# Chapter 18 - Superposition and Standing Waves

## Principle of Superposition

When more than one wave passes through a region in space at the same time the displacement is the sum of the separate displacements. This may be a vector sum!

## Reflection and transmission

A one dimensional pulse on a string which reaches the end of string will be reflected. The direction of the pulses displacement depends on the boundary condition where the reflection takes place, ie. whether the string is fixed or free.

This can be demonstrated on the Shive wave machine.

When a wave encounters a change in medium there will be some partial reflection with a phase change that depends on whether it is being reflected from a more or less resistive medium .

## Spatial Interference

The equation for a traveling wave

$D(x,t)=A\sin(\frac{2\pi}{\lambda}x-\frac{2\pi t}{T})=A\sin(kx-\omega t)$

tells us that if we are standing a distance $r_{1}(x)$ from a one dimensional wave the wave displacement at a time $t$ will be

$D_{1}(x,t)=A\sin(kr_{1}(x)-\omega t)$

if a second identical source is a distance $r_{2}(x)$ away

$D_{2}(x,t)=A\sin(kr_{2}(x)-\omega t)$

and the total wave displacement is

$D_{1+2}(x,t)=A\sin(kr_{1}(x)-\omega t)+A\sin(kr_{1}(x)-\omega t)$

Using $\sin\theta_{1}+\sin\theta_{2}=2\sin\frac{1}{2}(\theta_{1}+\theta_{2})\cos\frac{1}{2}(\theta_{1}-\theta_{2})$

$D_{1+2}(x,t)=2A\sin(\frac{k}{2}(r_{1}(x)+r_{2}(x))-\omega t)\cos(\frac{k}{2}(r_{1}(x)-r_{2}(x)))$

We thus hear a wave that has the same wavelength and frequency as that coming from the source, but the amplitude will depend on the distance between the sources, being maximum ($2A$) when $\frac{k}{2}(r_{1}(x)-r_{2}(x))$ is 0 or a whole number multiple of $\pi$ which corresponds to the waves at that point being in phase. The amplitude is minimum ($0$) when $\frac{k}{2}(r_{1}(x)-r_{2}(x))$ is a multiple of $\frac{\pi}{2}$ which corresponds to the waves at that point being out of phase.

## Interference between two speakers

If the waves propagate from the source in all 3 dimensions then we need to take in to account that as we showed in chapter 17, $A\propto\frac{1}{r}$. To determine perceived loudness we need to remember that it depends logarithmically on intensity. I have factored these considerations in the following calculations I performed in Maple. The patterns are for two speakers separated by 1m.

490Hz
686Hz
1143Hz
Ramp from 400 to 1200Hz

## Standing Waves on a String - Both ends fixed

When we have two waves traveling in opposite directions on path of fixed length and both ends are fixed

$D_{1}=A\sin(kx-\omega t)$ and $D_{2}=A\sin(kx+\omega t)$

their sum can produce a standing wave

$D=D_{1}+D_{2}=A\sin(kx-\omega t)+A\sin(kx+\omega t)$

Using $\sin\theta_{1}+\sin\theta_{2}=2\sin\frac{1}{2}(\theta_{1}+\theta_{2})\cos\frac{1}{2}(\theta_{1}-\theta_{2})$

$D=2A\sin kx\cos \omega t$

Recall $k=\frac{2\pi}{\lambda}$ and $\omega=2\pi f$

If the boundary condition is that both ends are fixed then $D$ must be zero at $l$ which means that

$kl=\frac{2\pi l}{\lambda}=\pi,2\pi,3\pi,4\pi..$ etc.

or $\lambda=2l,l,2/3l,l/2,..$ etc.

$f=\frac{v}{\lambda}=\frac{v}{2l},\frac{v}{l},\frac{3v}{2l},\frac{2v}{l},..$ etc.

or if we number the harmonics $n=1,2,3,4,..$

$\lambda=\frac{2l}{n}$ and $f=v\frac{n}{2l}$

When we refer to a harmonic, we are describing the frequency as a multiple of the fundamental frequency.

Recall that for waves on a string $v=\sqrt{\frac{F_{T}}{\mu}}$ so if you take a string and stretch it further you need to take in to account both changes in $l$ and $v$.

## Making Sound - String Instruments

The note in string instruments is generated by exciting a vibration and promoting a particular vibration in the string.

String instruments also use the body of the instrument to amplify the sound. We can see the standing wave patterns of objects with Chladni Patterns. Some examples on a violin and a guitar. And now with lasers.

For a great page on music acoustics: http://www.phys.unsw.edu.au/music/

## A flaming tube

Ruben's tube, invented by Heinrich Rubens in 1905, like the Shive Wave Machine, is really only useful at demonstrating wave concepts. But it's very good at that!

To understand why the flames are lower where the pressure varies more we need to realize that the velocity at which gas flows out is proportional to the square root of the pressure difference. This comes from Bernoulli's principle.

It's worth noting that as the speed of sound is $v=\sqrt{\frac{B}{\rho}}$ and the density $\rho$ can be approximated as the propane pressure the standing wave frequencies depend on the propane pressure and will not be the same as the frequencies when the tube is just full of air at external pressure.

## Beats

Beats occur when two waves with frequencies close to one another interfere.

If the two waves are described by

$D_{1}=A\sin2\pi f_{1}t$

and

$D_{2}=A\sin2\pi f_{2}t$

$D=D_{1}+D_{2}$

Using $\sin\theta_{1}+\sin\theta_{2}=2\sin\frac{1}{2}(\theta_{1}+\theta_{2})\cos\frac{1}{2}(\theta_{1}-\theta_{2})$

$D=2A\cos2\pi(\frac{f_{1}-f_{2}}{2})t\sin2\pi(\frac{f_{1}+f_{2}}{2})t$

A maximum in the amplitude is heard whenever $\cos2\pi(\frac{f_{1}-f_{2}}{2})t$ is equal to 1 or -1. Which gives a beat frequency of $|f_{1}-f_{2}|$.

## Can you hear the beat?

Typically when two tones are seperated by less than about 30-40Hz we hear beating, if the separation is more than that they tend to sound like to different tones. (You can try a similar experiment at a higher frequency at Beats from Physclips).

190Hz
191Hz
192Hz
194Hz
198Hz
206Hz
222Hz
254Hz