• Soren4
In summary: You should be able to see the correct pattern from that.In summary, the conversation discusses finding the wavefunction and distance between two points on an ideal rope with a wave moving at velocity 20 m/s. The wavefunction is given by ##\xi(x,t)=0.1 \mathrm{sin}(6t-0.3x )## and the distance between two points at a certain time is given by ##\frac{\pi-2\arcsin(0.2)}{0.3}##. The mistake in the derivation was not considering points where ##\xi = -0.02## m and a sketch can help visualize the correct pattern.
Soren4

## 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.

Apparently they want the smallest Δx between two points for which the absolute value of the displacement ##\xi## is 0.02 m.

Soren4

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..

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.

## What is the wavefunction?

The wavefunction is a mathematical function used in quantum mechanics to describe the quantum state of a particle or system of particles. It contains information about the position, momentum, and other properties of the particles.

## What does the wavefunction represent?

The wavefunction represents the probability amplitude of the particle or system being in a particular state. It is not a physical quantity, but rather a mathematical tool used to make predictions about the behavior of quantum systems.

## How is the wavefunction related to the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the wavefunction evolves over time. It is used to calculate the future state of a quantum system based on its current state and the potential energy of the system.

## Can the wavefunction be directly measured?

No, the wavefunction is a mathematical concept and cannot be directly measured. However, the predictions made using the wavefunction can be tested and verified through experiments.

## What are the units of the wavefunction?

The wavefunction does not have any physical units as it is a mathematical function. However, the square of the wavefunction has units of probability, typically measured in square meters or inverse seconds.

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