In Simple Harmonic Motion (SHM), a particle or mass oscillates about an equilibrium position and is subject to a linear restoring force. The direction of the restoring force is always towards the equilibrium position and the magnitude is directly proportion to the magnitude of the displacement.
When a spring with a mass attached to one is compressed/stretched and then released, it will oscillate with a restoring force of F = -kx (Hooke’s Law), where k is the spring constant.
The spring constant k is a measure of the spring’s stiffness.
A spring with a spring constant k and mass m, displaced x meters, will have an acceleration a = -(ω^2)x, where angular frequency ω = 2πf = sqrt(k/m).
Frequency f is measured in hertz (Hz), which is cycles of oscillation per second. Angular frequency ω is measured in radians per second and is a measure of the rate at which an oscillating object would move through an arc.
The position of a spring as it moves through its cycle is: x = X cos (ωt), where X is the amplitude.
Assume that the spring has maximum displacement at t = 0.
For small oscillations and short periods of time, springs can be treated as conservative systems: E = K + U = constant. Assume all potential energy is converted to kinetic energy as it oscillates.
The spring has potential energy U = (1/2)kx^2, where x is the displacement from equilibrium. The kinetic energy is K = (1/2)mv^2.
Maximum velocity is reached at equilibrium length, while velocity is zero at maximum displacement.
The restoring force of a simple pendulum is generated by gravity: F = -mg sin θ. For a pendulum of length L, the angular frequency is: ω = 2πf = sqrt(g/L).
A pendulum can also be approximated as a conservative force if air resistance and friction and negligible, where E = K + U = constant.
The pendulum has a maximum potential energy U = mgh, where h is the vertical height difference. Kinetic energy of the mass attached to the pendulum is: K = (1/2)mv^2.
Transverse waves are a waveform in which the direction of the particle oscillation is perpendicular to the movement (propagation) of the wave (e.g., visible light, microwaves and x-rays).
The longitudinal wave is a sinusoidal wave in which the particles of the wave oscillate along the direction of travel of the wave (e.g., sound).
The displacement of a particle in a wave is: y = Y sin (kx - ωt), where Y is the amplitude and k is the wavenumber (not the spring constant in Hooke’s Law!).
The distance between one maximum (crest) of the wave to the next is the wavelength λ. The speed of the wave is: ν = fλ.
Wavenumber is defined as: k = 2π/λ. The angular frequency is: ω = 2πf = 2π/T. Thus, wavespeed v = fλ = ω/k = λ/T.
Phase difference describes how “in step” or “out of step” two waves are with each other.
When waves are perfectly in phase, the phase difference is zero. If two waves occur such that the crests (maxima) of one coincide with the troughs (minima) of the other, the phase difference would be one-half of a wave or 180°.
The principle of superposition states that when waves interact, the result is a sum of the waves.
When waves are in phase, the resultant wave has greater amplitude (constructive interference). When waves are out of phase, the resultant wave’s amplitude is the difference between the amplitudes (destructive interference).
If a string fixed at one end is moved up and down, a traveling wave will form and propagate toward the fixed end. When the wave reaches the fixed boundary, it is reflected and inverted, interfering with the original wave.
When both ends of a string are fixed, certain wave frequencies result in standing waves, which appear to be stationary with only fluctuations of amplitude at fixed points.
Points that remain at rest are nodes, while points that are midway between nodes and fluctuate with maximum amplitude are antinodes.
The frequency or frequencies at which an object will vibrate when disturbed is the natural frequency.
If a periodically varying force is applied to a system, it will be driven at a frequency equal to the frequency of the force, known as forced oscillation. The amplitude of this motion will generally be small unless the frequency is close to the natural frequency.
If the frequency of the periodic force is equal to a natural frequency, the system is resonating and the amplitude of the oscillation is at a maximum.
Sound is transmitted by the oscillation of particles longitudinally, i.e., along the direction of motion of the sound wave. Sound waves propagate as mechanical disturbances through deformable media.
Sound travels fastest through a solid and slowest through a gas. In air, the speed of sound at 0°C
Audible waves have frequencies that range from 20 Hz to 20,000 Hz.
Sound waves whose frequencies are below 20 Hz are infrasonic waves, while those above 20,000 Hz are ultrasonic waves.
Intensity is the power transported per unit area, with SI units of watts per square meter, W/m^2. It is calculated as: I = P/A.
Intensity is proportional to the square of the amplitude and inversely proportional to the square of the distance from the source.
Sound level β, measured in decibels (dB), is: β = 10 log (I/I_0), where I_0 is a reference intensity set at the threshold of hearing, 1 x 10e-12 W/m^2.
When the intensity of sound is changed by some factor: β_f = β_i + 10 log (I_f/I_i)
The perception of frequency of sound is pitch, where higher-frequency sounds have higher pitch.
If two sound waves have nearly equal frequencies (in the audible range), the resultant wave will have periodically increasing and decreasing amplitude, which are heard as beats.
The beat frequency, perceived as a periodic variation in loudness and not pitch, is: f_beat = |f_1 - f_2|
The Doppler effect describes the difference between the perceived frequency of a sound and its actual frequency when the source of the sound and the sound’s detector are moving relative to each other
The perceived frequency is: f' = f [(v ± v_D) / (v ∓ v_S)], where v is the speed of sound, v_D is the speed of the detector and v_S is the speed of the source.
The upper sign is used when the detector and source are getting closer, while the lower is used when the detector and the source are going farther away.
Closed boundaries are those that do not allow oscillation and support nodes, while open boundaries are those that allow oscillation and support antinodes.
When strings are secured at both ends, the two points can only support a node of a standing wave. The wavelength of the standing wave for a string of length L is: λ = 2L / n, where n is a positive nonzero integer.
Possible frequencies are: f = nv / 2L.
The lowest frequency of a standing wave is known as the fundamental frequency (first harmonic). The frequency given by n = 2 is the first overtone or second harmonic.
The wavelength of a standing wave for an open pipe of length L is: λ = 2L / n, where n is a positive nonzero integer.
Possible frequencies are: f = nv / 2L.
The wavelength of a standing wave for a closer pipe of length L is: λ = 4L / n, where n is odd integers only.
Possible frequencies are: f = nv / 4L.