Waves and Sinusoidal Vibrations

Reading Graphs and Core Concepts in Sound

Purpose of the course emphasis on graph reading: you may not graph everything, but you will read graphs. The skill is widely transferable and essential for understanding news, literature, and research in clinical settings.
Repetition of the core message: reading graphs is a foundational, generalizable skill for sound, research, and real-world interpretation.

Sound waves are longitudinal, where air particles move parallel to the direction of wave propagation.
- Longitudinal vs. transverse: in a longitudinal wave, particles oscillate back and forth along the direction of travel; in a transverse wave, particle motion is perpendicular to the direction of travel.
Tuning fork demonstration:
- The tuning fork is a simple demonstration of how sound is generated by moving air and creating alternating regions of compression and rarefaction.
- Each air particle displaces in response to the vibrating tines, pushing air in front and pulling air behind, creating successive compressions and rarefactions.
Key conceptual takeaway from the tuning fork: all sound sources push air in a similar way to a tuning fork; the tuning fork diagram is not essential to memorize, but understanding the push/pull effect on air is.

Air molecules have two critical properties for sound propagation:
- Mass (though small for individual molecules)
- Elasticity (tendency to return to original position after displacement)
Because of elasticity and mass, air supports wave propagation: molecules compress (compression) and spread apart (rarefaction) as the wave travels.
Visualizing with a room full of slinkies: air behaves like a connected elastic medium, and sound waves propagate through this medium via successive compressions and rarefactions.

Propagation direction vs. particle displacement:
- Propagation: the wave moves through the medium in a given direction.
- Particle displacement: each air particle moves back and forth around its resting position.
In sound, the displacement of individual particles is a local effect; the overall wave propagation is a bulk phenomenon through the medium.
When discussing a single tone or wave, measurements are often easier to interpret in terms of pressure rather than displacement.

Temperature affects air properties in two main ways:
- It changes how close or far apart air molecules are (density changes).
- It changes elasticity (response of air to displacement).
The lecture avoids deep equations for temperature effects, instead adopting a practical simplification: assume a typical day (e.g., 70°F on the beach) to focus on core concepts.
Atmospheric pressure variation with altitude also affects sound propagation: closer to sea level, more air above you; higher altitudes mean less air above affecting pressure and propagation.

Periodic sounds:
- Repeating over time; can be graphed as repeating waveforms.
- Examples: regular sinusoids and other repeating patterns; a waveform that repeats over time is periodic.
Aperiodic sounds (noise):
- Do not repeat over time; waveform does not settle into a repeating pattern.
- In practice, noise often appears as irregular, non-repeating waveforms.
Simple vs Complex sounds:
- Simple sound: a single sinusoid (one frequency).
- Complex sound: contains more than one frequency; not a pure sine wave.
- Analogy: a sinusoid is like a primary color; complex sounds are composed of multiple sinusoids (like mixing colors to create other colors).

Phase: a measure of where a repeating wave is in its cycle, relative to a reference point in time.
Phase landmarks commonly used:
- 0 degrees, 90 degrees, 180 degrees, 270 degrees, 360 degrees (and equivalents like -90 degrees).
The relationship between displacement and pressure in air:
- Particle velocity (and pressure) are closely related; when particles move most, pressure changes are greatest.
- Displacement lags pressure by 90 degrees:
  ext{phase(displacement)} = ext{phase(pressure)} - 90^{\circ}
- For practical purposes, pressure is more intuitive to analyze than displacement.
How to think about phase in practice:
- When a wave is at a positive peak, phase is 90 degrees.
- When a wave is at a negative peak, phase is 270 degrees (or -90 degrees).
- Phase can be described relative to a reference wave; two waves can differ by 90 degrees if one leads or lags the other by 90°.
For complex sounds, the peak phase locations shift; landmarks like 90°, 180°, 270° remain useful as references but may not correspond exactly to peaks for non-sinusoidal waves.

Mass-spring system:
- Compression is a form of displacement; the restoring force arises from the spring stiffness that wants to return to equilibrium.
- When displaced, the spring exerts a restoring force that accelerates the mass back toward the equilibrium position.
Pendulum analogy:
- Gravity provides the restoring force, constrained by the string so the motion is back-and-forth rather than straight down.
Takeaways about vibrations:
- Restoring forces (springiness or gravity) drive the oscillation and determine the motion's nature.
- Amplitude reflects initial energy; larger initial displacement yields larger amplitude.
- Real systems experience damping due to gravity and friction, eventually reducing amplitude.

Amplitude measures the size of the vibration: the distance or displacement from rest to the extreme of the motion.
Peak amplitude vs. peak-to-peak amplitude:
- Peak amplitude: distance from rest line to the crest (or trough).
- Peak-to-peak amplitude: distance from crest to trough; equal to 2 × peak amplitude.
In teaching, friction and gravity cause damping, making real systems gradually stop vibrating; idealized cases ignore these to explore pure oscillation.

Relationship basics:
- Particle velocity and pressure are closely related in air; when particles move most, pressure changes are greatest.
- Displacement vs. velocity: frictionless, ideal cases link displacement to velocity, but pressure is often easier to measure and interpret.
Time-domain vs. space-domain graphs:
- Time-domain graph (typical classroom representation) shows how pressure at a fixed location changes over time.
- Space-domain (not drawn here) would show how pressure varies with position along the medium at an instant.
Phase relationship in time-domain graphs:
- Displacement lags pressure by approximately 90 degrees in simple harmonic motion.
- When reading graphs, if you see maximum pressure while displacement is zero, you are at a 90-degree phase difference.
Practical note: when analyzing sound, most measurements are effectively reflections of pressure variations (intensity is related to pressure squared over area).

Wave parameters and their relationships:
- Speed of propagation (speed of sound): v = f \lambda
- Frequency: f = \frac{v}{\lambda}
- Wavelength: \lambda = \frac{v}{f}
- Period: T = \frac{1}{f}
Example shown in lecture:
- Speed of sound at a beach day is approximated as v = 344\ \mathrm{m/s}.
- If the frequency is f = 2000\ \mathrm{Hz}, then the wavelength is
  \lambda = \frac{v}{f} = \frac{344}{2000} = 0.172\ \mathrm{m} = 17.2\ \mathrm{cm}
Time- and space-domain conversion:
- A given sinusoid with frequency f repeats every T = \frac{1}{f} seconds and covers one wavelength \lambda = v T in space.
- For the same tone, one repetition travels a distance of \lambda = 0.172\ \mathrm{m} in space.
Practical unit considerations:
- Converting time units: 1 ms