- #1
Soren4
- 128
- 2
Homework Statement
[/B]
Consider an ideal rope where there is a wave moving at velocity ##v=20 m/s##. The displacement of one end of the rope is given by
$$s(t)=0.1 \mathrm{sin}(6 t)$$
a) Find the wavefunction ##\xi(x,t)##, knowing that it is progressive
b) Find the distance ##\delta## (in absolute value) between two points of the rope, that, at a certain time istant, are displaced from Equilibrium position of ##0.02 \mathrm{m}##
##[\mathrm{Result} \, \delta=1.34 m]##
Homework Equations
[/B]
A general wavefunction is
$$\xi(x,t)=\xi_0 \mathrm{sin}( k x-\omega t +\psi)\tag{1}$$
The Attempt at a Solution
[/B]
a) My boundary condition for ##(1)## is that
$$\xi(0,t)=0.1 \mathrm{sin}(6 t)\tag{2}$$
Now is it correct to conclude that ##\xi_0=-0.1 m##, ##\omega=6 rad/s## and ##\psi=0##?
If so, then, considering also ##k=\frac{2 \pi}{\lambda}=\frac{\omega}{v}=0.3 \frac{1}{m}##
$$\xi(x,t)=0.1 \mathrm{sin}(6t-0.3x )\tag{3}$$
b)Here is the problem. I would say that
$$0.02=0.1 \mathrm{sin}(6t-0.3x )\implies (6t-0.3x)=\arcsin(0.2)+2n\pi \vee \pi-\arcsin(0.2)+2n\pi \implies |x_2-x_1|=\frac{\pi-2\arcsin(0.2)}{0.3}$$
But this does not give the correct result.
Where did I go wrong in this problem? Any suggestion is highly apprectiated.