Unit 1: Mechanical Vibrations and Waves in Extended Objects
The material in this unit is not directly related to electricity and magnetism, but it is the foundation of one of the most significant outcomes of Maxwell's theory – electromagnetic waves. In PHYS101: Introduction to Mechanics, we learned how to describe the motion of particle-like masses using classical mechanics. In this unit, we begin the transition from mechanics to electromagnetism by examining how objects of size – length, width, and depth – behave.
When we look at these types of extended objects, a mystery is hiding in plain sight: if you pull one end of a rope, how does the other end know? Your action somehow "propagates" from one end to the other. The answer is related to the invisible hand of electromagnetism that can transmit information between different locations.
In this unit, we focus on vibrating systems and the propagation of mechanical waves through media; think of ripples traveling outward from a stone dropped into water. We also lay the basic foundation for the development of a classical theory of mechanics for extended solids.
Completing this unit should take you approximately 7 hours.
1.1: Periodic Motion and Simple Harmonic Oscillators
Simple harmonic motion refers to a repetitive movement back and forth through an equilibrium, or central position, so the maximum displacement on one side of this position equals the maximum displacement on the other side. Therefore, the time interval of each complete vibration is the same.
Simple harmonic motion arises when a restoring force appears in a system, when that system is disturbed from equilibrium, and when the force is proportional to and has the opposite direction of the displacement.
This proportionality is called Hooke's Law. A typical example of a system obeying Hooke's Law is a point mass attached to the end of a massless spring. The spring exerts a restoring force on the mass that tries to return the spring to its natural, unstrained length. The length by which the spring is stretched or compressed is the displacement that Hooke's Law refers to. We also call this the deformation, or elongation. The strength of the restoring force for a given displacement is characterized by a spring constant, which is determined by the material of the spring and how tightly it is wound.
1.2: Simple Harmonic Motion
Oscillations can occur even when the restoring force does not obey Hooke's Law, or if a mass is attached to several different springs at once. The resulting motion can be pretty complicated, and you may have a hard time identifying its frequency. Think about how it is impossible to assign a single pitch to the sound of an orchestra playing a symphony. Each musical note, however, does have a unique frequency, and that makes them special. In close analogy, there is a special type of oscillation that we call "simple harmonic" (or just "harmonic" – there is no other kind than simple harmonic).
Simple harmonic oscillation is what you get when you combine the concept of oscillation with Hooke's Law. Any oscillation that shows a restoring force obeying Hooke's Law is called simple harmonic. The interplay between inertia and restoring force that underlies all oscillation phenomena becomes especially simple in this case because it turns out that the frequency becomes independent of the oscillation amplitude.
1.3: Oscillations and Energy
There is a continual exchange between kinetic and potential energy during any oscillation, as the system speeds up near equilibrium and slows down before it turns around. This is just as true for a playground swing as for a vibrating tuning fork. Since the concept of energy permeates all of physics from classical mechanics to electromagnetism and beyond, it is useful to describe oscillations in this vocabulary.
The total energy of an oscillation is determined at the moment you get it going. In an ideal simple harmonic oscillation, this total remains constant even as kinetic and potential energy individually change in synchrony with the motion. But, as you know from experience, oscillations do not usually last forever; they peter out because of friction, air resistance, or other forces. In this case, we speak of a damped oscillation instead of a harmonic oscillation.
1.4: Forced Oscillations and Resonance
How do you keep an oscillation going while it loses energy due to friction? You need to replenish the lost energy by doing work. For example, a wristwatch uses the period of a vibrating quartz crystal to keep time, but the crystal can only maintain that vibration with the help of a battery.
When an oscillator receives external energy, that is called a forced oscillation. A surprising effect arises when you try to supply the energy with an external driving force that is itself periodic. The rate that energy can be transferred to a harmonic oscillator maximizes if the periodicity of the driving force matches the period that the oscillator would move if left alone – its natural period.
1.5: Wave Motion
Waves come in various forms, such as sound, light, or ripples on the surface of a body of water. They all have the same underlying properties. The main difference between waves and the movement of matter particles is that waves can pass through each other without affecting each other's motion.
When more than one wave is present in the same place, they combine. This is known as superposition, and it creates wave patterns that are called interference. Waves also demonstrate diffraction, which is the ability to bend around an obstacle. When a wave encounters a boundary between two media, it undergoes reflection (traveling backward) and transmission (which goes along with refraction).
1.6: Superposition and Interference
When two or more waves travel through the same medium at the same time, the waves pass through each other without being disturbed. The net disturbance or displacement of the medium is simply the sum of the individual wave displacements. This is true of waves that are finite in length (wave pulses) or continuous sine waves.
When the crests and troughs of the two waves are precisely aligned, the superposition produces pure constructive interference. Pure constructive interference produces a wave that has twice the amplitude of the individual waves but has the same wavelength.
On the other hand, destructive interference occurs when two identical waves arrive at the same point and are exactly out of phase (aligned crest to trough). Because the disturbances are in the opposite direction for this superposition, the amplitude is zero for pure destructive interference: the waves completely cancel each other out.
1.7: Energy in Waves
All waves carry energy. We can see the destructive forces waves cause in many ways: earthquakes, ocean waves that cause erosion, and musicians who have hearing loss due to the sound waves that pound the nerve cells in their inner ears. However, we have also learned to harness the energy of waves, such as when physical therapists provide deep-heat ultrasound treatment to muscle strains or when surgeons use laser beams to burn away cancer cells.