Understanding Progressive Waves

• Oscillatory disturbance traveling through a medium without change of form is a wave.
• Water waves, light waves, sound waves, mechanical waves, and electromagnetic waves are types of waves.
• Waves carry or transfer energy from one point to another and are doubly periodic.

Progressive Waves

• A progressive wave travels through a medium continuously in a forward direction.
• All particles of the medium vibrate about their mean positions, performing Simple Harmonic Motion (SHM).
• All vibrating particles of the medium have the same amplitude, period, and frequency.
• The phase changes from one particle to another.
• No particle remains permanently at rest.
• Each particle comes to rest momentarily at the extreme positions of vibration.
• Particles attain maximum velocity when passing through their mean positions.
• Wave propagation transfers energy along the wave without matter transfer.
• Waves propagate through the medium with a certain velocity.

Types of Progressive Waves

• Transverse waves: Vibrations of particles are perpendicular to the direction of propagation, creating crests and troughs.
  • These can pass through solids.
• Longitudinal waves: Vibrations of particles are parallel to the direction of propagation, producing compressions and rarefactions.
  • These can pass through solids, liquids, and gases.
• Mechanical wave displacement of particles at a space point x at time 't' is given by y(x, t) = f(x - vt), where v is the speed of the wave.
• The factor (x - vt) indicates that the disturbance created at point x = 0 at time t reaches the point x = x at time t.
• This equation represents a progressive wave traveling in the positive x direction with constant speed v.
• If the source of disturbance performs SHM, the wave is represented as a sine or cosine function.
• y(x, t) = A sin(kx - ωt) is + direction of x-axis expression, where A is the amplitude of the wave.
• k = 2π/λ is the wave number, where λ is wavelength.
• ω = 2πf is the angular frequency.
• If the wave travels to the left (negative x - direction), then y(x, t) = A sin(kx + ωt).

Reflection of Waves

• When a wave travels through a medium, part of its energy reflects back into the same medium and changes direction.
• Reflection occurs when a sound wave travels from one medium to another and returns to the original medium with slightly different intensity and energy.

Reflection of a Transverse Wave

• Crest reflects as a trough when a wave pulse is sent as crest from a rarer medium to a denser medium.

Example 1

• Take a long light string AB, attach one end to a rigid support B.
• A crest is generated in the string by giving it a jerk at end A.
• The string is pulled downwards, so a crest reflects as a trough with a phase change of π.

Example 2

• A long light string AB is used, with end B attached to a ring that can slide on a vertical rod without friction
• Wave pulse sent as crest from a denser medium to a rarer medium
• Crest reflects as a crest, meaning no phase change

Example 3

• A heavy string P and a light string Q are joined
• Wave pulse produced as crest on P, moving towards junction O; crest produced on light sting Q, moves towards junction point O
• Crest reflected as a crest from the lighter string, but when crest travels from lighter string to heavy string, it is reflected as a trough

Reflection of a Longitudinal Wave

• When longitudinal wave is incident on a denser medium such as rigid wall, the wall exerts an equal and opposite reactive force
• Compression is reflected as a compression and rarefaction as rarefaction, resulting in phase change of π
• When longitudinal wave is incident on a rarer medium, it exerts force on particles of rarer medium particles get displaced in the direction of the wave
• Compression is reflected as rarefaction and rarefaction as compression, because there is no phase change

Superposition of Waves

• When two or more waves travel through a medium and pass through a common point, each wave produces its own displacement.
• Resultant displacement at that point is the vector sum of individual wave displacements.
• Superposition doesn't change individual wave shapes and nature.
• Superposition applies to sound waves, light waves, and waves on a string.

Case I: Superposition of Two Wave Pulses of Equal Amplitude and Same Phase Moving Towards Each Other

• Shows these two wave pulses of equal amplitude and phase moving towards each other
• As they approach each other, their superposition results in a larger amplitude.
• At the point of complete overlap, the resultant displacement is the sum of their individual displacements, constructive interference results
• After crossing, both wave pulses continue to maintain their shapes.

Case II: Superposition of Two Wave Pulses of Equal Amplitude and Opposite Phases Moving Towards Each Other

• In Case II, pulses superimpose, zero resultant displacement, phase difference by 180° (destructive interference)
• After crossing, both wave pulses maintain individual shapes.

