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# 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!

## 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.