Question

Consider a rope of length $L$ is held fixed at one end but free to slide up and down a frictionless rod at the other end. The appropriate boundary condition can be obtained by thinking about the forces on the free end. If the tension acts in any direction except perpendicular to the rod, the free end will accelerate instantly and move to a position such the vertical component of the force is zero. This means that \\ $\frac{\partial y}{\partial x}(L, t) = 0$ (1)\\ where $y(x, t)$ is the position of the rope.\\ (a) Find the general solution to wave equation\\ (b) Find $y(x, t)$ for $t > 0$ for an initial configuration $y(x, 0) = \alpha x$ and $\frac{\partial y}{\partial t}(x, 0) = 0$

          Consider a rope of length $L$ is held fixed at one end but free to slide up and down a frictionless rod at the other end. The appropriate boundary condition can be obtained by thinking about the forces on the free end. If the tension acts in any direction except perpendicular to the rod, the free end will accelerate instantly and move to a position such the vertical component of the force is zero. This means that \\
$\frac{\partial y}{\partial x}(L, t) = 0$ (1)\\
where $y(x, t)$ is the position of the rope.\\
(a) Find the general solution to wave equation\\
(b) Find $y(x, t)$ for $t > 0$ for an initial configuration $y(x, 0) = \alpha x$ and $\frac{\partial y}{\partial t}(x, 0) = 0$

Consider a rope of length L is held fixed at one end but free to slide up and down a frictionless rod at the other end. The appropriate boundary condition can be obtained by thinking about the forces on the free end. If the tension acts in any direction except perpendicular to the rod, the free end will accelerate instantly and move to a position such the vertical component of the force is zero. This means that

(∂ y)/(∂ x)(L, t) = 0 (1)

where y(x, t) is the position of the rope.

(a) Find the general solution to wave equation

(b) Find y(x, t) for t > 0 for an initial configuration y(x, 0) = α x and (∂ y)/(∂ t)(x, 0) = 0

Added by Lori N.

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