Oscillatory Motion and Waves
Oscillatory Motion
Periodic Movement
Oscillatory motion, or periodic movement, occurs when a particle experiences displacement from its equilibrium position. When the disturbance is small, the restoring force is proportional to the displacement from equilibrium (x): F = -Kx. The negative sign indicates that the force opposes the displacement.
Simple Harmonic Motion
Knowledge of the force allows us to use Newton’s second law to obtain the differential equation that governs the particle’s trajectory: x(t) = A*sin(Wt-φ).
Simple Harmonic Motion Energy
The total energy of a particle undergoing simple harmonic motion remains constant in an isolated system. This energy comprises two components: kinetic energy (Ec) associated with particle velocity and potential energy (Ep) due to the restoring force.
Waves
A wave is a physical disturbance that transmits energy and momentum but not matter. In material waves, particles oscillate around their equilibrium positions without propagating. Only the disturbance propagates, described by a function of position and time, f(x, t).
Nature of Waves
Material Waves
These waves require a material medium for propagation. Individual particles within the medium oscillate around their equilibrium positions without propagating.
Electromagnetic Waves
These waves are variations of electric and magnetic fields perpendicular to the direction of propagation, making them transverse waves. The electric and magnetic fields are also perpendicular to each other.
Wave Intensity
The total energy of a wave on a rope, for example, is the sum of the energies of each particle due to its vibrational motion.
Dimensionality
One-Dimensional Waves
These waves propagate in one dimension, such as a wave on a string.
Two-Dimensional Waves
These waves propagate in two dimensions, such as waves on a surface. The initial disturbance is transmitted to all surface points without mass transport.
Three-Dimensional Waves
These waves propagate in three dimensions, such as sound waves.
Direction of Oscillation
Longitudinal Waves
In longitudinal waves, the direction of oscillation coincides with the direction of wave propagation.
Transverse Waves
In transverse waves, oscillations occur perpendicular to the direction of wave propagation.
Huygens’ Principle
This principle states that particles in a wavefront act as sources of secondary waves, and the envelope of these secondary waves forms a new primary wavefront.
Reflection
Reflection occurs when waves traveling through a homogeneous medium encounter an obstacle, causing them to change direction. It can be visualized as the wavefront rebounding at the interface between two media.
Refraction
Refraction is the change in direction of waves as they pass obliquely from one medium to another with different propagation speeds. The refracted ray, the normal, and the incident ray all lie in the same plane. The ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for two given media and equal to the ratio of their wave propagation velocities (v1 and v2).
Wave Interference
Wave interference is a special case of superposition that occurs when two or more waves coincide in a region of space. It can occur between an incident wave and a reflected wave. The principle of superposition states that the resulting displacement at a point is the vector sum of the displacements caused by each individual wave.
Diffraction
Diffraction is the change in shape of a wave as it passes through a narrow opening or around an obstacle.
Polarization
A transverse wave is linearly polarized when its oscillations occur in a single direction. Ordinary light can be linearly polarized using polarizers, which only allow light with a specific polarization direction to pass through, or through reflection.
Doppler Effect
The Doppler effect is the shift in frequency experienced by a wave when the source, receiver, or both are moving relative to the propagation medium. It’s a general wave phenomenon with various applications.
Superposition Principle
The superposition principle states that when two or more waves overlap, the resulting disturbance at a point is the sum of the individual disturbances. This principle applies when the disturbances have small amplitudes.