# Standing waves along trolleys

##### Demonstration

Standing waves along a line of dynamics trolleys linked by springs make quite a sight - some trolleys stationary while nearby trolleys move quickly.

#### Apparatus and materials

Dynamics trolleys, 11

Spring holders, 20

Springs, 20

#### Health & Safety and Technical notes

If the model is set up on a table with no raised edge, it is easy for part of it to run off the side of the table. The rest of the model inexorably follows! So it is best set up on a smooth floor, or on a surface provided with barriers along its edges.

Use fairly long dowels and keep the springs down at the bottom of them. Otherwise, the springs have a tendency to fly off and the trolleys scatter.

#### Procedure

a For transverse waves, clamp a trolley to one end of a smooth bench and connect the others side by side with springs, as shown. The trolleys are separated so that they can move without hitting each other.

Oscillate the trolley at the unclamped end continuously, to generate a continuous transverse wave down the line of trolleys. Continue and the reflected wave will build up a standing wave pattern.

b For longitudinal waves, connect the trolleys end to end, as shown. (Some trolleys have a projecting front wheel. Such trolleys may have to be linked between their towing pegs with springs.) The springs snap shut when released, so you will need to hold the whole model in tension.

Oscillate the trolley at the unclamped end continuously, to generate a continuous longitudinal wave down the line of trolleys. Continue and the reflected wave will build up a standing wave pattern.

#### Teaching notes

1 These arrangements provide a good model of a pulse travelling through a medium in which masses are connected together by spring-like connections. Atoms, at an equilibrium distance, are ‘connected’ by electrical forces, though of course these increase whether the material is compressed or extended.

2 You could investigate how the wave speed is affected by changes in mass and in tension. The mass of each trolley can be doubled by adding loads, and the tension can be doubled by adding extra springs. Doubling the mass of each trolley reduces the wave speed; doubling the tension raises it.

Both modifications change the speed by the same factor (actually 2) and both made together will restore the speed to its original value. Clearly, the wave speed depends upon how long it takes each part of the model to acquire some speed when forces act upon it, as the wave front arrives.

Using a simple mathematical wave model for the trolley system you could predict the wave speed, then measure the speed and compare them. You would find that they fit quite closely.

3 A system like this, with the mass of the wave medium concentrated in discrete lumps with forces between each, does not behave in all respects like a smoothly spread-out medium would do. The system is dispersive: the speed depends upon the wavelength when the wavelength is not much larger than the spacing between parts of the lumped medium. It exhibits 'cut off’: waves of high frequency are not propagated at all.

4 Try moving an end trolley very rapidly to and fro. The next-door trolley oscillates a little, the next oscillates less, and there is something like an exponential decrease of amplitude along the system. No wave energy propagates down the system. (These problems are discussed in The Berkeley Physics Course, vol. 3 Waves.)

This experiment was safety-checked in February 2006