**Waves and Interference**

Created by D. Acosta, 9-22-98

This page was created using MapleV. The worksheet used to create these plots is available here:

(Use "shift-click" to save the file)

Here is an example of a wave with no time dependence:

`> `
**y := sin(kx);**

Below is a plot of this wave, where k*x is in units of Pi. Two complete wavelengths are shown.

`> `
**plot(sin(kx*Pi),kx=0..4);**

Two waves which will completely interfere destructively are shown next:

`> `
**plot([sin(kx*Pi),sin(kx*Pi+Pi)],kx=0..4);**

Both waves have the same wavelength, but one has a phase shift of 180 degrees (Pi radians)

Two waves with the same wavelength but a 90 degree phase shift are shown next:

`> `
**plot([sin(kx*Pi),sin(kx*Pi+Pi/2),sin(kx*Pi)+sin(kx*Pi+Pi/2)],kx=0..4,color=[red,green,blue]);**

The resultant (shown in blue) is also a wave with the same wavelength but different amplitude.

Now let's see what happen when we superpose two waves with slightly different wavelengths:

`> `
**plot([sin(k1x*Pi),sin(k1x*Pi/1.0625)],k1x=0..51);**

Beats are formed when these two waves interfere:

`> `
**plot([sin(k1x*Pi)+sin(k1x*Pi/1.0625),2*cos(0.058824/2*k1x*Pi),-2*cos(0.058824/2*k1x*Pi)],k1x=0..51,color=[red,green,green]);**

The intensity of the resultant wave is given by the square of the amplitude:

`> `
**plot([(sin(k1x*Pi)+sin(k1x*Pi/1.0625))**2,4*cos(0.058824/2*k1x*Pi)**2],k1x=0..51,color=[red,green]);**

Now let's consider
*travelling waves*

The next figure shows a travelling wave moving to the
*right.*

`> `
**animate(sin((kx-wt)*Pi),kx=0..8,wt=0..8,frames=50);**

The next figure shows a travelling wave moving to the
*left.*

`> `
**animate(sin((kx+wt)*Pi),kx=0..8,wt=0..8,frames=50);**

Now let's see what happens when two travelling waves interfere:

`> `
**animate(sin((k1x-w1t)*Pi)+sin((k1x-w1t)*Pi/1.0625),k1x=0..51,w1t=0..70,frames=51,numpoints=200);**

Click here for a larger version (860kB)

In this case the phase velocity and the group velocity are the same. Both the carrier wave and the modulation envelope move at the same speed.

In the next case the carrier wave and modulation envelope move at different speeds:

`> `
**animate(sin((k1x-w1t)*Pi)+sin((k1x/1.0625-w1t/1.045)*Pi),k1x=0..51,w1t=0..51,frames=51,numpoints=200);**

Click here for a larger version (860kB)

In this last case, the group velocity (the beat velocity) is much less than the phase velocity (the carrier wave velocity).

Now let's see how the group velocity can be larger than the phase velocity:

`> `
**animate(sin((k1x-w1t)*Pi)+sin((k1x/1.0625-w1t/1.1)*Pi),k1x=0..51,w1t=0..45,frames=51,numpoints=200);**

Click here for a larger version (860kB)