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Physics

Wave Motion and Waves on a String

Wave Motion

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⚡ Quick Summary
Wave motion is a way to transport energy from one place to another without moving matter in bulk. A disturbance is created and propagates through a medium (or without a medium in some cases), carrying energy with it.
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When a particle moves through space, it carries kinetic energy with itself. Wave motion is another way to transport energy from one part of space to the other without any bulk motion of material. When you create some disturbance in a medium, energy is transferred to the particles, which then exert force on the next layer, transferring the disturbance. The disturbance produced in the air travels, but the air itself does not move. Waves requiring a medium are called mechanical waves, and those which do not require a medium are called nonmechanical waves.

Wave Propagation

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⚡ Quick Summary
Describes how a disturbance (wave) travels along a string, transferring energy without the string itself moving significantly along the direction of the wave. The shape of the wave is represented mathematically using functions.
['y = g(x + vt)', 'f(t) = A sin ωt', 'T = 2π/ω', 'ν = 1/T = ω/2π', 'y = A sin ω(t - x/v)', '∂y/∂t = A ω cos ω(t - x/v)']
  • If the displacement of different particles at t=0 is g(x), the displacement of the particle at x at time t will be y = g(x + vt)
  • The function f in equations (15.1) and (15.2) represents the displacement of the point x = 0 as time passes and g in (15.3) and (15.4) represents the displacement at t = 0 of different particles.
  • When the person vibrates the left end x = 0 in a simple harmonic motion. The equation of motion of this end may then be written as f(t) = A sin ωt, where A represents the amplitude and ω the angular frequency.
  • The time period of oscillation is T = 2π/ω and the frequency of oscillation is ν = 1/T = ω/2π. The wave produced by such a vibrating source is called a sine wave or sinusoidal wave.
  • Since the displacement of the particle at x = 0 is given by f(t) = A sin ωt, the displacement of the particle at x at time t will be y = f(t - x/v) or, y = A sin ω(t - x/v).
  • The velocity of the particle at x at time t is given by ∂y/∂t = A ω cos ω(t - x/v).

Velocity of a Wave on a String

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⚡ Quick Summary
The speed of a wave on a string depends on how tightly the string is stretched (tension) and how heavy it is (mass per unit length). A tighter string or a lighter string results in a faster wave.
['v = √(F/μ)', 'Where:\nv = wave velocity\nF = tension in the string\nμ = mass per unit length (linear mass density)']
The velocity of a wave traveling on a string depends on the elastic and inertia properties of the string. * **Elastic Property:** Measured by the tension (F) in the string. * **Inertia Property:** Measured by the mass per unit length (μ). Derivation: 1. Consider a small element of the string of length Δl at the highest point of a crest, forming an arc of radius R. 2. The element moves in a circle with speed v. 3. The resultant force on the element due to tension is Fr = F(Δl/R). 4. The mass of the element is Δm = Δlμ. 5. The downward acceleration of the element is a = Fr/Δm = F/(μR). 6. Since the element is moving in a circle, its acceleration is also a = v²/R. 7. Equating the two expressions for acceleration, we get v²/R = F/(μR). 8. Solving for v, we get the velocity of the wave on the string. Approximation: The derivation assumes that the tension F remains almost unchanged as the string vibrates up and down, valid for small amplitudes.

Superposition of Waves

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⚡ Quick Summary
When two or more waves overlap, the resulting displacement at any point is the sum of the displacements each wave would cause individually. This principle is generally valid for small disturbances (linear waves). When waves overlap, it's called interference. Each wave contributes to the disturbance independently of the others.
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Principle of Superposition

  • When two or more waves simultaneously pass through a point, the disturbance at the point is given by the sum of the disturbances each wave would produce in the absence of the other wave(s).
  • This principle is generally valid for small disturbances only (linear waves).
  • Nonlinear waves do not obey the superposition principle.

Interference of Waves

  • When two or more waves pass through the same region simultaneously, we say that the waves interfere or the interference of waves takes place.
  • Each wave makes its own contribution to the disturbance, no matter what the other waves are doing.

