# Exercise about the wavefunction

## 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.

## Answers and Replies

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
Thanks for the reply!

Yes that's part of the question I guess, could you suggest where I went wrong in my derivation? I still do not see the mistake..

TSny
Homework Helper
Gold Member
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.