Interesting Articles on Waves
Good Vibrations
by Stephen Cohen
Science in Life
The vibrations that the Beach Boys were singing about were probably the kind felt emotionally for something or someone else. Vibrations, however, exist in many forms. Engineers are often concerned with the mechanical kind, which manifest in all kinds of structures, such as bridges and tall buildings. These vibrations are caused by an external excitation, such as an earthquake or high winds. Musicians are interested in transmitting sound waves, and causing vibrations in their listeners’ ear drums. In the most general sense, a vibration can be thought of as something that moves back and forth repetitively, such as a buoy bopping up and down in a lake, a stock price during the course of a week, or a person’s mood during the course of a day.
Vibrations occur around a specific ‘average value’. In science, this value is known as equilibrium. If left untouched, all mechanical items find and stay in their equilibrium positions. If a spectator at a Tennis match is twisting his head back and forth to follow the ball, his equilibrium position occurs when he is looking straight ahead. The vibration from side to side can be thought of as the search for equilibrium. It is interesting that this search exists for living and non-living things alike. When a guitar string is pulled on and then let go, it then vibrates back and forth hundreds of times per second (its frequency) until the vibration subsides, and the string returns to equilibrium. A person can also be stretched out of his or her comfort zone. When some external stress is applied to someone, the natural tendency for this person is to seek out the balance that existed before the stress occurred. This search for balance is what most people struggle with throughout life – some succeed, others do not. And the funny thing about succeeding in this balancing act we call life is that it is so fleeting; there is always a new challenge around the corner threatening our personal equilibrium. If you hit a bump when driving along in your car, the car’s chassis will bounce due to this external excitation. The job of the car’s shocks is to cause this bounce, or vibration, to dissipate, or dampen. Nearly all vibrating systems have some kind of damping associated with them. The more damped that a system is, the quicker the system will return to equilibrium. I would argue that people too have a damping coefficient associated with them. Some people bounce back quickly from an insult and resume their day, while others brood over it for a long time.
Another property that vibrating systems have is a specific natural frequency. This value defines the rate of vibration. The natural frequency of a simple pendulum is governed by its length and the surface gravity of the Earth, whereas that of a block of wood is governed by the wood’s elastic Modulus (material property), geometry, and how it is constrained. If a system is purposely vibrated at its natural frequency, a special phenomenon known as resonance occurs. Resonance can be devastating to a structure, as it causes the amplitude of its vibration to grow a great deal (the extent to which the amplitude increases depends on how damped the system is). If a metallic structure is excited at its natural frequency, the input vibration can be amplified up to one hundred times, so that a slight input vibration can quickly output a disaster (youtube the famous collapsing Tacoma Narrows Bridge for an illustration).
Returning to our people analogy, I believe that they too have a natural frequency. When a person is stimulated in just the wrong way, we often call it ‘pushing their buttons’. This specific code that ticks off a person can be thought of as their natural frequency. It is unwise to stimulate someone’s natural frequency... I’ve seen insufficiently damped individuals stuck in traffic – they are on the verge of snapping. If you wish to see resonance manifest within a person, just attend a sporting event, and watch them lose it on a referee that makes a few wrong calls.
You can only stretch a spring so far and expect it to return to its equilibrium. An airplane’s wing can only vibrate so much before it cracks due to fatigue. People are equally fragile. We can be taken far out of our comfort zone due to some outside stimulus. If we are stretched too far from our personal equilibrium, we risk never finding it again. Sometimes these stimuli are simply bad choices we make that lead to an addiction like drugs or alcohol. Sometimes we are being stretched by something internally, as in many cases of mental illness. Such changes can cause our equilibrium position to shift. In more unfortunate cases, such changes can cause exponentially increasing vibrations in our lives: the very definition of instability.
We must remember that vibrations are a very natural part of life. Without some kind of stimulation from the outside, life would be boring. A tree never wishes for a day without wind so that the incessant swaying will stop. When confronted with a problem, we can accept it, roll with it, and we can return to equilibrium afterwards. With enough damping in our system (patience), we can even learn to enjoy challenges, and take on more than one at a time. We are more in control of our destiny than we may think. A pendulum does not get to choose its natural frequency, but people do. We can choose whether or not to be bothered by external excitations.
When a structure remains static, without any external excitation, its motion (or lack thereof) is sometimes referred to as ‘the trivial solution’. Such a designation is equally fitting for a person who coasts through life without any challenge at all.