Demo #4: Standing Waves

Purpose:
To demonstrate that the interference of P and S waves causes standing waves such as Surface Waves.
Supplies:
Slinky
Background and Demonstration:
In the Earth, changes in incompressibility and rigidity occur primarily in the vertical direction. This is due both to the predominantly horizontal sedimentary strata near the surface, and to the increase in pressure and temperature with depth in the Earth. The previous demonstration illustrated that seismic P and S waves will reflect from materials with different elastic properties. Therefore in the Earth, P waves and S waves are bouncing back and forth in the vertical direction, even while the waves are moving continuously outward in the horizontal direction. A slinky cannot represent this 3-dimensional wave propagation, so we will concentrate only on the vertical portion of this phenomenon.

Ask a helper to hold one end of a stretched-out slinky without moving it (your helper may have difficulty doing this, but he/she should try). Send an S-wave pulse down the slinky (by moving your end quickly in an up-down direction). Note the traveling wave and its reflection. Now send two pulses down the slinky by repeating this motion with a slight delay in-between. After the first pulse reflects from your helper, it interferes with the second pulse. In an instant in time the two pulses superpose, with some parts adding together to generate a larger pulse, while others subtract such that the slinky moves hardly at all at that point. Now send a continuous series of pulses down the slinky by oscillating your end up and down with a constant frequency (rate). Now you will see a complicated interference, which may look like a jumbled mess. However, if you have chosen an appropriate frequency, a standing wave will develop, in which waves no longer appear to travel along the slinky, but some portions of the slinky seem to oscillate up and down while other portions remain stationary. If you experiment with this, you will find that there are very distinct frequencies that will set up standing waves on your slinky, and if you change to a different frequency, you will revert to the jumbled mess of traveling waves.

At the lowest frequency (slowest oscillation) for which a standing wave develops, the center of the slinky oscillates up and down (like a jump rope). This is the fundamental mode of oscillation. If you shake your end twice as fast (double the frequency), another standing wave develops with two large oscillations separated by a stationary point in the center. This point is called a "node", and this is the first higher mode of oscillation (the first overtone or harmonic). Shake your end with somewhat higher frequency and you can find the third higher mode, which has three oscillating sections and two nodes. Keep increasing the frequency to find higher modes. Usually, the amplitude of your shaking will decrease with increasing frequency and at about the eighth or ninth mode, you will feel as though your arm is ready to fall off!

Nevertheless, what you have demonstrated is that, for a given length of slinky, there are discrete frequencies of waves that will generate the various modes of standing waves. For the next important point, slow down so that you can, once again, produce the fundamental mode (like a jump rope) standing wave. Now shorten your slinky by collecting half or so of the spring in your hand and notice the frequency of shaking now required to generate the fundamental mode; it is higher (or faster). You can reinforce this idea by chosing shorter or longer lengths of slinky. For a given mode (the fundamental in this case), the discrete frequency required to generate a standing wave is lower for a longer slinky and higher for a shorter slinky.

Now, returning to the idea of P and S waves bouncing around vertically in the Earth, we can see that these traveling waves will generate standing waves (in the vertical direction) with discrete frequencies for each mode. In addition, for a given mode, if the vertical distance over which the waves interfere to generate the standing wave (which is analogous to the length of the slinky) is large, the frequency of that mode will be low. Conversely, if the depth range of interference is small (only shallow bouncing around), the frequency of that mode will be higher. In the Earth, these vertically standing waves are called Surface Waves. High-frequency surface waves involve interference from the surface down to shallow depths, while low-frequency surface waves includes the interference of deeper-penetrating waves.

Recall that, in the Earth, waves are also traveling outward horizontally while they are interfering vertically. Since the P and S wave speed in the Earth generally increases with depth, and these are the waves that are interfering, we observe that high-frequency surface waves travel outward at a slower speed than low-frequency surface waves; this is called "dispersion". By measuring the arrival times of different frequencies of surface waves, we can obtain indirect estimates of the increase in the speed of P and S waves with depth in the Earth.


Jeffrey S. Barker (SUNY Binghamton) Demonstrations of Geophysical Principles Applicable to the Properties and Processes of the Earth's Interior, NE Section GSA Meeting, Binghamton, NY, March 28-30, 1994.
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Last modified: March 18, 1996 (jsb)