Chapter 6 problem 6.6

python.assignment

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name: Li FangYing
number: 2013301020027

Abstract

In this chapter we will consider topics about wave motion.To be beginning, we simulate waves on a perfectly flexble, frictionless string, then we go to more complicated situation. Some interesting phenomena will be observed such as half loss and superposition of waves.

Introduction

The central equation of wave motion is

$$\frac{\partial^2 y }{\partial t^2}=c^2\frac{\partial^2 y}{\partial x^2}$$
This equation can be applyed to many situation, including electronmagnetic waves, waves on water surfuce, sound waves and so on. We start with the simplest wave, the wave on a string which is fixed on the both ends.Even if this is a simple system, it can reveal many essential properties of waves, such as transmition, refection, half loss and superposition principle.

Nummerical Approach

We treat both $x$ and $t$ as discrete variables with $x=i\Delta x$ and $t=n\Delta t$, the wave equation can be written as finite difference form, for example,

$$\frac{\partial y^2}{\partial t^2}=\frac{y(i,n+1)+y(i,n-1)-2y(i,n)}{(\Delta t)^2}$$
So the whole equation can transfer to $$y(i,n+1)=2[1-r^2]y(i,n)-y(i,n-1)+r^2[y(i+1,n)+y(i-1,n)]$$
in which $r$ means $c\Delta t/\Delta x$.
Then we fix the both ends with $y(0,n)=y(M,n)=0$ for all time steps n, in which we apply a fixed boundary condition.

Data analysis

At first, we simulate a wave on a perfectly flexible, frictionless string with fixed ends. The string has a length of 1m and wave velocity is 300m/s, and we take $r=c\Delta t/\Delta x=1$:

You can see a wave package will automatically decomposes to two packages, and they propagate to opposite directions.
What's more, when a wave package reaches the end of the string, its vabrition direction will inverse.

superposition principle

Then we add one more wave package:

Two wave packages will propagate towards each other and collide, but they are unaffected by each other, that's to said their shape and speed are unchanged. This result is reasonable beacuse the $wave$ $equation$ is linear, and all linear equations observe superposition principle.

half loss

Wave propagates on a composite string,transmits to a denser medium and reflects at the surface, with $c=300m/s$ in the first medium and $c=150m/s$ in the second medium:

If we make the second medium even more denser, that is make $c=125m/s$:

When $c=15m/s$

When the second medium become denser, the amplitudes of transitted waves become smaller,and you can see the so-called half-loss phenomena:
because the incident angle is zero, when wave propagate from a sparse medium to a denser one, reflected wave has a phase mutation of $\pi$, which equals to half of the wave length of the opitical path is lost.
So the vabrition direction of the reflected wave is changed, but the transmitted wave not.

Conclusion

We simulate wave packages propagate on flexible, frictionless uniform string and composite string, and observe packages decomposition, reflection and inversion. Then we verify superposition principle of waves and we also analyse half loss phenomenon. Although the model is simple, it reveals many intrinsic properties of waves.

program codes

wave.py