Interference of Waves Going in the Same Direction

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⚡ Quick Summary
When two sinusoidal waves with the same frequency travel in the same direction, their amplitudes and phase difference determine the resulting wave. The principle of superposition is used to find the resultant wave, which is also a sinusoidal wave with a possibly different amplitude and phase.
y = y1 + y2 y1 = A1 sin(kx - ωt) y2 = A2 sin(kx - ωt + δ) y = A sin(kx - ωt + ε) A cos ε = A1 + A2 cosδ A sin ε = A2 sinδ

Interference of Waves Going in the Same Direction

Consider two identical sources sending sinusoidal waves of the same angular frequency ω in the positive x-direction. The wave velocity and wave number k are the same for both waves.

Let the amplitudes of the two waves be A1 and A2, and let them differ in phase by an angle δ. Their equations can be written as:

  • y1 = A1 sin(kx - ωt)
  • y2 = A2 sin(kx - ωt + δ)

According to the principle of superposition, the resultant wave is:

y = y1 + y2 = A1 sin(kx - ωt) + A2 sin(kx - ωt + δ)

Expanding and combining terms:

y = sin(kx - ωt) (A1 + A2 cosδ) + cos(kx - ωt) (A2 sinδ)

Let:

  • A cos ε = A1 + A2 cosδ
  • A sin ε = A2 sinδ

Then, the resultant wave can be written as:

y = A sin(kx - ωt + ε)

Reflection of Waves

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⚡ Quick Summary
Waves can be reflected at boundaries. Reflection at a free end doesn't invert the wave, while reflection from a fixed end does. When a wave moves from a less dense to a denser medium, the reflection is inverted, and vice versa. Transmitted waves are never inverted.
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  • When a wave pulse reaches a free end, the lack of restoring force causes the end to overshoot, which is equivalent to an extra force acting from the right, sending a wave from right to left with the same shape as the original. The incident and reflected waves add, resulting in reflection without inversion.
  • When a wave pulse is produced on a light string moving towards a junction with a heavier string, part of the wave is reflected and part is transmitted. The reflected wave is inverted.
  • If the wave is produced on the heavier string moving towards the junction with a lighter string, part will be reflected and part transmitted, without inversion.
  • If a wave enters a region where wave velocity is smaller, the reflected wave is inverted. If it enters a region where the wave velocity is larger, the reflected wave is not inverted. The transmitted wave is never inverted.

Standing Waves

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⚡ Quick Summary
Standing waves are formed when two identical waves travel in opposite directions. They have nodes (points of zero displacement) and antinodes (points of maximum displacement). Energy cannot be transmitted across nodes.
['y = (2A cos kx) sin ωt', 'kx = (n + 1/2)π (for nodes)', 'x = (n + 1/2)λ/2 (for nodes)']
  • Standing waves are produced by the interference of two sine waves with equal amplitude and frequency propagating in opposite directions. The equations of the two waves are given by:
    • y1 = A sin(ωt - kx)
    • y2 = A sin(ωt + kx + δ)
  • When δ = 0, the resultant displacement is: y = (2A cos kx) sin ωt
  • Each particle vibrates in simple harmonic motion with an amplitude |2A cos kx|.
  • Nodes are points where the amplitude is zero (|2A cos kx| = 0). This occurs when kx = (n + 1/2)π, or x = (n + 1/2)λ/2, where n is an integer.
  • Antinodes are points where the amplitude is maximum (|cos kx| = 1).
  • At a given time, all particles with positive cos kx reach their positive maximum displacement, and those with negative cos kx reach their negative maximum displacement.
  • The separation between consecutive nodes or antinodes is λ/2.
  • Energy cannot be transmitted across nodes.

