Physics 204: Spring 2008

Standing Waves

Most vibrating systems – musical instruments for example – can be understood in terms of their normal modes of vibration, or standing waves. These standing waves show up in two apparently different ways. Spatially they have a fixed pattern of strong vibration in some places, separated by nodes in other places, places that don’t vibrate at all. Temporally they are characterized by definite frequencies, peculiar to the system, called characteristic frequencies, or normal frequencies. For the simplest such system, a vibrating string, there is a nice relationship between the spatial behavior and the temporal behavior. Each standing wave has a spatial frequency, and the spatial frequency and the temporal frequency are proportional. In symbols, if is the temporal frequency and k is the spatial frequency, then

What is more, the constant of proportionality in this relationship is the speed of the wave on the string -- call it v. Thus

Note: both of the frequencies above are angular frequencies.

We may express this relationship in other ways too. Using the relationship between (angular) frequency and the period T of the vibration, namely , and the analogous relationship between spatial frequency k and wavelength , namely , we have

Using the relationship between period T and frequency f (not angular frequency), namely T=1/f, we have

These are all different ways of saying the same thing, but (1.1) is perhaps the simplest way. Remembering the dimensions of the quantities will help you put things in the right place. Temporal frequencies are inverse times, period T is a time, spatial frequency k is an inverse length, wavelength is a length, and speed v is a length divided by a time. If you haven’t already done it, check the above relationships for dimensional consistency!

In this lab we will determine and k for several different standing waves on the same string and check (1.1) experimentally. The technique will be to drive a string with an oscillator at various frequencies f , tuning the frequency until we find a normal mode (standing wave). For this mode we will note the frequency f, calculate the angular frequency , note the wavelength , and calculate the spatial frequency k. Then we will increase the frequency looking for the next normal mode, and so on. We should be able to find 4 or 5 modes, each one having its characteristic spatial frequency and temporal frequency. Do they increase together? Are they proportional? The way to decide that is to plot these data on x-y axes and see if they lie on a straight line through the origin. If they do, the slope is the wave speed v (pay attention to units and to pesky factors of ). If you haven’t used Excel to do this kind of analysis before, refer to ExcelHandout.pdf. Represent the data in a neat table:

f k

and make an appropriate graph to find v.

The vibrating string is under tension, because a weight W hangs on the end of it. It is a familiar fact that if the tension on such a string increases (if more weight is added, for example), then the characteristic temporal frequencies will go up. This is how you tune stringed instruments – you adjust the tension in the string to change the temporal frequency. On the other hand the spatial frequencies stay the same, because they are built into the instrument, and correspond to its physical size, which doesn’t change. Thus when the tension goes up, the wave speed v must go up. In the preceding paragraph we described how to measure the wave speed v on a string for one tension W. Repeat this measurement for several weights W, making neat tables of mode frequencies and spatial frequencies for each W, and thus determine for each W a wave speed v as the slope of an appropriate graph.

Finally we ask about the relationship between wave speed v and tension W. There are reasons to think that it is a power law, and that the power is ½ (i.e. a square root). The way to check for a power law relationship, you should recall, is to graph log(v) vs log(W). This is another case of looking for a straight line relationship in the data. If these data lie on a line with slope ½, it means that . Use your data to check for a power relationship between v and W.

As always, summarize what you have done, and what you have found out, in a paragraph or two, and talk it over with the lab instructor before leaving. This is the place to be sure you are very secure on some key concepts and typical phenomena, things we will meet literally everywhere this semester!