# Doppler teachers notes

## Not very angry birds

I've never played angry birds but know it's something to do with throwing birds, I guess that's what makes them angry. We are not going to throw the birds but we'll release them one at a time from a moving truck, this will make them a bit cross but not exactly angry.

The birds in question can fly at 5 ms-1 in still air. They are released one at a time from a truck travelling due North along  a straight road at 2ms-1 . Half of the birds are released from the front the others from the back. The birds travel along the road, the ones at the front forwards and the ones at the back backwards (they travel backwards but fly forwards).

Two twitchers (bird watchers) are waiting by the road about 1km ahead of and 1km behind the truck.

• If the birds are released one per second how often will they pass each observer?

• What would happen if the truck increased it's speed?

The birds would arrive more regularly ahead, less regularly behind.

• What if the truck went at 5ms-1?

The birds going forwards will all arrive at the same time.

The last bird will arrive first (ahead).

The birds are released f times per second, the bird speed is c and the speed of the truck is v.

• Write an expression for the frequency ahead and behind the truck.

The truck now stops but keeps releasing the birds. Calculate the rate at which the birds pass the two observers if they each start to walk towards North at 1 ms-1.

• If the walkers speed is v write expressions for the rate at which the birds pass.

Explain why this car horn sounds like it does.

The compressions and rarefactions in the sound wave are like the birds. When the source moves towards the observer they get closer together when it moves away they get further apart.

Can you find the velocity of the car?

One way of doing this is to use Audacity to measure the frequencies then calculate the source velocity.

Try to build a simulation of the moving truck example in Geogebra.

This was much more difficult than I thought it would be.

• everything starts moving at t=0
• the initial displacement of the birds is set so that they have the same displacement as the truck a time Δt apart.
• The display conditions are set so that they are not visible until they pass the truck

On another day the experiment with the birds is ruined by an old lady feeding the birds at a point 1 km along a line perpendicular to the road. Instead of flying forwards all the birds fly to the food.

• How will the arrival time of the birds be affected?

This one is quite difficult but again Geogebra can be used to help understand what is going on.

• The birds will follow a line joining their starting point to the seeds, this line can be defined if you know the stating point which you do, it's the point where the path of the bird and truck coincide.
• The bird's speed will be c, the path of the bird can be plotted by using a circle with radius ct centred on the starting point previously defined.

To graph the change as the truck moves along the road is much easier to do without the animation. I calculated the difference in  time for the birds to arrive using the triangles then plotted the coordinate (a) of the point as it changed position.

Note that although I am moving point A it is actually the truck that is moving, point A is easier to move.

This Excel version uses the Doppler equation to calculate the frequency at different points. using the horizontal component of the velocity.

I cheated to get the sign change at the passing point and removed 0 because it messed things up.

The truck stops and the old lady goes home because the wind has started to blow at 3 ms-1

• How does the wind affect the arrival times ahead and behind the truck?

The birds will be further apart but also travelling faster f = v/λ so the frequency remains the same. You can try this in Geogebra by varying the bird speed with the truck stationary.

#### Sound example

Calculate the frequency observed of a 400 Hz car horn travelling towards an observer at 30 ms-1

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