Normal Modes, Coupled Oscillators, and Waves — Lecture Notes

Simple Harmonic Oscillator (SHO)

A single simple harmonic oscillator has a frequency determined by its own properties, not an external force.
- Frequency depends on stiffness (restoring force) and mass: for a mass-spring system,
- Angular frequency: $\omega_0 = \sqrt{\dfrac{k}{m}}$
- Linear frequency: $f0 = \dfrac{\omega0}{2\pi} = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}$
- Period: $T = \dfrac{2\pi}{\omega_0} = 2\pi\sqrt{\dfrac{m}{k}}$
Amplitude changes the energy, not the frequency. Larger amplitude means more energy, but the period (and thus frequency) remains the same.
Heuristic summary: the frequency is a property of the system, set by stiffness and mass; energy and amplitude trade off (more energy, bigger swing, but same period).

From One to Two: Coupled Oscillators

When you couple two SHO via a shared spring, they interact: a force on one affects the other.
With coupling, the motion can become quite complex, but it can be understood in terms of simpler pieces.
Example intuition: energy can transfer back and forth between the two oscillators through the connecting spring. One setup might have one oscillator dominating at times, then energy shifts to the other, and so on.
Question posed: how many distinct motions exist for two coupled oscillators?
- Answers discussed: mathematically there are infinitely many possible motions (by choosing initial energies), but there are only two fundamental frequencies (two normal modes) for the linear, identical-oscillator case.
- Any complicated motion is a superposition (sum) of these two simple motions (normal modes).

Normal Modes and Superposition

Key terms:
- Superposition: you can add simple motions to build more complex ones.
- Normal modes: simple motions with a single frequency. For two coupled identical oscillators, there are two normal modes.
Two basic normal modes for the two-oscillator system:
- Mode 1 (in-phase): both masses move together; the connecting spring does not stretch much, so the motion is basically the same as a single SHO.
- This mode has the same frequency as the original simple oscillator:
- Frequency: $f1 = f0 = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}.$
- Mode 2 (out-of-phase): the masses move in opposite directions; the connecting spring stretches and provides extra restoring force.
- Increased effective stiffness: the mode behaves as if stiffness is doubled, giving
- Angular frequency: $\omega2 = \sqrt{\dfrac{2k}{m}} = \sqrt{2}\,\omega0$
- Linear frequency: $f2 = \dfrac{\omega2}{2\pi} = \dfrac{1}{2\pi}\sqrt{\dfrac{2k}{m}} = \sqrt{2}\, f_0$
Conceptual takeaway:
- The two normal modes provide a complete basis for describing any motion of the two-oscillator system.
- Any complicated motion can be decomposed into a sum of these two simple motions (a superposition of normal modes).
Why we care: normal modes are the building blocks to understand more complex systems and to simplify analysis by focusing on independent simple motions.

Spectrograms, Sound, and the Physical Meaning

A spectrogram displays how much amplitude there is at each frequency in a motion or signal:
- For a single SHO, you see a single dominant frequency (one line).
- For a coupled oscillator, you see two frequencies corresponding to the two normal modes.
Sound connection:
- If you combine the two normal modes with varying amplitudes, you get different audible sounds.
- Example: adjusting how much energy you put into the lower mode vs the higher mode changes the timbre while keeping the same pitch, akin to how vowels are formed in speech.
Vowels and formants (in speech) conceptually link to normal-mode amplitudes:
- With a fixed vocal tract geometry, different vowel sounds correspond to different distributions of amplitude among the normal modes.
- The spectrogram of speech shows multiple peaks (formants) whose relative strengths shape the perceived vowel.
Practical demonstration idea described:
- If you give roughly equal energy to the two modes, you might get a sound approximating a particular vowel (e.g., an “ah”). If you shift more energy to the second, you might get closer to another vowel (e.g., an “ee”).
- The analogy helps explain how vocal tract configuration (mouth, throat, nasal passages) shapes vowel sounds by redistributing energy among the normal modes.
For musical instruments:
- Different instruments can play the same fundamental pitch but sound different because they excite different distributions of normal-mode amplitudes (timbre).

Two-Mode Motion and Mode Frequencies (In-Depth)

Recollection of the two fundamental modes for the pair of equal masses connected by a spring:
- In-phase mode (Mode 1): frequency equals the single SHO frequency f0.
- Out-of-phase mode (Mode 2): frequency is higher, approximately sqrt(2) times f0.
Visualization idea:
- Standing-wave-like patterns appear for the coupled system, with nodal points where motion is zero (nodes).
Node concept:
- Mode 1: zero nodes (all masses move together).
- Mode 2: one node (e.g., center remains stationary while ends move in opposite directions).
- Higher modes (Mode 3, 4, …) add more nodes, corresponding to more complex partitions of the system into moving regions.
- Pattern: Mode n has n−1 nodes (for the chain/string model discussed).
General takeaway about higher modes:
- Higher mode number means more “wiggliness” and higher frequency due to more restorative interactions across smaller segments.