Standing Waves and Resonance

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⚡ Quick Summary
When a string fixed at both ends is vibrated at specific frequencies, it forms standing waves with nodes and antinodes. Resonance occurs when the length of the string is a multiple of half-wavelengths, leading to large amplitude vibrations.
['λ = v/ν (Relationship between wavelength, velocity, and frequency)', 'L = nλ/2 (Resonance condition for a string fixed at both ends)', 'ν = n v / 2L = n / 2L * √(F/µ) (Natural frequencies of a string)', 'ν₀ = 1 / 2L * √(F/µ) (Fundamental frequency of a string)', 'F = Tension in the string', 'µ = Linear mass density (mass per unit length)']
Standing waves are formed when waves propagating in opposite directions interfere. For a string fixed at both ends, resonance occurs when the length (L) is an integral multiple of half the wavelength (λ/2), i.e., L = nλ/2, where n is an integer. This allows constructive interference and builds up the amplitude. At resonance, the string absorbs energy efficiently from the source. The frequencies at which resonance occurs are called natural/normal/resonant frequencies.

Normal Modes of a String Fixed at One End

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⚡ Quick Summary
A string fixed at one end can vibrate in specific patterns called normal modes, each with a distinct frequency. Only odd harmonics are allowed as overtones.
kL = (n + 1/2)π ; ν = ((n + 1)/2) * (v/2L) = ((n + 1)/2) * (√(F/µ) / L) where n = 0, 1, 2,... ; ν₀ = v/4L (fundamental frequency) ; νₙ = (2n+1)ν₀ where n = 1, 2, 3,... (overtones)
Standing waves can be produced on a string fixed at one end (x=0) and free at the other (x=L). The displacement is given by y = 2A sin(kx) cos(ωt). The boundary condition at x=0 (fixed end) is a node, and at x=L (free end) is an antinode. This leads to the condition sin(kL) = ±1 or kL = (n + 1/2)π, where n is an integer. This gives the allowed frequencies. When a string of a musical instrument such as a sitar is plucked aside at some point, its shape does not correspond to any of the normal modes discussed above. In fact, the shape of the string is a combination of several normal modes and thus, a combination of frequencies are emitted.

Normal Modes of a String Fixed at Both Ends

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⚡ Quick Summary
A string fixed at both ends can vibrate in specific patterns called normal modes, each with a distinct frequency.
ν = ((n + 1)/2) * (v/L) = ((n + 1)/2) * (√(F/µ) / L) where n = 0, 1, 2,...
For a string of length L fixed at both ends, the nodes occur at x = 0, L/3, 2L/3, and L for the third harmonic. In general, for the nth overtone, there are n nodes between the ends and n+1 antinodes midway between the nodes.

Laws of Transverse Vibrations of a String (Sonometer)

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⚡ Quick Summary
The fundamental frequency of a vibrating string depends on its length, tension, and mass per unit length.
ν ∝ 1/L ; ν ∝ √F ; ν ∝ 1/√µ
The fundamental frequency of vibration of a string fixed at both ends is given. From this, the Laws of Transverse Vibrations of a String are derived: (a) Law of length: ν ∝ 1/L if F and µ are constants. (b) Law of tension: ν ∝ √F if L and µ are constants. (c) Law of mass: ν ∝ 1/√µ if L and F are constants.

Law of Length

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⚡ Quick Summary
The frequency of vibration is inversely proportional to the length of the vibrating string when tension is constant.
ν ∝ 1/l, l₁ν₁ = l₂ν₂
Frequency (ν) is inversely proportional to length (l). Mathematically, ν ∝ 1/l. Therefore, l₁ν₁ = l₂ν₂ where l₁ and ν₁ are the initial length and frequency, and l₂ and ν₂ are the final length and frequency.

Law of Tension

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⚡ Quick Summary
The frequency of vibration is directly proportional to the square root of the tension in the string when length is constant.
ν ∝ √T, l' ∝ 1/√T, l'√T = constant
Frequency (ν) is proportional to the square root of tension (T). Mathematically, ν ∝ √T. Thus, l' ∝ 1/√T, implying l'√T is constant.