Amplitude of Resultant Waves

• Consider same frequency waves, different amplitudes A₁ and A₂, differing in phase by φ
• Displacement is a sum of each wave at x = 0: y = y₁ + y₂
• If source of disturbance is performing SHM, waves are written as: y₁ = A₁ sin ωt and y₂ = A₂ sin (ωt + φ)
• According to superposition, y = A₁ sin ωt + A₂ sin (ωt + φ).
• Can be simplified to y = (A₁ + A₂ cos φ) sin ωt + A₂ cos ωt sin φ, where A₁ + A₂ cos φ = A cos θ and A₂ sin φ = A sin θ
• So, y = A cos θ sin ωt + A sin θ cos ωt which simplifies to y = A sin(ωt + φ)
• Equation of resultant wave, same frequency as interfering waves
• Squaring and adding modified equations using trigonometric identities give resultant amplitude is : A = √(A₁² + A₂² + 2A₁A₂ cos φ)

Special Cases

• Case I: When φ = 0; i.e., the waves are in phase with each other.

• When φ = π, the waves are out of phase, cos π = -1. Amplitude is minimum when φ = π.

• Note: Intensities of waves are proportional to squares of amplitudes.

Stationary Waves (Standing Waves)

• Two identical progressive waves with same amplitude, frequency and speed travel through a medium along same line but in opposite directions superimpose

Formation of Stationary Waves

• When the string is pulled at the middle and released, we get stationary waves.
• These waves are reflected at the fixed ends.
• The wave produced in the string initially and their reflected waves combine to produce stationary waves
• Certain points, half wavelength apart
• There are some mid-way between the nodes where the displacement is maximum known as anti-nodes

Equation of Stationary Wave on a Stretched String

• In a string, consider simple harmonic progressive waves having same amplitude (A), wavelength (λ) and frequency (n) propagating on a long uniform string in opposite directions
• Equation of wave traveling along positive x direction: y₁ = A sin[2π(nt - x/λ)]
• Equation of wave traveling along negative x direction: y₂ = A sin[2π(nt + x/λ)]
• By superposition: y = y₁ + y₂
• Simplifies to: y = 2A sin[2πnt/τ] cos[2πx/λ]
• Which results in y = R sin(2πnt/τ).
• The wave does not contain a term (nt - x/λ).
• Hence resultant wave is called stationary wave

Case I: Condition for Node

• Nodes are points of minimum displacement

Case-ll: Condition for Antinode

• Antinodes, points of maximum displacement

Properties of Stationary Waves

• Stationary waves form due to superposition of two identical traveling waves through environment in opposite directions.
• Every medium particle oscillates with the same SHM period.
• Medium particle oscillation amplitude is different.
• Stationary waves never transfer energy through environment.

Free And Forced Vibrations

Free Vibrations

• In free vibration, the body first undergoes initial displacement, then the force withdraws.
• The body vibrates and continues on its own because no external force acts on it.
• Free vibrations occur when system vibrates at its natural frequency..
• Object's natural frequency is vibration it produce it when hit, plucked, or otherwise disturbed.
• A simple pendulum, a tuning fork, and an oscillating spring are examples

Forced Vibrations

• Forced vibrations are vibrations of object influenced by external periodic force.
• Vibration frequency equals external force frequency.
• Drill machine and washing machine vibrations exemplify
• Holding vibrating tuning fork against the table causes forced table vibrations called forcing frequency.

Harmonics

• Integral multiple of fundamental frequency.
• Harmonic is combined term for describing the fundamental with all of its integral multiple frequencies
• The harmonic concept applies to strings and air columns by the following: lowest frequency with which string or air column vibrate is the fundamental frequency

Overtone

• Notes higher, more frequent in harmonic progressions than the original vibration are called overtones
• Fundamental tone and multiples form overtones.

End Correction

• Air column vibrates in pipe that is
• Open at either end
• Closed at one end
• Boundary conditions dictate anti-node that can be located near open end of chamber.

Beats

• Beats arise by wave superposition.
• Slightly different wavelengths between sound waves produce periodic intensity variations known as beats which are the principle of musical chords
• Beats help adjust frequency of two different yet similar wavelength based instruments in live performance, as well as sonometer experiments

Characteristics of Sound:

• Sound has three characteristics: Loudness, Pitch and Quality
• Loudness: Loudness is the sensation received by the ear. Loudness is related to intensity of sound.
• Pitch: The sensation of shrillness produced by a sound note is known as the perception of frequency
• Quality: Quality of sound note are dependent number of overtones along with frequency.