We make use of resonant behaviour in many ways. For example, resonance chambers are used to amplify sound in acoustic instruments; resonant behaviour of electrical circuits is used in the transmission and reception of the wireless communications of radio, television and cellphones; lasers operate on the principle of resonance.
Resonant systems can sometimes create situations over which we have no control. There have been some devastating consequences of uncontrolled resonant oscillations.
One of the most spectacular mistakes in the history of civil engineering was the design of the Tacoma Narrows bridge, built at a cost of seven million dollars. It was the third-longest suspension bridge in the world when it opened and the supporting towers were separated by more than half a mile.
Well in advance of the opening, spectacular motions of the bridge had earned it the nickname of Galloping Gertie. Oscillations with an amplitude of about 1 metre were a regular feature and attracted sightseers.
On 7 November 1940, in steady winds of about 40 mph, the bridge began to twist violently and eventually collapsed, shedding a 183-m-long section into the valley below. The only casualty was a dog.
The 1941 report of the investigating committee attributed the disaster to some form of resonance. It was not a simple case of forced resonance (such as , J, . ,. J. n A violet twisting motion.
when the periodic motion of sol-
^ Courtesy of University of diers marching in step over a Washington Libraries.
Official opening on 1 July 1940.
Courtesy of University of Washington Libraries.
suspension bridge sets it into resonant oscillation). The wind could not have maintained a sufficiently large regular stimulus for simple resonant oscillation.
There has been a lengthy debate as to the exact cause of the collapse, but what is certain is that a full understanding of what happened can only be gained through a rigorous mathematical analysis of the sequence of events.
'Considerable liquefaction and damage to new buildings occurred in Mexico City during the Great Earthquake of 19 September 1985. Although the epicentre was more than 300 km away, the valley of Mexico experienced acceleration of up to 17% g, with peaks concentrated at 2-sec periods. The extreme damage in Mexico City was attributed to the monochromatic type of seismic wave, with this predominant period causing 11 harmonic resonant oscillations of buildings in downtown Mexico City. The ground accelerations were enhanced within a layer of 30 ft of unconsolidated sediments underneath downtown Mexico City, which had been the site of a lake in the 15th century, causing many buildings to collapse.' (Pararas-Carayannis, G. 1985*)
6.7 Diffraction — waves can bend around corners
All waves can spread around corners — a phenomenon called diffraction. A classic example of diffraction is where incoming waves pass between rocks or other obstacles close to the shore. The photograph shows a series of stone breakwaters along the shore at Kingsmill in Chesapeake Bay, Virginia. The breakwaters were constructed as part of a coastal erosion control programme in the Bay. Incoming wave energy is sapped by frictional effects, refraction and diffraction. The semicircular way in which the sea has eroded the shoreline behind the breakwaters is remarkably clear evidence of the effect of diffraction.
6.8 The magic of sine and the simplicity of nature
A question which might spring to mind at this stage is why we should have chosen the simple sine wave as the means of describing natural waves which are frequently far from sinusoidal in their behaviour. The mathematics of a simple sine wave is relatively easy to handle, but there is a much more compelling reason.
A young cellist, apprehensive about giving her very first performance, decides to practise in a secluded corner of a local park, early one morning.
The wave form is far from sinusoidal but surprisingly enough, provided that it is periodic, we can describe it as the sum of a number of sine waves of different amplitudes and frequencies.
To illustrate how complex wave forms may be produced we can take the example of a square wave form. We can graphically illustrate how adding the amplitudes of sine waves of certain frequencies and amplitudes results in a wave which becomes increasingly like a square wave.
Sets of 2, 3 and 7 sine waves are shown in Figure 6.9 together with the wave form which results from adding them. As we add more sine waves, we can see that the resultant wave becomes closer and closer to a square wave.
3 sine waves
7 sine waves
3 sine waves
7 sine waves
01234 01234 01234
Figure 6.9 The evolution of a square wave.
The frequencies of the component sine waves are all harmonics of the frequency of the square wave (see Appendix 6.3). The mathematical method for the decomposition of any harmonic wave into its component sine waves is called Fourier analysis.
So the simple sine wave proves to be the basis for the mathematical description of any periodic wave, even complex ones such as the sound waves from musical instruments. The key to the process of describing such waves is in knowing the relative amplitudes and frequencies of the component sine waves. This is the essence of the true harmony and simplicity of nature, which becomes more apparent as we learn what to look for.
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