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.
Upon successful completion of this unit, you will be able to:
- state Hooke's Law;
- describe the properties of simple harmonic motion and provide examples;
- describe the forms energy is contained in, and can be lost to, in oscillatory motion;
- define resonance;
- define terms related to wave motion: frequency, wavelength, diffraction, and interference; and
- solve problems using simple harmonic motion.
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.
Before we begin, watch this video that explains why it is important to study electromagnetism. There are two types of charges, and like charges repel, while unlike charges attract. When equal amounts of both charge types are combined in an object, it becomes electrically neutral. The forces from balanced charges of opposite types inside a neutral object cancel out when viewed from the outside. This explains why we do not often notice the effects of electromagnetism directly, which creates the impression that it must be a weak kind of force. In reality, electric forces are much stronger than gravity! We will come back to this in later lectures.
Read this introduction to Hooke's Law that covers the concept of energy. Using the language of energy, deformed springs store potential energy, which can be converted into kinetic energy when the spring is released. In this process, the spring force causes acceleration by doing mechanical work.
When a spring causes acceleration, the motion often overshoots the equilibrium that it wants to return to because the oscillating mass has inertia. After it overshoots, the restoring force reverses direction and causes an opposite acceleration. This is how oscillations are created: the motion keeps reversing and overshoots its equilibrium every time.
Read this text to learn how we characterize oscillations quantitatively. The standard unit of frequency is called hertz (abbreviated Hz). It is no coincidence that you find the same unit labeling your radio dial: later, we will see that radio stations transmit electromagnetic waves of specific frequencies.
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.
This text describes simple periodic motion by making graphs of the spring deformation over time. The graphs show a universal shape no matter what the specific oscillator looks like, and it is described mathematically by a sine or cosine function.
In the description of harmonic motion, the choice between sine and cosine functions depends on how the oscillation was launched at the initial time. Since a harmonic oscillation repeats exactly after one period, different launch conditions only have two possible effects: they determine the amplitude (the maximal deformation during the oscillation) and also the times at which that maximum is reached. In a graph, changing the time of maximum deformation corresponds to shifting the plot left or right along the time axis without changing its overall shape.
A sine function can be mathematically thought of as a cosine function that is shifted by a specific amount.
This video discusses this idea mathematically, but allows the shift along the time axis to have an arbitrary value to account for arbitrary launch conditions at a chosen time (usually called t = 0). This introduces the concept of the phase of oscillation.
Simple harmonic motion does not just happen with springs. For example, an old-fashioned grandfather clock makes use of simple harmonic motion to keep accurate time. A simple pendulum is defined as having an object with a mass of small size (the pendulum bob) that is suspended from a light wire or string. Examples include the pendulums that guide the movement of time on a clock, a child's swing, a wrecking ball, or even a sinker or weight at the end of a fishing line.
This text explores the conditions where a pendulum performs simple harmonic motion and derives an interesting expression for its period. For small displacements, a pendulum is a simple harmonic oscillator. As with all harmonic motion, the period of a pendulum is independent of the amplitude. This robustness of the period is what makes pendulum clocks work with great accuracy.
If you are familiar with rotational motion, you will recognize that the motion of a pendulum is really a back-and-forth rotation around the point of suspension. For a discussion of the pendulum from this point of view, watch this optional video.
Read this optional text to dive deeper into the relationship between rotational motion and harmonic oscillations in general.
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.
In terms of energy, damped oscillations cannot maintain a constant balance of kinetic and potential energy because some of the mechanical energy is drained away in the form of thermal energy. This eventually causes the mechanical motion of the oscillator to return to equilibrium and stay there. At that point, all the energy of the oscillation has been converted to microscopic, invisible motion at the molecular level in the surrounding air or the oscillator itself.
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.
Read this text which explains how a dramatic increase in the oscillation amplitude as the driving period is adjusted can lead to resonance. Notice the frequency where the driven oscillation occurs. Is it the natural frequency of the oscillator, or is it the frequency with which the external force is applied? If you try the experiment in Figure 16.26, you will see that you control the frequency – not the paddle ball. However, the response of the oscillation amplitude depends dramatically on which frequency you choose.
