In-Depth Notes on Vibrations, Waves, and Sound
Learning Outcomes for Vibrations, Waves, and Sound
Students should be able to:
Explain oscillations.
Explain simple harmonic motion (SHM).
Identify the displacement equation for SHM, including amplitude, phase constant, and phase.
Relate kinetic and potential energy for a simple pendulum.
Describe frequency in SHM.
Describe damped oscillations.
Describe waves in terms of wavelength, amplitude, frequency, speed, and intensity.
Describe wave speed.
Recall that all electromagnetic waves are transverse waves traveling at the speed of light in free space.
Explain phase difference.
Distinguish between transverse and longitudinal waves.
Explain the energy carried by a wave.
Explain the intensity of a wave.
Oscillations
Definition: Oscillation is a repetitive motion where an object moves back and forth around an equilibrium position.
When stopped, the object returns to equilibrium.
Periodic Motion: If the oscillation takes the same amount of time, it is periodic (Giancoli 2013).
Sound: Movement of air molecules in oscillations transfers sound energy.
Measurement Example: ECG records heart electrical signals as it beats.
Simple Harmonic Motion (SHM)
Definition: A special type of oscillation where acceleration is proportional and opposite to displacement from equilibrium.
Key Requirements:
A mass oscillating.
An equilibrium position.
Positive displacement right; negative left.
A restoring force proportional to displacement towards equilibrium.
Displacement Function:
x(t) = A ext{cos}( heta t + \Phi)where:
x(t) = displacement at time t
A = amplitude
heta = angular frequency
t = time
\Phi = phase constant/angle
Examples of Simple Harmonic Motion
Mass on a spring
Vibrations in the ear
Pendulum movement
Walking leg motion
Plucked guitar string
Pendulum Dynamics
Characteristics:
Velocity changes constantly during the swing.
Deceleration occurs as it approaches the peak positions.
Exhibits continuous acceleration-deceleration: characteristic of SHM.
Frequency for Simple Harmonic Motion
Formula: f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}
where k = spring constant, m = mass.
Damped Oscillations
Definition: Damping leads to reduced amplitude over time due to forces like friction or air resistance.
Frequency and period:
Determined by mass and stiffness of the spring.
Not dependent on amplitude.
Waves
Amplitude:
Maximum displacement from undisturbed position (measured in meters).
Wavelength (\lambda):
Distance from one similar point on a wave to another (e.g., crest to crest).
Wave Characteristics:
Displacement (x): Distance from equilibrium.
Period (T): Time for one complete oscillation (seconds).
Frequency (f): Oscillations per time unit (Hertz, Hz).
f = \frac{1}{T}
Wave Speed (v): Speed of wave fronts, e.g., v = \frac{d}{t}.
Types of Waves
Transverse Waves: Displacement is perpendicular to direction of travel.
Examples: Ocean waves, electromagnetic waves.
Longitudinal Waves: Particles displaced parallel to wave direction.
Examples: Sound waves.
Energy Transport:
Energy is proportional to the square of amplitude.
Mechanical waves need a medium.
Wave Intensity
Definition: Rate of energy transmitted per unit area perpendicular to wave velocity (measured in W m^{-2}).
Intensity Variance: Generally decreases as waves travel, spreading out reduces amplitude.
Learning Outcomes for Vibrations, Waves, and Sound
Students should be able to:
Explain oscillations.
Explain simple harmonic motion (SHM).
Identify the displacement equation for SHM, including amplitude, phase constant, and phase.
Relate kinetic and potential energy for a simple pendulum.
Describe frequency in SHM.
Describe damped oscillations.
Describe waves in terms of wavelength, amplitude, frequency, speed, and intensity.
Describe wave speed.
Recall that all electromagnetic waves are transverse waves traveling at the speed of light in free space.
Explain phase difference.
Distinguish between transverse and longitudinal waves.
Explain the energy carried by a wave.
Explain the intensity of a wave.
Oscillations
Definition: Oscillation is a repetitive motion where an object moves back and forth around an equilibrium position.
When stopped, the object returns to equilibrium.
Periodic Motion: If the oscillation takes the same amount of time, it is periodic (Giancoli 2013).
Sound: Movement of air molecules in oscillations transfers sound energy.
Simple Harmonic Motion (SHM)
Definition: A special type of oscillation where acceleration is proportional and opposite to displacement from equilibrium.
Key Requirements:
A mass oscillating.
An equilibrium position.
Positive displacement right; negative left.
A restoring force proportional to displacement towards equilibrium.
Displacement Function:
x(t) = A \, \cos(\theta t + \Phi)
where:
x(t) = displacement at time t
A = amplitude
\theta = angular frequency
t = time
\Phi = phase constant/angle
Examples of Simple Harmonic Motion
Mass on a spring
Vibrations in the ear
Pendulum movement
Walking leg motion
Plucked guitar string
Pendulum Dynamics
Characteristics:
Velocity changes constantly during the swing.
Deceleration occurs as it approaches the peak positions.
Exhibits continuous acceleration-deceleration: characteristic of SHM.
Damped Oscillations
Definition: Damping leads to reduced amplitude over time due to forces like friction or air resistance.
Frequency and Period:
Determined by mass and stiffness of the spring.
Not dependent on amplitude.
Waves
Amplitude:
Maximum displacement from undisturbed position (measured in meters).
Wavelength (\lambda):
Distance from one similar point on a wave to another (e.g., crest to crest).
Wave Characteristics:
Displacement (x): Distance from equilibrium.
Period (T): Time for one complete oscillation (seconds).
Frequency (f): Oscillations per time unit (Hertz, Hz).
f = \frac{1}{T}
Wave Speed (v): Speed of wave fronts, e.g., v = \frac{d}{t}.
Types of Waves
Transverse Waves: Displacement is perpendicular to direction of travel.
Examples: Ocean waves, electromagnetic waves.
Longitudinal Waves: Particles displaced parallel to wave direction.
Examples: Sound waves.
Energy Transport:
Energy is proportional to the square of amplitude.
Mechanical waves need a medium.
Wave Intensity
Definition: Rate of energy transmitted per unit area perpendicular to wave velocity (measured in W m^{-2}).
Intensity Variance: Generally decreases as waves travel, spreading out reduces amplitude.