A random vibration problem
(10 pts)
Problem is good to go!!
In this problem, we're going to use our one-dimensional wave explorer to answer a specific vibrational question. The general question looks like so:
A string of length 1 lies along the unit interval with its left endpoint fastened at $x=0$ and its right endpoint fastened at $x=1$. The string is then displaced thrown so that it's initial position and velocity satisfy
$$u(x,0) = \sin(m\pi\,x) \: \text{ and } \: u_t(x,0)=\sin(n\pi\,x).$$
Use the one-dimensional, damped wave equation
$$u_{tt}+d\,u_t = c^2u_{xx}$$
to determine the displacement from equilibrium of the midpoint of the string at a given time.
Note that the parameters $m$, $n$, $c$, and $d$ will vary from student to student. You can find your exact problem on our class webpage.
Since this tool is rather qualitative in nature, an estimate is fine. Once you have your estimate, response to this post with the numerical value of your answer and a screenshot of your string.
Comments
The displacement from equilibrium of the midpoint of the string at t=1.2 s is -0.280.
The displacement from equilibrium of the midpoint of the string at time $t=1.2$ seconds into the vibration is $-0.28$.
The displacement of the midpoint from equilibrium at time t=1.9s is -0.028.
The displacement of the midpoint from equilibrium at time t=1.1s is 0.010.
The displacement of the midpoint from equilibrium at time $t=1.5$ seconds is approximately $-0.376$
The distance of the midpoint from the equilibrium at time t=1.3 is approximately .265
The displacement of the midpoint from the equilibrium is approximately 0.358 when t=1.7
The displacement of the midpoint from the equilibrium is approximately -0.324 when t=1.8
The displacement from the equilibrium of the midpoint of the string at time t=1.3 seconds into the vibration is approximately -0.351.
The displacement for my particular string set up at t = 1.3s at the midpoint was -.632
The displacement is t=1.7s and the midpoint is approximately -0.271.
Based on my assigned parameters, at t=1s the displacement at the midpoint is approximately .004.
The displacement of the midpoint from the equilibrium at time, $t = 1.1,$ is approximately $-0.616.$
The displacement of the midpoint from the equilibrium at the time 1.1s is -0.121. (redo)
The vertical displacement of the midpoint from equilibrium at time t = 1.6 [seconds] is approximately equal to -0.16 [unit lengths(?)].