.) Since z = 3t, the curve spirals upward around the cylinder as t increases. The curve, shown in the figure below, is called a helix. ZA (6, 0, 0) (0, 6, 37) I the curve must lie on the circular cylinder x² + y² =