Question

4.4: Misc. separation of variables on wave equation problems 1. Consider the wave equation $u_t = c^2 u_{xx}$ for vibrations in a tightly stretched string of length $L$ with initial conditions $u(x, 0) = f(x)$ and $u_t(x, 0) = g(x)$ and different boundary conditions listed below. For each boundary condition listed below, solve the wave equation (provide coefficient formulas as well). Also in each case, what is $lim_{t o infty} u(x, t)$? (If this limit does not exist, write DNE. If this limit is infinite, write infinite). (a) $u(0, t) = u(L, t) = 0$ (We did this one in the lecture video, so you may omit it from your homework submission. It is good practice, though.) (b) $u_x(0, t) = u_x(L, t) = 0$. Hint: be careful to consider the $h$-ODE in the special case $lambda = 0$. Also, under what condition on $g(x)$ does the displacement remain bounded, i.e. finite? (c) $u_x(0, t) = u(L, t) = 0$ (d) $u(0, t) = u_x(L, t) = 0$

          4.4: Misc. separation of variables on wave equation problems
1. Consider the wave equation $u_t = c^2 u_{xx}$ for vibrations in a tightly stretched string of length $L$
with initial conditions $u(x, 0) = f(x)$ and $u_t(x, 0) = g(x)$ and different boundary conditions listed
below. For each boundary condition listed below, solve the wave equation (provide coefficient
formulas as well). Also in each case, what is $lim_{t 	o infty} u(x, t)$? (If this limit does not exist, write DNE.
If this limit is infinite, write infinite).
(a) $u(0, t) = u(L, t) = 0$ (We did this one in the lecture video, so you may omit it from your
homework submission. It is good practice, though.)
(b) $u_x(0, t) = u_x(L, t) = 0$. Hint: be careful to consider the $h$-ODE in the special case $lambda = 0$. Also,
under what condition on $g(x)$ does the displacement remain bounded, i.e. finite?
(c) $u_x(0, t) = u(L, t) = 0$
(d) $u(0, t) = u_x(L, t) = 0$

4.4: Misc. separation of variables on wave equation problems
1. Consider the wave equation ut = c^2 uxx for vibrations in a tightly stretched string of length L
with initial conditions u(x, 0) = f(x) and ut(x, 0) = g(x) and different boundary conditions listed
below. For each boundary condition listed below, solve the wave equation (provide coefficient
formulas as well). Also in each case, what is limt o infty u(x, t)? (If this limit does not exist, write DNE.
If this limit is infinite, write infinite).
(a) u(0, t) = u(L, t) = 0 (We did this one in the lecture video, so you may omit it from your
homework submission. It is good practice, though.)
(b) ux(0, t) = ux(L, t) = 0. Hint: be careful to consider the h-ODE in the special case lambda = 0. Also,
under what condition on g(x) does the displacement remain bounded, i.e. finite?
(c) ux(0, t) = u(L, t) = 0
(d) u(0, t) = ux(L, t) = 0

Added by Silvia B.

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