- #1

Soren4

- 128

- 2

## Homework Statement

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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

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A general wavefunction is

$$\xi(x,t)=\xi_0 \mathrm{sin}( k x-\omega t +\psi)\tag{1}$$

## The Attempt at a Solution

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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.