8 Harmonics and Waves
How can regular, repeating motion be described with waves?
Contents
- Simple Harmonic Motion and Springs
- Simple Pendula
- Energy in Harmonic Oscillators
- Sine functions
- Wave Motions
Harmonics
Many objects oscillate - an object on the end of a spring, a pendulum, a tuning fork, electrons in atoms and molecules, and the molecules themselves.
When an object vibrates or oscillates back and forth, over the same path, each oscillation taking the same amount of time, the motion is periodic.
Simple Harmonic Motion
An oscillating or vibrating object undergoes simple harmonic motion (SHM) if the restoring force is proportional to the (negative of) its displacement (Hooke’s Law).
$F = -kx$ (where $k$ is the spring constant and $x$ is displacement)
Hooke’s Law accurately describes springs, but other oscillating solids as well.
Restoring Force
Properties of Oscillations
Oscillations can be described by three properties:
- Amplitude ($A$) - the maximum displacement from equilibrium
- Period ($T$) - the time required to complete one full cycle
- Frequency ($f$) - the number of cycles per second
Frequency and Period
The unit of frequency is the Hz or $s^{-1}$ and represents the number of cycles that occur each second.
- Period ($T$) - the time required to complete one full cycle
- Frequency ($f$) - the number of cycles per second
Frequency and Period
The period of oscillation for a mass $m$ on the end of a spring with spring constant $k$ is:
$T$ is the period of time for the mass to complete one cycle of movement.
Frequency and Period
Strange though it seems, the period is not dependent on amplitude.
Displacing a spring farther increases its velocity, and therefore the period is kept constant.
Transformation in SHM
During simple harmonic motion, the total energy of the system is conserved but it is continually changing from kinetic to potential (and back).
Simple Pendulum
When friction is present, the motion is damped.
The maximum displacement of the pendulum decreases with time, mechanical energy is eventually transformed into thermal energy (friction).
Simple Pendulum
A simple pendulum with length $l$ approximates SHM if the amplitude is small and friction is negligible (when displacement angles are below $15^\circ$).
For small amplitudes, its period of a pendulum is given by:
Note that the period depends only on length, not the mass or “bob”.
This is the length of the pendulum.
Measures of SHM
Wave Motions
A wave is the motion of a disturbance.
Most waves require a medium, or material through which the disturbance travels.
- Waves that require a medium are called mechanical waves.
- Waves that do not require a medium are electromagnetic waves.
Particles in Waves
Particles in the medium oscillate, tracing a returning path but not traveling with the wave.
As the wave passes, particles return to their original positions.
Particles in Waves
Particles in the medium oscillate, tracing a returning path but not traveling with the wave.
Transverse and Longitudinal Waves
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Transverse waves are waves where the particles move perpendicular to the wave. Example: dropping a pebble in a pond moves the water up and down, but the waves radiate outward laterally (to the sides).
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Longitudinal waves are waves that travel parallel to the wave motion. Example: compressing and releasing a spring.
Creating Waves
Vibrating objects act as sources of waves that travel outward from the source.
Examples include waves in water or a string or cord.
Wave Motions
The rate at which waves transfer energy depends on the amplitude at which the particles in the medium vibrate.
For mechanical waves, the energy transferred is proportional to the square of the amplitude.
When the amplitude of a wave is doubled, its energy increases by a factor of four.
Creating Waves
A pulse is a single wave traveling through a medium.
Creating Waves
Creating Waves
Sine waves describe particles vibrating within harmonic motion.
- Wavelength ($\lambda$) - the distance a wave travels in one cycle or the distance between crests or troughs on a wave.
The trigonometric function $y = \sin x$ produces the curve when plotted.
Modeling Waves
- Crest - the highest point above the equilibrium position.
- Trough - the lowest point below the equilibrium position.
Moving Waves
A traveling wave can be represented mathematically.
The velocity of a wave is its wavelength divided by its period.
Interacting Waves
The waves interact to form an interference pattern.
As waves move outward from their respective sources, they pass through one another.
Colliding Waves
When waves come together, they do not collide as particles.
Because mechanical waves are not matter, only its displacement, two waves can occupy the same space at the same time.
Superposition is the combination of two overlapping waves at once.
Colliding Waves
At each point in the wave, the displacements of the particles due to each wave are added.
The resulting sum is the displacement of the resultant wave.
- Displacements in the same direction produce constructive interference.
- Displacements in opposite directions produce destructive interference.
Colliding Waves
Displacements in the same direction produce constructive interference.
Colliding Waves
Displacements in opposite directions produce destructive interference.
Resultant displacement at each point is zero; pulses cancel each other out.
After collision
After the two pulses pass through each other, each pulse has the same shape it did before the waves met, and is still traveling in the same direction.
This is true for sound waves, water waves, light waves, and many others.
Waves maintain their own characteristics after interference.
After collision
- Constructive
- Destructive
- Partial
Reflection
When a pulse reaches a boundary, two changes can occur:
- If it is a free boundary, the wave reflects, changing direction and maintaining its amplitude and orientation.
Reflection
When a pulse reaches a boundary, two changes can occur:
- If it is a fixed boundary, the wave reflects, changing direction and is also inverted.
Standing Waves
If a medium is vibrated at exactly the right frequency, a standing wave can be formed.
A standing wave is a wave that appears to stand motionless due to alternating regions of constructive and destructive interference.
- Nodes are points of complete destructive interference (displacement = 0).
- Antinodes are points of largest amplitude, caused by perfectly aligned constructive interference.
Standing Waves
- Nodes are points of complete destructive interference (displacement = 0).
- Antinodes are points of largest amplitude, caused by perfectly aligned constructive interference.
Only certain frequencies (and therefore wavelengths) produce standing waves.
Standing Waves
Only certain frequencies (and therefore wavelengths) produce standing waves.
For any given wave of period $T$, four standing waves can exist:
The Scales of Things
Key Equations Summary
Simple Harmonic Motion:
- Hooke’s Law: $F = -kx$
- Spring Period: $T = 2\pi\sqrt{\frac{m}{k}}$
- Pendulum Period: $T = 2\pi\sqrt{\frac{l}{g}}$
Waves:
- Wave Speed: $v = \frac{\lambda}{T}$ or $v = f\lambda$
- Frequency-Period Relationship: $f = \frac{1}{T}$
- Wave Energy: $E \propto A^2$
Example 1 | A pendulum clock
A pendulum clock is being designed to have a period of exactly 1.0 s. How long should the pendulum be?
Example 2 | Suspension on a car
The body of a 1 275 kg car is supported on a frame by four springs. Two people riding in the car have a combined mass of 153 kg. When driven over a pothole in the road, the frame vibrates with a period of 0.840 s. For the first few seconds, the vibration approximates SHM. Find the spring constant of the suspension springs.
Example 3 | A piano string
A piano string tuned to middle C vibrates with a frequency of 262 Hz. Assuming the speed of sound in air is 343 m/s, find the wavelength of the sound waves produced by the string.