Sine and Cosine Functions, Simple Harmonic Motion, and Sound Wave Parameters

Overview

This study guide introduces the mathematical and physical foundations of wave motion, focusing on the sine and cosine functions, simple harmonic motion (SHM), and the physical parameters of sound waves. It connects these mathematical concepts to physical phenomena and their perceptual correlates, providing a toolkit for understanding oscillatory systems in physics.

Sine and Cosine Functions

Unit Circle Definitions

• Sine and cosine functions are defined using the unit circle:

  • The y-coordinate of a point on the unit circle is sine of the angle.

  • The x-coordinate is cosine of the angle.

• Plotting these coordinates as the angle increases traces out the familiar sine and cosine waveforms.

• This geometric definition connects directly to the physical motion of oscillating systems.

Phase Relationship

• Sine and cosine waves are 90° out of phase with each other.

• This phase difference is fundamental and recurs in many physical systems (e.g., displacement vs. velocity in SHM).

Simple Harmonic Motion (SHM)

Definition and Examples

• Simple harmonic motion is projected uniform circular motion.

• Uniform circular motion: a body moves at constant speed around a circle; projecting its position onto one axis yields SHM.

• Physical examples:

  • Mass on a spring

  • Pendulum

  • Tuning fork

  • Vibrating guitar string

  • Air particles in a sound wave

Constancy of Sine and Cosine

• Scaling the amplitude of a sine or cosine wave does not affect its phase or zero-crossings.

• This property allows sine and cosine to serve as universal descriptors of oscillatory motion.

General Equation for Sinusoidal Motion

Mathematical Formulation

• The general equation for sinusoidal displacement as a function of time is:

• Where:

  • D(t): instantaneous amplitude (displacement)

  • A: maximum amplitude

  • f: frequency (Hz)

  • t: time (s)

  • \theta: starting phase (radians or degrees)

• Angular frequency , so the equation can also be written as:

Physical and Psychological Correlates

Sound Transmission and the Need for a Medium

Propagation of Sound

• Sound requires a medium (solid, liquid, or gas) to propagate; it cannot travel through a vacuum.

• Sound transmission involves the oscillatory motion of particles in the medium:

  • Displacement (x)

  • Velocity (v)

  • Acceleration (a)

  • Pressure (P)

• Without particles to displace, there is no sound ("in space, no one can hear you scream").

Piston-and-Balloon Model

• Illustrates how sinusoidal motion of a source creates alternating regions of compression and rarefaction in the medium.

• All related quantities (displacement, velocity, acceleration, pressure) vary sinusoidally.

Phase Relationships

• Displacement and velocity: 90° out of phase

• Velocity and pressure: in phase (0° apart)

• Displacement and pressure: 90° out of phase

• Displacement and acceleration: 180° out of phase

These relationships are crucial for understanding oscillatory systems and wave propagation.

Sound Amplitude (A)

Definition and Measurement

• Amplitude is the maximum displacement from equilibrium, proportional to the applied force (Hooke's Law):

• Amplitude is a vector quantity (has magnitude and direction).

• Amplitude is independent of frequency.

Types of Amplitude Measurement

• Peak Amplitude (Apeak): Maximum displacement from zero.

• Peak-to-Peak Amplitude (AP2P): Total distance from maximum positive to maximum negative displacement.

• Root Mean Square (RMS) Amplitude (ARMS): Standard deviation of all instantaneous amplitudes; a more general measure for arbitrary waveforms.

For a pure sinusoid:

• RMS amplitude is preferred for comparing signals with different shapes or energy distributions.

Sound Frequency (f)

Definition and Units

• Frequency (f): Number of cycles per unit time.

• Units: cycles per second (cps), Hertz (Hz), or revolutions per minute (rpm).

• 1 Hz = 1 cycle per second = 60 rpm

• Frequency is a scalar quantity.

Calculation Example

• If a signal completes one cycle every 200 ms:

Sound Period (T)

Definition and Relationship to Frequency

• Period (T): Time required to complete one cycle.

• Frequency and period are reciprocals:

• High-frequency waves have short periods; low-frequency waves have long periods.

Sound Wavelength (\( \lambda \))

Definition and Properties

• Wavelength (\( \lambda \)): Physical distance between successive compressions or rarefactions.

• Depends on the medium and the frequency of the sound.

• Higher frequency → shorter wavelength; lower frequency → longer wavelength.

• Relationship to speed of sound (c):

Phase (\( \theta \) or \( \phi \))

Starting Phase

• Starting phase (\( \theta \)): The point in the cycle where the wave begins, measured in degrees (0°–360°) or radians (0–2π).

• All sinusoids have a starting phase; it determines the initial position in the cycle.

Phase Difference

• Two sinusoids of the same frequency but different starting phases are out of phase by a constant amount.

• Phase difference is the same at all corresponding points along the waveform.

• Example: If one wave starts at 90° and another at 180°, the phase difference is 90°.

Instantaneous Phase

• Instantaneous phase: The phase angle at a specific point in time.

• For same-frequency sinusoids, the instantaneous phase difference is constant.

• For different-frequency sinusoids, the phase difference varies over time.

• A cosine wave is a sine wave with a 90° phase offset:

Synthesis and Key Takeaways

• Sine and cosine functions describe oscillatory motion and are always 90° out of phase.

• Simple harmonic motion is the projection of uniform circular motion onto one axis.

• The general sinusoidal equation encapsulates amplitude, frequency, time, and phase.

• Sound propagation involves sinusoidal variations in displacement, velocity, acceleration, and pressure, each with specific phase relationships.

• Amplitude can be measured as peak, peak-to-peak, or RMS; RMS is most robust for arbitrary waveforms.

• Frequency and period are reciprocals; wavelength depends on both frequency and the speed of sound.

• Phase describes the starting point and relative alignment of waves.

Self-Check Questions

1. Explain, using the unit circle, why sine and cosine are always exactly 90° out of phase with each other.

2. Define simple harmonic motion using the term “projected uniform circular motion.” Give two physical examples besides a sine wave.

3. Write out the general equation for sinusoidal motion, , and identify the psychological/perceptual correlate of each of A, f, and t.

4. A sound wave has a peak amplitude (Apeak) of 20 µPa. What is its RMS amplitude? What is its peak-to-peak amplitude?

5. Why does taking a plain arithmetic average of a sinusoidal waveform fail to capture its amplitude, and how does the RMS calculation solve this?

6. A signal completes one full cycle every 25 ms. Calculate its frequency in Hz and its period in seconds ().

7. If the speed of sound in a given medium is 1500 m/s and a tone has a frequency of 500 Hz, what is its wavelength? ()

8. During sound transmission, list the phase relationship (in degrees) between each of the following pairs: (a) displacement and velocity, (b) pressure and velocity, (c) displacement and pressure, (d) displacement and acceleration.

9. Why can't sound travel through a vacuum? Connect your answer to particle displacement, velocity, acceleration, and pressure.

Additional info: The notes reference a textbook erratum regarding phase relationships in Yost (2000), and emphasize the importance of memorizing the general sinusoidal equation and the mapping between physical and perceptual sound parameters.