## 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.

Related Introductory Physics Homework Help News on Phys.org
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.

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.