This is the third post in my series on self studying physics. The first post can be found here and the previous post can be found here.
In the prior posts I have already described how I began this journey self studying physics by following the excellent guide “So You Want to Learn Physics” by Susan Rigetti. The material in this post corresponds to the third unit in the guide. While the first two units, mechanics and electrostatics, were both in the book “University Physics with Modern Physics” by Freedman and Young, this unit used two separate textbooks. I will discuss the choice of textbooks in more detail in the content section of this post.
I was apprehensive about starting this unit since during my math degree the material on oscillations was some of my least favorite material. In hindsight, I believe my distaste came more down to the circumstances of the time rather than the material itself. My first exposure to the simple pendulum was in my differential equations course. It was a four week summer course so I believe I had all of two days to learn the topic and prepare to take a test on it.
Prerequisites
If you have made it through the material from the first two units, mechanics and electrostatics, then you should be prepared for this unit. For extra preparation you could return to chapters 15 and 16 in the book “University Physics” by Young and Freedman which was used in the previous units. Those two chapters cover most of the same material found in this course, but in a condensed form with better pictures. Mathematically, you must be fluent with trigonometry and basic calculus (no major use of vector calculus here). The material makes ample use of complex numbers and differential equations, but not to a depth that would require a full course on either.
Content
Following the guide, the books I used for this unit were “Vibrations and Waves” by George C. King and “Vibrations and Waves” by A.P. French. Although the books cover roughly the same material and could each be used alone, they do complement each other. If you have to choose just one I recommend choosing the book by King. I preferred the explanations given in King as well as the accompanying diagrams.
This unit, as the name implies, covers the physics of vibrations and waves. Vibrations refer to objects moving in a repetitive, oscillating motion. For example, a swinging pendulum, a mass on a spring, and a vibrating molecule are all examples of this type of vibrational motion. If you have ever studied the harmonic oscillator in other physics or math courses then you have seen a representative example of this kind of motion. Waves come in many forms, from waves on the ocean to sound waves to electromagnetic waves to even gravitational waves. Understanding them is critical to everything from electrical engineering to astronomy to quantum mechanics. As you might have guessed, there are deep connections between the study of vibrations and the study of waves.
Assuming you bought both books, I recommend starting with King and then alternating back and forth each chapter. I’m not going to describe the content chapter by chapter, but will give a broad outline.
The books begin with a coverage of simple harmonic motion, also called the simple harmonic oscillator. This involves a system that has an equilibrium state and a restoring force that acts on the system (typically following Hooke’s Law). For example, consider a mass on a spring. There is an equilibrium position where the spring is neither stretched or compressed and is therefore not exerting a force on the mass. If one were to pull (or push) the mass away from this position, the spring will cause the mass to move back towards equilibrium. However, by the time the mass gets back to the equilibrium position it will have gained a velocity and overshoot before being slowed and then reversed. This leads to oscillating behavior where the mass will bounce back and forth, possibly forever. The same behavior can describe many circumstances, from pendulums to atoms to electric circuits. Initially, the systems are simple with no friction or other external forces. If you have taken a differential equations course then you have likely seen this circumstance before.
King uses the more familiar approach using sines and cosines to solve the differential equation while French spends time to introduce the complex exponential form of the solution. Both techniques have their value (and are used by both books, just at different times). They are equivalent through the use of Euler’s identity and you should certainly be comfortable with both approaches.
In the next several chapters, the books add additional complexity to the systems such as friction (or other damping forces), driving forces (which input addition energy into the system), and more moving pieces (coupled oscillators). You will learn how to quantify the amount of energy in a system, and predict the behavior of coupled oscillators by looking at the normal modes of the systems.
At this point, you will have learned how to study systems of coupled oscillators (such as many masses connect by springs). However, what happens if you take the number of coupled masses to approach infinity? In this way, the study of harmonic oscillators (vibrations) leads naturally to the study of waves such as on a string or through a material. In the later chapters, you will see the derivation of the wave equation and ways of finding solutions to it. One powerful technique is called Fourier Analysis, which involves breaking waves down into infinite sums of sines and cosines. It is not for the feint of heart and I distinctly remember struggling in it in my advanced engineering mathematics course (the second semester of which was basically a course in Fourier Analysis).
Many facets of the study of waves are explored, such as their energy, wavelength, velocity, and other features. Also, behaviors such as propagation, reflection, and interference are studied in detail. These topics are important for everything from the obvious fluid mechanics to acoustics, to optics and quantum mechanics. All of our wireless communications infrastructure relies on the transmission and reception of electromagnetic waves of different sorts. Our whole understanding of the universe beyond the solar system comes from our analysis of the various waves that reach us and our telescopes. In short, it is critical material for physics (and all of science).
How I Studied
I supplemented my reading with watching the MIT 8.03 lectures on vibrations and waves with Walter Lewin. The videos were a fantastic supplement and I highly recommend them. They are available (at the time of writing) for free on YouTube. My study methods were quite similar to those described in the previous posts. The largest difference is that these textbooks did not have the same sort of example problems as the University Physics book did, so I often just read through the example problems rather than stopping and fulling completing them. The examples in these books often introduced new material and were not suitable for use as knowledge checks. I spent about 70 hours on vibrations and waves material, making it the shortest unit so far. Part of the reason is that I have had prior exposure to much of the material (even if I disliked it) and could move quickly. I did about half the problems overall.
As always, what you get out of a course if related to how much time you put into it. Only you truly know your goals. If you want a deep level of understanding and the ability to apply these concepts to more complicated systems you will need to put in a large amount of time. This would include doing most if not all of the problems in both books. If your aim is to gain a solid grasp of the core concepts while being able to apply that knowledge to basic situations you wouldn’t need as much time or as many problems. The wonderful thing about self study is that you get to choose your goals.
Final Thoughts
Overall, this course was a positive experience. Facing down material which I previously disliked forced me to grow. It was satisfying to see the initially simple building blocks come together to provide deep insight into a wide range of phenomenon. Although the material was not as directly related to applied technology as the last unit, the application of waves is deeply fundamental to everything from chemistry to communication. My one quibble is with the timing of the material in relation to the rest of the guide. I have been slow to publish these posts so I can speak with some foreknowledge of what is to come.
The guide spends the majority of the first four units in one book: the University Physics book by Young and Freedman. Introductory mechanics covers chapters 1-14. Electrostatics jumps forward to chapters 21-32. The guide then switches to the two books mentioned in this guide for an in depth study of vibrations and waves before going back to University Physics for the modern physics material (chapters 17-20, 37-44). There is nothing wrong with this ordering, but in my view it can be simplified. There is enough coverage of vibrations and waves material in chapters 14 through 16 of University Physics to understand everything in the book including the modern physics section near the end. Going through this material at the end of the mechanics content would also mean the student would have a good understanding of waves by the time they get to the material on alternating current and electromagnetic waves.
Really, I think the first four units of the guide (mechanics, electrostatics, vibrations and waves, and modern physics) could be rephrased into a single sentence: Get University Physics by Young and Freedman and work through it cover to cover, all 44 chapters. The ordering is more logical, you don’t have to buy two extra books, and there is no worrying about jumping around or skipping chapters. I will return to this point in my lower division wrap up post.
The next unit is on modern physics, and I will place a link to that post here when it is finished.
Happy studying, and thanks for reading!