Find a piece of thin rope or a slinky and a stopwatch. If you have a friend nearby, have them hold one end of the rope or slinky. If not, tie or connect one end of the rope or slinky to a door knob. Stand a distance apart so that there is a bit of tension on the rope or slinky. Send a pulse down the rope or slinky and watch what happens when the pulse reflects off your friend or the doorknob. Next, start creating a sine wave motion with your hand, see Fig. 2. Try to move your arm up and down so that the rope or slinky creates the shapes seen in Fig. 1.
Fig. 1 The first three standing waves of a one dimensional rope or slinky. Each standing wave is drawn with the same amplitude A. Interestingly, the frequency of a standing wave is independent of the amplitude of the standing wave
Fig. 2 A visual description of wavelength and frequency. Wavelength is the distance between two “like” points on the wave, for example the distance between two adjacent peaks or two adjacent troughs. If the wave is moving towards the right, frequency is how many peaks pass through the dashed line every second. If both waves are moving with the same speed, the upper wave has a higher frequency than the lower wave since more peaks pass the dashed line in 1 second
You will find that your hand has to move up and down with a very specific frequency to create these shapes, which are called standing waves. When you produce standing waves, the wave that you are creating with your hand is constructively interfering with the reflected wave. No other frequencies produce perfect constructive interference. Using your stopwatch, find the time for ten full oscillations (the time it takes for your hand to move up and down ten times) for each of the first three standing waves. The period of the standing wave is that time divided by 10. If you count twenty full oscillations, you would divide the time by 20 to find the period. Next use the formula f = 1/T to find the frequency needed to produce that standing wave. The frequency of the standing wave with one “loop” (the left picture) is called the fundamental frequency. You should find that the second picture with two loops has twice the fundamental frequency. The third picture with three loops should have a frequency that is three times the fundamental frequency. In general, fn = nf1, where n is how many loops the standing wave has. Notice that the higher mode standing waves (the waves with more loops) require more shaking energy! (1)
This will be important later. (2)
- We conclude from this experiment that standing waves occur only at specific resonant frequencies, which are integer multiples of the fundamental frequency.
- Because standing waves are fundamental to understanding resonance and harmonics, concepts that play an important role in musical instruments, sound waves, and many physical systems
