Oscillations and Waves

The One Thing

Oscillations are repeated motions about an equilibrium position, and simple harmonic motion is the ideal case in which restoring acceleration is proportional to displacement and directed toward equilibrium. Waves are propagating oscillations that transfer energy and information without net transport of matter.

Reactions, Processes and Experiments

Key Terms

• Periodic motion: Motion that repeats itself after equal intervals of time.
• Oscillatory motion: To-and-fro motion of an object about a mean or equilibrium position.
• Equilibrium position: The position at which the net force on an object is zero.
• Simple harmonic motion: Periodic motion in which acceleration is directly proportional to displacement from equilibrium and always directed toward equilibrium.
• Displacement: The instantaneous distance and direction of a particle from its equilibrium position.
• Amplitude: The maximum displacement of an oscillating particle from equilibrium.
• Time period: The time required to complete one full oscillation, represented by T.
• Frequency: The number of complete oscillations per second, represented by f; f = 1/T.
• Angular frequency: The rate of change of phase, represented by omega; omega = 2 pi f = 2 pi/T.
• Phase: The state of oscillation of a particle at a particular instant, usually expressed as an angle.
• Restoring force: A force that acts toward equilibrium and tends to return a displaced object.
• Spring-mass oscillator: An oscillator consisting of a mass attached to a spring, with restoring force F = -kx.
• Simple pendulum: A small bob suspended by a light, inextensible string; for small angular displacements, it performs approximately SHM.
• Wave motion: Propagation of a disturbance through a medium or space, transferring energy without net transport of matter.
• Mechanical wave: A wave requiring a material medium for propagation, such as sound or waves on a string.
• Electromagnetic wave: A wave consisting of changing electric and magnetic fields that can travel through vacuum, such as light.
• Transverse wave: A wave in which particle vibration is perpendicular to the direction of propagation.
• Longitudinal wave: A wave in which particle vibration is parallel to the direction of propagation.
• Wavelength: The distance between two successive points in the same phase, such as two consecutive crests or compressions.
• Wave speed: The speed at which a particular phase travels; v = f lambda.
• Crest and trough: The highest and lowest points of a transverse wave, respectively.
• Compression and rarefaction: Regions of maximum and minimum particle density in a longitudinal wave, respectively.
• Progressive wave: A wave that travels continuously through a medium, transferring energy from one region to another.
• Stationary wave: A pattern formed by superposition of two identical waves travelling in opposite directions, producing nodes and antinodes.
• Superposition principle: When waves overlap, the resultant displacement is the algebraic sum of the individual displacements.
• Resonance: A condition of very large amplitude when the external driving frequency equals or is close to the natural frequency.
• Damping: The gradual decrease in amplitude and energy of oscillations due to resistive forces such as friction.

Easily Confused

• Periodic motion and oscillatory motion: Periodic motion repeats after equal time intervals, whereas oscillatory motion specifically involves to-and-fro movement about an equilibrium position.
• Oscillation and SHM: An oscillation need not be simple harmonic; SHM requires acceleration to be proportional to displacement and directed toward equilibrium.
• Amplitude and displacement: Displacement is the instantaneous position relative to equilibrium; amplitude is the maximum possible displacement.
• Frequency and angular frequency: Frequency is measured in hertz, while angular frequency is measured in radians per second, with omega = 2 pi f.
• Wave speed and particle speed: Wave speed is the speed at which a phase or disturbance travels; particles of the medium oscillate about mean positions rather than travelling with the wave over long distances.
• Transverse and longitudinal waves: Particle vibration is perpendicular to propagation in a transverse wave but parallel to propagation in a longitudinal wave.
• Progressive and stationary waves: A progressive wave transfers energy from one region to another, whereas a stationary wave has fixed nodes and antinodes.
• Wavelength in transverse and longitudinal waves: It is measured between successive crests or troughs in a transverse wave, and between successive compressions or rarefactions in a longitudinal wave.
• Damping and resonance: Damping decreases amplitude through resistive forces, whereas resonance produces large amplitude when driving and natural frequencies are equal or nearly equal.
• Open and closed pipes: An open pipe has displacement antinodes at both ends and permits all harmonics; a pipe closed at one end has a displacement node at the closed end and permits only odd harmonics.

What Gets Asked

• Define or identify SHM using the condition that restoring acceleration is proportional to displacement and opposite in direction: a proportional to -x. A common error is to describe any repeated motion as SHM.
• Derive or apply the periods T = 2 pi square root of (m/k) for a spring-mass system and T = 2 pi square root of (l/g) for a simple pendulum. Marks are lost by treating the pendulum period as dependent on bob mass or by applying the formula outside small angular displacements.
• Calculate displacement, velocity, acceleration, or energy in SHM using x = A sin(omega t + phi), v = plus or minus omega square root of (A^2 - x^2), a = -omega^2 x, and E = 1/2 kA^2 = 1/2 m omega^2 A^2. The key distinction is between conditions at equilibrium and at extreme positions.
• Use the wave relations f = 1/T, omega = 2 pi f, and v = f lambda = lambda/T. A frequent slip is confusing the source-controlled frequency with wave speed, which depends on the medium and wave type.
• Identify transverse and longitudinal waves, including crests and troughs versus compressions and rarefactions. Wavelength must be measured between successive points in the same phase.
• Determine stationary-wave patterns and frequencies for strings and pipes. The examined distinctions include node–antinode boundary conditions, lambda/2 between consecutive nodes or antinodes, lambda/4 between a node and nearest antinode, and the occurrence of only odd harmonics in a pipe closed at one end.
• Explain superposition, beats, resonance, and damping. The relevant equations and conditions are f_beat = absolute value of (f1 - f2), resonance at driving frequency equal or close to natural frequency, and decreasing amplitude due to damping.