Terms and concepts:
• Equilibrium position
• Simple harmonic motion
• Potential energy
• Kinetic energy
• Hooke’s Law
• Natural frequency
• In phase
As we said earlier, music, like life, is an art of motion, but a specific motion; one that begins from a fixed, stable centre in which all forces balance out – the equilibrium position – and rapidly moves back and forth. This motion is a vibration and it is the principle on which all musical instruments rely. In fact, all musical instruments are vibrating systems. The following components and combinations are the building blocks of musical sound.
Periodic Motion is any movement that repeats itself. A vibrating string, the earth circling the sun, waves lapping onto the shore, all are periodic and constantly in motion. In the Indigenous worldview, this would include the moon’s phases and a woman’s monthly cycle. Periodic motion also includes the coming and going of communities and generations; of the cycles of every life that is born, grows old and dies.
In physics, the applications are concrete. Many of the components we discussed earlier create and affect periodic motion, including the displacement of a body from its equilibrium position and related quantities such as mass, force and speed. However, periodic motion also has its own distinctive features and measurements.
The period is the time needed for the body to travel to its farthest displacement on both sides of the starting point and then back to the starting point again. In musical bodies, the starting point is the body’s equilibrium position.
Equilibrium position: Equilibrium means balanced, or “equalled out.” It is the point at which all forces that can affect an object have equalized, or to put it simply (perhaps too simply) – have zeroed each other out. This is the balance point between all forces. As long as the forces remain constant, the object appears to be at rest.
This, too, has a spiritual application. My Lakota Elder, Tunkasila (Grandfather) Sydney Keith taught that when people reach 40, they should return without angst to where their lives were balanced and stable (equilibrium points). People then should examine their pasts; keeping that which is good and discarding the rest. This, he said, prepares the way for the wisdom that includes true Eldership.
A cycle is one completed motion in which a body returns to the point from which the motion began. The time it takes to do so is the cycle’s period. Expressed as a verb, we say the vibrating body oscillates. Physicists call the full cycle one oscillation.
The motion’s frequency is the number of full cycles it completes in one second and is thus the inverse quantity of the period. We can express this value using the equation:
Formula for Frequency:
F = 1 / t
F = frequency (measured in Hz)
t = period
Where F designates frequency and t the period of the motion. We measure frequency in oscillations, or cycle per second (Hertz). One Hertz (Hz) means one cycle per second.
The final feature of periodic motion is the extent, or amplitude, of the displacement. This represents the farthest distance the body (or source of sound) oscillates. A sound wave’s pressure amplitude is the maximum increase of the air pressure as the sound wave passes.
Graphically, amplitude is the distance between the high point and the low point (see the arrows on the left). The bigger the distance, the louder the sound.
Note on Amplitude:
• We measure the pressure amplitude of a sound wave in newtons per square metre (N/m2). The unit measuring how loud the wave sounds to our ears is the decibel.
• The sound of a whisper measures about 10 decibels.
• The sound of an ordinary conversation measures about 40-50 decibels.
• The sound of thunder right overhead measures about 120 decibels.
Simple Harmonic Motion, a special periodic motion, is fundamental to all musical sounds. Simple harmonic motion repeats itself symmetrically around an equilibrium position. As we discuss above, equilibrium means “balance,” so the equilibrium position is the point at which all forces that affect an object are balanced out and the object appears to be at rest. Consider a plucked guitar string. The string’s speed varies throughout the cycle and it always returns to the equilibrium position. The restorative force that directs the string back to equilibrium is proportional to how strongly (how far) the musician has plucked it.
However, the string does not return to its equilibrium position and stop there right away. It overshoots it and keeps doing this over and over again. Scientists call this phenomenon Hooke’s law, named after an English scientist, Robert Hooke, who discovered it about 350 years ago.
Essentially, Hooke’s law says the restoring force acting on an elastic body is directly proportional to the amount that it has been displaced from its initial position of equilibrium and the amount that it will be displaced on the other side of the equilibrium, is directly proportional to the force trying to restore it back to equilibrium.
We can write Hooke’s Law mathematically like this:
F = -KX
F = force acting on an elastic body (see note1) trying to restore it back to equilibrium
K = spring constant (see note 2)
X = displacement of the spring from the equilibrium position
1 An elastic body is an object that when stretched or compressed, will have a restoring force act upon it, tending to bring it back to equilibrium position. An example is an ordinary spring. If you pull it one way the spring’s tension increasingly resists you and makes it want to go back the other way. Another example is a rubber band. Stretch it out, and it wants to snap back. (*Remember from Module One elasticity is a property of a physical body.) Besides springs and rubber bands, elastic objects could include piano wires, guitar strings and drum hides.
