Waves on a string

程熹 2013301020038
Homework

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Abstract

This article investigate the wave motion on a string using wave function,how the length of the rope and initial position influence the wave. Besides, how two different disturbances on the string meet,join and seperate. We study them with numerical method

Key Word

Laplace's equation,boundary conditon,electric field

Introduction

Wave is an oscillation accompanied by a transfer of energy that travels through medium . Frequency refers to the addition of time. Wave motion transfers energy from one point to another, which displace particles of the transmission medium — that is, with little or no associated mass transport. Waves consist, instead, of oscillations or vibrations , around almost fixed locations.
A single, all-encompassing definition for the term wave is not straightforward. A vibration can be defined as a back-and-forth motion around a reference value. However, a vibration is not necessarily a wave. An attempt to define the necessary and sufficient characteristics that qualify a phenomenon to be called a wave results in a fuzzy border line.
Wave is often intuitively understood as referring to a transport of spatial disturbances that are generally not accompanied by a motion of the medium occupying this space as a whole. In a wave, the energy of a vibration is moving away from the source in the form of a disturbance within the surrounding medium . However, this motion is problematic for a standing wave , where energy is moving in both directions equally, or for electromagnetic waves in a vacuum, where the concept of medium does not apply and interaction with a target is the key to wave detection and practical applications. There are water waves on the ocean surface; gamma waves and light waves emitted by the Sun; microwaves used in microwave ovens and in radar equipment; radio waves broadcast by radio stations; and sound waves generated by radio receivers, telephone handsets and living creatures (as voices), to mention only a few wave phenomena.

Main Features

The one dimensional wave equation:

$$\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}$$
we add the damping term to it and then it become: $$\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}-\epsilon L^2\frac{\partial^4 u}{\partial x^4}$$
we solve this and get the equation: $$y(i,n+1)=2(2-2r^2-6\epsilon M^2)y(i,n)-y(i,n-1)+r^2(1+4\epsilon M^2)y(i+1,n)+y(i-1,n)]-2r^2\epsilon M^2[y(i+2,n)+y(i-2,n)]$$
The equation of the disturbance is: $$y_0(x)=\exp[-k(x-x_0)^2]$$
Then we can use this to design program to get different results in different x and l

Result

1.l=1m,x=0.6m and k=1500

2.Two different disturbances on the string

3. The triangular wave's power spectrum

Conclusion

1. From diagram 3 and diagram 4 we can see that the waves caused by the disturbances are completely independent to each other. The properties of the waves such as shape and velocity remain the same before and after they join togather. They satisfy the linear superposition principle.
2. From diagram 5 and diagram 6 we can find that the peaks decrease when the frequency becomes larger.

Acknowledgement

I have used the origin code of 陈天懿，and run his program to get some of the upper result. So thanks to him here.

Reference

Computational Physics(second edition) Nicholas J.Giordano and Hisao Nakanishi
Wikipedia wave
https://en.wikipedia.org/wiki/Wave