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).
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).
Are waves a type of motion? Yes, if they exist in a medium – such as sound waves in the air or ripples on a pond. The building blocks that the medium is made of (for example, air molecules) perform a coordinated motion when a wave passes over them. However, they return to their original positions when the wave is gone. This is something waves have in common with oscillators: for all the building blocks of the medium, there is an equilibrium position that can be perturbed (for example, the flat surface of a still pond), and there is a restoring force similar to that of a spring. Just as in a simple harmonic oscillator, mass gives the medium inertia, so it reacts to its own internal restoring forces with some delay.
But waves are not simply collections of oscillators, all doing their own little dances. The spring-type forces inside the wave medium have one additional function, which is to couple neighboring oscillators to each other. This is what creates the highly coordinated motion that we identify as a wave. This is something waves have in common with forced oscillations. Each oscillating part of the wave medium feels forces from its neighbors and can receive and transmit energy. As a result, waves can transport energy across a medium, even if the medium as a whole stays put!
Pay attention to the difference between transverse and longitudinal waves. The oscillations of a transverse wave are perpendicular to the direction of the wave's advance. A longitudinal wave travels in the direction of its oscillations.
When you play this demonstration, you will see that the frequency you use to jiggle the end will also determine the frequency of the wave. After trying this manually, click "Oscillate" at the top right to see what happens if you excite a wave by making the end follow a simple harmonic motion described mathematically by the sine function. That is called a periodic wave.
Periodic waves are characterized by their wavelength, which is the distance the wave travels during one period – the time it takes for the oscillation to go through a complete cycle. Watch this video to see how the wavelength and period are related to the speed of the wave propagation.
To illustrate wave speed in the simulator, it was essential to use a very long string available so that there are no reflections at the other end. They would obscure the pattern that we are trying to describe, which is called a traveling wave. As with the harmonic oscillator, we can also write down a mathematical description of the wave patterns shown above. You can see this in this video.
Watch this video, which shows how to put this mathematical function into the context of the physical property of the wave we are trying to describe (its wavelength).
Watch this video to review how the concepts fit together. It demonstrates another graphical representation of a traveling wave.
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.
Read this text to learn more about standing waves and musical beats and see an interactive demonstration of wave interference on a water surface.
Watch this video to see how constructive and destructive interference is created by adding two waves while taking the sign of the wave's displacement into account at every point.
The superposition principle for waves may still seem a little mysterious. To avoid misconceptions, be aware that not all waves obey this principle! When you see two boats leaving wakes on a still lake, you can observe the wakes passing through each other to form interference patterns, just as the superposition principle predicts. But for a ship in stormy seas, the wake will be obliterated by the ocean's crashing waves, which is because waves of large amplitude usually fail to form superpositions. Waves obey the superposition principle only if the restoring forces in the medium obey Hooke's Law!
Hooke's Law is a proportionality between force and displacement that breaks down when the displacement from equilibrium is too large. A simple example of this breakdown is the pendulum we studied earlier. The component of the gravitational pull that acts to restore the pendulum to its vertical equilibrium ceases to be proportional to the angle with the vertical when that angle becomes too large. When we talk about wave interference, we assume that the wave amplitudes are small enough to make Hooke's Law valid. Watch this video, which summarizes the relationship between interference and beats.
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.
Read this text to learn more about energy in waves. The amount of energy in a wave is related to its amplitude. When we talk about the amplitude of a wave, we refer to deviations from the equilibrium of the medium carrying the wave. Large-amplitude earthquakes produce large ground displacements. Loud sounds have higher pressure amplitudes and come from larger amplitude source vibrations than soft sounds. Large ocean breakers churn up the shore more than small ones.
At the level of the wave medium, a wave is a displacement that is resisted by forces that have the dual tendencies of restoring equilibrium and coupling neighboring regions. The speed of the wave also depends on the mass contained in the medium because this implies inertia. Inertia governs the delay with which the energy put in at one end reaches the other end. The medium stores the energy of the wave as the potential energy of the coupling forces and as the kinetic energy of the oscillating mass. Since the mass only oscillates in place, a wave can transport energy over large distances without any mass being transported overall.