Waves with trolleys
Lines of dynamics trolleys connected by springs make good models for both longitudinal and transverse waves. They are best used at advanced level, when investigating factors affecting wave speed.
Apparatus and materials
Spring holders, 20
Dynamics trolleys, 11
Health & Safety and Technical notes
1 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.
2 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.
a 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.
b Give the trolley at the unclamped end a sudden deflection to one side and back, to show a pulse wave travelling along the line of trolleys.
a 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.
b Roll the trolley at the unclamped end a short distance and back, suddenly, to show a pulse traveling along the line of trolleys.
1 Unless the bench is very rough, if you try to produce continuous waves you will get reflections from the clamped end. These are likely to cause standing waves. This may be a distraction worth avoiding.
2 Longitudinal pulses can be produced much more successfully if the trolleys are connected with compressive springs (available as trolley accessories), rather than the extension springs shown in the diagram. One end of the line of trolleys can be left free and a pulse sent down the trolleys from the other end. The reflected pulse from this ‘open reflection’ can be compared with the reflected pulse when the end of the line of trolleys is firmly fixed, a ‘closed reflection’.
3 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.
4 A system like this, with the mass of the wave medium concentrated in discrete lumps with forces between each, behaves differently than a smoothly spread-out medium would do. This system is dispersive: when the wavelength is not much larger than the spacing between parts of the lumped medium, wave speed depends on wavelength. It also exhibits 'cut off': waves of high frequency are not propagated at all.
Try moving an end trolley rapidly to and fro. The neighbouring trolley will oscillate a little, the next trolley oscillates less, and there is something like an exponential decrease in amplitude along the system. Wave energy cannot be propagated through the line of trolleys.
5 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.
6 Another way to model transverse waves is using a wave machine (cheap and simple to construct), as this video  from the National STEM Centre eLibrary shows.
This experiment was safety-checked in February 2006