# 4.1 Oscillations

• Periodic Motion:

• Motion that repeats itself after equal intervals of time.

• Examples include the motion of a loaded spring, an object moving in a circle, and a simple pendulum.

• Terms Related to Periodic Motion:

• Amplitude (A): Maximum displacement from the equilibrium position.

• Time period (T): Time taken for a complete oscillation.

• Frequency (f): Number of oscillations per unit time. (f = 1/T)

• Angular frequency (𝓌): Equivalent of frequency. (𝓌 = 2(π)f)

• Phase (θ): If the motion starts away from equilibrium, it leads or lags by θ.

• Isochronous oscillations: Maintain a constant time period regardless of amplitude changes.

• Simple Harmonic Motion (SHM):

• A type of periodic motion where the restoring force is proportional to the negative displacement from the equilibrium position.

• Examples include a spring-loaded with a mass and a simple pendulum with a small amplitude.

• Equation defining SHM: a - kx where a is accelerated, k is a constant, and x is displacement.

• Units of constant k:

• K = - a/x, so the unit of k is m x s^-2 / m = s^-2.

• Difference in oscillations of two systems S1 and S2.

• If S1 has frequency f, S2 with 4k frequency has a frequency of √4f = 2f

• Describing Simple Harmonic Motion:

• Equation for SHM: x  = Asin(2πft + θ) = Asin(wt + θ), where θ is π/2

• Velocity (v):

• v = (dx)/(dt) = 𝓌Acos(𝓌t + θ)

• Acceleration(a):

• a = (dv)/(dt) = -𝓌^2(A)(sin(𝓌t+θ))

• Phase difference between displacement-time graphs:

• Circular Motion and SHM:

• The projection of an object in circular motion on a diameter follows simple harmonic motion.

• Energy Changes in Simple Harmonic Motion:

• Kinetic energy: KE= ½(m𝓌^2A^2)

• Total energy remains constant in the absence of dissipative forces.

• Waves and their types:

• Mechanical waves: require a material medium to travel

• Electromagnetic waves: can travel through a vacuum.

• Describing Waves:

• Wavefront: A surface perpendicular to the direction of wave travel.

• Amplitude (A): Maximum displacement from equilibrium.

• Wavelength (⅄): Shortest distance between two points in phase on a wave.

• Period (T): Time for a complete wavelength to pass a fixed point.

• Frequency (f): Number of wavelengths passing through a fixed point per unit time.

• f = 1/T

# 4.2 Traveling waves

• Transverse waves: Direction of vibration perpendicular to the direction of propagation.

• Longitudinal waves: Direction of vibration parallel to the direction of propagation.

• Wave Equation:

• The velocity of a wave (c) is given by c = f⅄

• Electromagnetic waves:

• Travel with varying electric and magnetic fields at 3 x 10^8 m/s in a vacuum.

# 4.3 Wave characteristics

• Intensity of Waves:

• Intensity (I) is power received per unit area. I = (P)/(4πr^2) and is proportional to the square of amplitude (A^2).

• Example:

• Intensity at 120m from source: 3 x 10^-6 W/m^2.

• Principle of Superposition:

• When two waves meet, the total displacement is the vector sum of their individual displacements.

• Polarization:

• Restriction of oscillation direction to a plane perpendicular to the direction of propagation. Result: Plane-polarized light.

• Malus’s Law:

• Intensity (I) transmitted by an analyzer is proportional to cos^2(θ) where θ is the angle between the polarizer and the analyzer.

# 4.4 Wave Behaviour

• Laws Of Reflection And Refraction:

• Incident, reflected, and refracted rays, and normal lie on the same plane.

• The angle of incidence equals the angle of reflection.

• (sinθ1)/(sinθ2) = 1/n (Snell’s Law)

• Reversibility of Light:

• (sinθ1)/(sinθ2) = 1/n_2 for light going from medium 1 to medium 2, and (sinθ1)/(sinθ2) = 1/n_1 for light traveling in the opposite direction.

• Critical Angle And Total Internal Reflection:

• The angle of incidence for which the angle of refraction reaches the right angle is the critical angle.

• Total internal reflection occurs when the angle of incidence is greater than the critical angle.

• Double-Slit Interference:

• Two coherent sources create interference patterns. Constructive interference occurs at nλ and is destructive at (n + ½)λ.

• Example:

• The path difference at point P is 7λ. The nature of the fringe at P is bright, and there are 7 dark fringes between O and P.

• Diffraction:

• Wave passed through a narrow gap forms bright and dark fringes. Angular position of minima given by θ = (nλ/a).

• Example:

• Path difference at point P is 7λ.

• The nature of fringe at P is bright, and there are 7 dark fringes between O and P.

• Interference With Multiple Slits:

• More slits result in sharper and more intense maxima and minima.

• Dispersion:

• Different wavelengths of light refract at different angles. White light disperses into its constituent wavelengths.

• Resolution:

• Rayleigh's criterion states two points are just resolved if the central maximum of the first point falls on the first minimum of the second point.

• Diffraction Grating:

• For a grating with N slits, R = λ/change in λ = mN

• Reflection Of Light Off Thin Films:

• Reflected light undergoes a phase change of 180∘ if reflected off a denser medium.

• A thin film of thickness t, refractive index n, and incident wavelength λ exhibits interference.

• Doppler Effect In Light:

• The change in frequency of the light wave is (v/c)(f_0).

• Water Waves:

• Follow laws similar to light. Exhibit reflection, refraction, interference, and diffraction.

• Wave Propagation:

• Wavefront consists of infinite new disturbance centers.

• Successive wavefronts result from wavelets from these disturbances.

• Reflection Of Water Wave:

• When a wave hits a barrier, it behaves as if a similar wave is coming from the barrier in the opposite direction.

• Doppler Effect In Sound:

• The frequency of a moving source changes for an observer at rest or moving toward/away from the source.

# 4.5 Standing waves

• Boundary Conditions:

• Reflected off a fixed boundary suffers a phase change of 180∘.

• No change in the phase of a free boundary.

• Standing Waves:

• Formed when two waves of equal amplitude and frequency traveling in opposite directions are superimposed.

• Positions of crests and troughs do not change with time.

• Nodes and Antinodes:

• Nodes are points with zero displacement, antinodes are points with maximum displacement.

• Harmonics On A String:

• The string is tied at one end and connected to a vibration generator at the other.

• Harmonics formed with increasing loops at n times the frequency of the first harmonic.

• Displacement of string at different times:

• Quarter of a cycle: t = 1/4f

• Half of a cycle: t = 1/2f

• Frequency of vibration of the spring:

• Wavelength 2L, wave velocity 240 m/s, frequency 120 Hz.

• Harmonics In A Pipe:

• Harmonics formed with one end open or both ends open. Nodes form at closed ends and antinodes at open ends.

• Explanation regarding refraction of light:

• The speed of light is faster in a vacuum than in water, bending away from normal.

• The critical angle for total internal reflection:

• sinፀ_c = 1/n

• where n is the relative refractive index of denser material with respect to rarer material.

• Frequency of the first harmonic if both ends are open:

• Twice the frequency of the first harmonic when one end is closed