5.3 Vibrating string: standing waves on a string
We still haven't answered the question of how standing waves are set up on a string. To do so we need to return to our string, fixed at one end and held in someone's hand at the other end. Imagine now that instead of sending a single pulse along the string, the person flicks their hand up and down periodically and sends a sinusoidal wave along the string. This wave gets reflected and inverted at the fixed end and travels back towards the person holding the string. There are now two waves of the same wavelength travelling in opposite directions on the string. Just the condition needed to set up a standing wave!
But why should standing waves be set up only at certain frequencies? The answer to this question lies in what happens when our reflected wave gets back to the hand holding the string. Here it is reflected and inverted again and travels back along the string towards the fixed end. So, there are now three waves travelling along the string. Of course, when the third wave reaches the fixed end it will be reflected and inverted, resulting in a fourth wave. This will carry on until eventually there are hundreds of waves travelling backwards and forwards along the string.
So won't there end up being a messy jumble of waves on the string? Well, most of the time this will be exactly the case. However, there are certain frequencies at which the person can flick their hand up and down that will cause the third wave to be in phase with the initial wave (Figure 9). That is, the peaks and troughs of the third wave will line up with the peaks and troughs of the initial wave. Under such circumstances, it will follow that all the waves travelling towards the fixed end will be in phase with each other. And, indeed, all the waves travelling back from the fixed end will be in phase with each other too. The waves travelling towards the fixed end will look and behave like one wave, and the same will be true for the waves travelling back towards the person holding the string. Therefore, only at these certain frequencies will there be the equivalent of just two travelling waves moving in opposite directions on the string. These travelling waves will combine to produce a standing wave.
Run the Flash animation below. It should help you see how a standing wave can be set up on a string if the string is driven at the correct frequency.
Use the controls below (Play/Pause, Step, Reset) to run flash animation.