Standing Waves, Nodes, and One-Dimensional Normal Modes

Standing waves in a one-dimensional medium (like a string or chain of masses):
- Energy does not propagate indefinitely; it stays in the system due to reflections at boundaries.
- Normal modes are standing-wave patterns with fixed spatial patterns and time-dependent amplitudes.
- Each mode has a distinct frequency and a specific nodal pattern.
Visual patterns and counting nodes:
- Mode 1: all points move together; 0 interior nodes.
- Mode 2: pattern with 1 interior node (e.g., a midpoint with zero amplitude).
- Mode 3: 2 interior nodes; pattern divides the string into 3 moving segments moving in alternating phases.
- In general, Mode n shows n−1 interior nodes and n moving segments.
Practical note:
- If you want to analyze a drum head or a two-dimensional plate, the normal-mode patterns become more complex to determine, but the same principle applies: a spectrum of modes with increasing frequency as mode shape becomes more intricate.

Waves and Media: Traveling vs Standing, and Media Dependence

Waves as energy transport through a medium:
- A disturbance in a medium (air, a string, the Earth) propagates as a wave, carrying kinetic and potential energy.
- In a string or air, local displacements propagate energy to neighboring regions.
- For standing waves, energy stays in the system and forms fixed patterns (nodes).
Traveling waves vs standing waves:
- Traveling waves move energy from one location to another; they can be periodic (repeating) or nonperiodic (a single event).
- Standing waves are formed by the superposition of two traveling waves moving in opposite directions with the same frequency and amplitude.
Energy components:
- Kinetic energy: due to motion of the medium’s elements.
- Potential energy: stored in the restoring forces (e.g., tension in a string, compression in a spring).
Medium dependence and exceptions:
- Most waves require a medium (air, string material, Earth) to propagate.
- Light is exceptional: it can travel in a vacuum, which significantly broadens its propagation properties.
Analogies and examples mentioned:
- Seismic waves: P-waves (compressional) and S-waves (shear) propagate through Earth, illustrating different wave modes in a real medium.
- Audio vocal tract and singing: vocal cords and throat shape create disturbances in air that are carried to the ear as sound waves.

Putting It All Together: Why Normal Modes Matter

Core theme: Coupled oscillators can behave in highly complex ways, but any motion can be decomposed into a sum of simple motions (normal modes).
Normal modes are the fundamental building blocks for describing motion in coupled systems, vibrations, sound production, and wave phenomena.
Practical takeaway statements:
- Any complex motion of a system of coupled oscillators can be described by choosing appropriate amplitudes for the normal modes.
- In acoustics and speech, the distribution of energy among normal modes shapes what we hear (pitch, timbre, vowel quality).
- In wave phenomena, standing waves and their nodal patterns guide our understanding of resonances, instruments, and structural vibrations.

Key Equations and Concepts to Remember

Simple harmonic oscillator:
- $\omega0 = \sqrt{\dfrac{k}{m}},\quad f0 = \dfrac{\omega0}{2\pi} = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}},\quad T = \dfrac{2\pi}{\omega0} = 2\pi\sqrt{\dfrac{m}{k}}.$
Coupled two-oscillator normal modes (identical masses and spring between them):
- Mode 1 (in-phase): $f1 = f0 = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}.$
- Mode 2 (out-of-phase): $f2 = \dfrac{1}{2\pi}\sqrt{\dfrac{2k}{m}} = \sqrt{2}\, f0.$
General two-mode motion (superposition):
- $x(t) = A\cos(\omega1 t + \phi1) + B\cos(\omega2 t + \phi2),\quad \omega1 = 2\pi f1,\ \omega2 = 2\pi f2.$
Node concept in standing-wave patterns:
- Mode n has (n−1) nodes; lowest mode (n=1) has 0 nodes; second mode (n=2) has 1 node, etc.
Waves and medium concepts:
- Travel vs standing waves, energy transfer, and the role of formants in speech.
- Light as an exception traveling in vacuum.

Quick Takeaway

Coupled oscillators reveal a structured way to understand complexity: decompose into a small set of normal modes, each with a single frequency, and then build any motion by adjusting the amplitudes of those modes.
The same idea underpins how we analyze sound, speech, and many wave phenomena in physics and engineering.