2 The spring constant of a body is its amount of stiffness. The larger the spring constant, the stiffer the body.
3 Negative sign before –KX. Another way to word this is to say that the negative sign indicates that the force acting on the elastic body is a restoring one. It is like the pulled spring and the stretched rubber band. It always wants to go back in the opposite direction of the force that initially caused the displacement.
Let’s consider a rock suspended from a coiled spring. Gravity pulls the rock downward, but the spring is elastic, so it pulls the rock upward. The rock’s equilibrium position is a balance between the spring’s upward tension and the gravitational pull from below. When we use force to pull the rock downward, the spring’s tension gets stronger and stronger. This imbalance charges the rock with potential energy.
When we release the rock, the spring’s elasticity causes the rock to accelerate upward towards its equilibrium position. However, due to the rock’s great kinetic energy, it overshoots the equilibrium position and travels upward, proportional to how far we pulled it (Hooke’s Law).
As the rock approaches its full displacement (amplitude), its speed begins to slacken. This allows the spring’s restorative force to start pulling the rock down. Gravity is at work in this, too. When the rock is higher than the equilibrium point, both gravity and the spring’s tension pull the rock in the same direction: back down again.
When the rock is below the equilibrium point, gravity, which is the same everywhere, still pulls the rock down. But, the spring now works against gravity to pull the rock back up.
Both above and below the equilibrium point, the rock’s kinetic energy translates once more into potential energy when its acceleration slows, then back into kinetic energy as gravity and the spring’s tension keep pushing and pulling the rock up and down.
Were it not for one thing, the rock would vibrate back and forth above and below the equilibrium point forever. But, friction with the air is always at work and eventually the system will lose all its energy as heat. The frequency and period remain the same, but if we do not apply extra force, the oscillating cycle’s amplitude gradually will get smaller and smaller. Eventually, the rock will come to rest at the equilibrium point where the spring’s tension and gravity’s pull balance each other out.
Let’s apply this to the guitar string again. The string is elastic, so when a musician pulls it back, the string stores the energy the finger supplies. When the musician releases the string, it accelerates past its equilibrium position until it reaches a point on the other side. According to Hooke’s law and the law of conservation of energy, the string travels the same distance beyond the equilibrium point as it did when the musician pulled it back.
It then accelerates backward through the equilibrium point again. If we lived in a world where the string lost no energy to heat or acoustic energy (sound waves), it would keep bouncing back and forth forever. But, like the previous example, the string’s energy does gradually bleed off. These heat and sonic transforms cause the string’s cycles to become smaller and smaller until the string finally stops vibrating.
We refer to this gradual slow-down, or decay, as damping. However, we can offset damping if we repeatedly apply the external force at the proper moments. In other words, if an outside periodic disturbance matches the body’s natural frequency (its innate vibration rate) we can boost the amplitude. And, if we keep applying the force at the same point, the vibrations will get stronger and stronger.
In physics, a common example is an adult pushing a child on a swing. Each time the swing goes up (Or, using physics’ jargon, the amplitude of its displacement increases). At the peak, gravity pulls it back down again. The swing moves backward past its equilibrium position, hits an opposite high point and begins moving forward again. If the adult pushes at precisely the right moment, and if the push is stronger than all counteracting forces, the swing keeps going higher and higher. Some youngsters want Papa to push them all the way over the top bar, but we know better!
This reinforcement is resonance. This synchronized movement in a vibrating system is in phase when the pusher moves in the same direction as the movement. Technically, in-phase motion is a fraction of a completed cycle as reckoned from a chosen starting point. This is how musical instruments, such as violins and guitars, make the strings sound louder. Without resonance and acoustic coupling, we would barely hear a guitar’s strings or a drum beat. The effect combines with the way an instrument’s large surface area transfers its vibrations to the air. In a violin, the sound box and the air inside the instrument’s body resonates with the string. The same is true for drums.
Vibrations must be synchronized to obtain resonance in a musical instrument. When they are, we say the two resonating forces are in phase with one another, but hold this thought. It will become more important in the following section on waves.
The Indigenous way also seeks in-phase resonance, but of another kind. When to hunt, when to create new life, when to live and when to die; all these require careful timing. We must learn to co-exist in harmony with the cycles of Earth, community and family. This often needs the Elder’s wisdom, one who has experienced the interdependence of many cycles. Like the child’s swing, the drum and the vibrating string, to work with Nature’s cycles at the right moment strengthens life’s power; to work against Nature diminishes or even cancels it.