## 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.

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TSny
Homework Helper
Gold Member
Apparently they want the smallest Δx between two points for which the absolute value of the displacement $\xi$ is 0.02 m.

• Soren4
It looks like your solution only considers values of $x$ where $\xi = +0.02$ m. What about points where $\xi = -0.02$ m? It helps to make a sketch of the sine wave at some instant of time and mark the points on the x axis where $\xi = \pm 0.02$ m.