Transverse Waves

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Waves in which the displacement of the particles is perpendicular to the direction of wave propagation.
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In transverse waves, the disturbance or displacement of the particles of the medium is perpendicular to the direction in which the wave travels. Examples include waves on a string and light waves (electric field oscillations).

Longitudinal Waves

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Waves in which the displacement of the particles is along the direction of wave propagation.
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In longitudinal waves, the particles of the medium are displaced parallel to the direction of wave propagation. Sound waves are longitudinal.

Law of Mass

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⚡ Quick Summary
The frequency of vibration is inversely proportional to the square root of the mass per unit length of the string when length and tension are constant.
ν ∝ 1/√µ, l' ∝ √µ, l'/√µ = constant
Frequency (ν) is inversely proportional to the square root of mass per unit length (µ). Mathematically, ν ∝ 1/√µ. Thus, l' ∝ √µ, implying l'/√µ is constant.

Wave Speed on a String

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⚡ Quick Summary
The speed of a transverse wave on a string depends on the tension in the string and the linear mass density of the string.
['v = sqrt(F/μ)', 'Y = (F/A) / (ΔL/L)', 'ΔL = (FL) / (AY)']
The speed of a transverse wave traveling on a string is given by the formula v = sqrt(F/μ), where F is the tension in the string and μ is the linear mass density (mass per unit length) of the string.

Young's Modulus and Extension

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⚡ Quick Summary
Young's modulus relates stress (force per unit area) to strain (fractional change in length). It can be used to calculate the extension of a wire under tension.
['Y = (F/A) / (ΔL/L)', 'ΔL = (FL) / (AY)']
Young's modulus (Y) is defined as the ratio of tensile stress to tensile strain. Tensile stress is the force (F) applied per unit area (A), and tensile strain is the change in length (ΔL) divided by the original length (L). Therefore, Y = (F/A) / (ΔL/L). This formula can be rearranged to calculate the extension: ΔL = (FL) / (AY).

Wavelength and Wave Speed Relationship

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⚡ Quick Summary
The speed of a wave is related to its frequency and wavelength by the equation v = νλ. In a medium with varying tension, the wavelength changes while the frequency remains constant.
['v = νλ', 'v ∝ sqrt(F) (if μ and ν are constant)', 'sqrt(F) / lambda = constant (if μ and ν are constant)']
The relationship between wave speed (v), frequency (ν), and wavelength (λ) is given by v = νλ. When a wave travels through a medium with varying properties (like a rope with changing tension), the wave speed changes, and consequently, the wavelength also changes to maintain a constant frequency (assuming the source frequency is constant). If frequency is constant, then v is proportional to lambda.

Wave Properties and Formulas

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⚡ Quick Summary
This section provides formulas for calculating fundamental frequencies in strings, relating frequency, length, tension, and linear mass density. It also covers harmonic frequencies and their relationship to the fundamental frequency and string length.
['ν = (1 / 2L) * √(F/µ)', 'ν_n = n * (1 / 2L) * √(F/µ) where n = 1, 2, 3...']
  • Fundamental Frequency: The lowest frequency at which a string can vibrate.
  • Harmonics: Integer multiples of the fundamental frequency. A string can vibrate at these higher frequencies as well.
  • Relationship between frequency, length, tension, and linear mass density: The frequency of vibration of a string is directly proportional to the square root of the tension and inversely proportional to the length and the square root of the linear mass density.

Sonometer Wire

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⚡ Quick Summary
A sonometer wire's vibration frequency is affected by its length, tension, and density. Changing these properties alters the wire's natural frequency and how it vibrates when excited by a tuning fork.
f ∝ 1/l, f ∝ √T, f ∝ 1/√ρ (where f is frequency, l is length, T is tension, and ρ is density)
  • The fundamental frequency of a sonometer wire is inversely proportional to its length.
  • The frequency is directly proportional to the square root of the tension.
  • The frequency is inversely proportional to the square root of the density.
  • When a tuning fork is used to vibrate a sonometer wire, the wire will vibrate at the frequency of the tuning fork if the tuning fork's frequency is close to one of the natural frequencies of the wire. This phenomenon is called resonance or forced vibration.
  • Standing waves can be produced on a string clamped at one end and free at the other, and the length of the string must be an odd integral multiple of λ/4 (λ being the wavelength).

