Waves:The Ideal Case

物基一班　张志城 　2013301110121
Exercise:6.6

Contents

Abstract

　　In this passage,I discretize the central equation of wave motion and adopt the end-fixed condition to research the evolution of Gaussian wave packet,triangular wave packet and square wave packet.Besides,I compare the differences between the result of r=1.2 and the result of r=0.8,and I find that r=1.2 makes the result diverge drastically with time and r=0.8 can make the result relatively stable.
　　

Background

　　The central equation of wave motion is
　　　　$\frac{\partial^2 y}{\partial t^2}=c^{2}\frac{\partial^2 y}{\partial x^2}$　　(6.1)
　　which is usually referred to as the wave equation.This equation arises in many situations,including waves on a string,electromagnetic waves,waves on the surface of a lake,and sound waves.Although our methods and conclusions apply to other conditions as well,here we will use a language appropriate for waves on a string.
　　It doesn't hurt to derive the equation from the wave motion on a string.The following diagram demonstrate the force on the segment.
　　　　
　　According to Newton's second law,we get
　　　　$(\mu \Delta x)\frac{\mathrm{d} ^{2}y_{i}}{\mathrm{d}^{2} x}=Tsin\theta_{i+1}-Tsin\theta_{i}$　　(6.2)
　　and then use a finite difference approximation for the angles:
　　　　$sin\theta_{i}\approx\frac{y_{i}-y_{i-1}}{\Delta x}$　　(6.3)
　　We obtain
　　　　$\frac{\mathrm{d}^{2}y_{i} }{\mathrm{d}^{2} x}\approx(\frac{T}{\mu})\frac{y_{i+1}-2y_{i}+y_{i-1}}{(\Delta x)^{2}}\approx \frac{T}{\rho}\frac{\partial^2 y}{\partial x^2}$　　(6.4)
　　which is precisely of the form of (6.1).
　　
　　Our main goal is to develop a numerical scheme for sovling (6.1). We can finally get the approach:
　　　　$y(i,n+1)=2[1-r^{2}]y(i,n)-y(i,n-1)+r^{2}[y(i+1,n)+y(i-1,n)]$　　(6.6)
　　in which $r=c\Delta t/\Delta x$.
　　Since our algorithm requires only the current value and the value at previous time step,we thus need only three values of the time index.And we let y1 correspond to y at previous time step,y2 correspond to y at current time step and y3 correspond to y at next time step. After each iteration,we let y1 get the values of y2 and y2 get the values of y3 to realize the update of values.
　　

Source Codes

• the evolution of different wave packets
• the effect of r on wave evolution
• the superposition of two waves
  　　

Results

Evolution of different wave packets ($r=1$)

　　First,if the initial wave packet is a Gaussian wave packet of the form
　　　　$y_{0}=exp[-k(x-x_{0})^{2}]$
　　we can get following result:
　　　　
　　
　　Second,if the initial triangular wave packet is of the form:
　　　　$y=0.01x \quad 0<=x<500$
　　　　$y=10-0.01x \quad 500<=x<1000$
　　we can get following result:
　　　　
　　
　　Third,if the initial square wave packet is of the form:
　　　　$y=0 \quad 0<=x<250,400<=x<1000$
　　　　$y=1 \quad 250<=x<400$
　　we then can get following result:
　　　　
　　　　

The effect of r on evolution

　　When $r=0.8$,we can respectively get the evolution pattern of Gussian,triangular and square wave packets.
　　　　
　　　　
　　　　
　　
　　When r=1.2,we can respectively get the evolution pattern of Gussian,triangular and square wave packets.
　　　　
　　　　
　　　　

Superposition of Waves

　　　　
　　　　We can find that the two waves collide with each other and then pass through each other without changing the shape or speed.

Conclusion

　　1、We can find from the evolution of the three wave packets (r=1) that the wave packet will split to two separate wave packets towards oppisite direction,and when they encounter the ends they will reflect with negative values,which is half-wave loss.
　　2、From the three pictures of $r=0.8$ we find that the result is relatively stable,and differ a little with the result of $r=1$. However,for r=1.2,the results are not stable,and we can see the amplitudes diverge rapidly with time.
　　3、We can find from the gif that the wavepackets are unaffected by collisions.The independence of waves is consistent with the feature that the sum of two solutions of a linear equation is still a solution of the equation.
　　

References＆Acknowledgements

[1] Nicholas J.Giordano. 计算物理. 北京：清华大学出版社,2007.
[2] Thanks to the codes of animation written by Wentao Liu(刘文焘) in his homework:https://www.zybuluo.com/Canonvar/note/394434