Wave Properties

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⚡ Quick Summary
Waves can be transverse or longitudinal. Transverse waves involve particle motion perpendicular to the wave's direction, while longitudinal waves involve particle motion parallel to the wave's direction. Waves transmit energy and have properties like wavelength, frequency, and amplitude.
v = fλ, y = A sin(ωt + kx), v = ω/k
  • Mechanical waves propagate in a medium and can be transverse or longitudinal.
  • Transverse waves: Particles of the medium move perpendicular to the direction of wave propagation. Example: waves on a string.
  • Longitudinal waves: Particles of the medium move parallel to the direction of wave propagation. Example: sound waves in a gas.
  • Longitudinal waves cannot be polarized, while transverse waves can.
  • A wave is represented by the equation y = A sin(ωt + kx), where A is the amplitude, ω is the angular frequency, t is time, k is the wave number, and x is the position.
  • Wave velocity (v) = ω/k = fλ (where f is frequency and λ is wavelength).
  • Phase difference of π between two particles implies they move in opposite directions.
  • In a traveling wave, energy is transmitted along the wave. The energy of small parts of the string does not remain constant.
  • In a stationary wave, all the particles between consecutive nodes vibrate in phase.

Transverse Wave on a String

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⚡ Quick Summary
This section deals with the properties of transverse waves traveling on a string, including speed, tension, linear mass density, power, superposition and interference.
['v = sqrt(T/μ)', 'μ = m/L', 'k = 2π/λ', 'ω = 2πf', 'P_avg = (1/2) * μ * v * ω^2 * A^2']
  • Speed of a Transverse Wave on a String: The speed (v) of a transverse wave on a string is determined by the tension (T) in the string and the linear mass density (μ) of the string.
  • Linear Mass Density: Linear mass density (μ) is defined as the mass per unit length of the string.
  • Wave Equation: A general form of a transverse wave equation can be expressed as y = A sin(kx - ωt + φ), where:
    • y is the displacement of the string at position x and time t,
    • A is the amplitude of the wave,
    • k is the wave number (k = 2π/λ, where λ is the wavelength),
    • ω is the angular frequency (ω = 2πf, where f is the frequency),
    • φ is the phase constant.
  • Power Transmission in a Wave: The average power (P) transmitted by a wave on a string is proportional to the square of the amplitude, the square of the frequency, the wave speed, and the linear mass density.
  • Superposition of Waves: When two or more waves travel through the same region of space, the displacement of the medium at any point is the sum of the displacements due to each individual wave.
  • Interference: Interference occurs when two or more waves superpose to create a resultant wave of greater, lower, or the same amplitude. If two waves have a phase difference, the resultant amplitude depends on this phase difference.

Standing Waves on a String

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⚡ Quick Summary
Deals with standing waves, harmonics, and related calculations like frequency, wavelength, and wave speed. Problems involve strings fixed at both ends and concepts like Young's modulus and density.
['Equation of standing wave: y = (Amplitude) * sin(kx) * cos(ωt)', 'Frequency of nth harmonic: f_n = n * f_1 (where f_1 is the fundamental frequency)', 'Wave speed: v = sqrt(Tension / Linear mass density)', "Young's modulus: Y = (Stress / Strain)"]
Standing waves are produced by the interference of two waves traveling in opposite directions. The equation for a standing wave can be expressed in terms of sine and cosine functions, incorporating parameters like amplitude, wavelength, and frequency. Harmonics are integer multiples of the fundamental frequency and determine the modes of vibration. The frequency of vibration depends on the length of the string, tension, and mass per unit length. Problems involve finding frequencies, wavelengths, positions of nodes, and relating these parameters to physical properties like Young's modulus and density.