Mechanical, Electromagnetic, and Matter Waves
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Questions and Answers

Which of the following best distinguishes matter waves from mechanical waves?

  • Matter waves are associated with moving particles, while mechanical waves involve the disturbance of a medium. (correct)
  • Matter waves are governed by Newton's laws, while mechanical waves are not.
  • Matter waves always travel at the speed of light, whereas mechanical waves travel at varying speeds.
  • Matter waves require a medium, whereas mechanical waves can travel through a vacuum.

A wave is traveling through a medium. If the particles of the medium oscillate perpendicular to the direction of wave propagation, the wave is classified as:

  • Transverse (correct)
  • Longitudinal
  • Electromagnetic
  • Mechanical

Which of the following media can support the propagation of longitudinal waves?

  • Solids, liquids, and gases (correct)
  • Gases only
  • Liquids only
  • Solids only

What is the primary restoring force for capillary waves?

<p>Surface tension (B)</p> Signup and view all the answers

A wave is described by the equation $y(x, t) = a \sin(kx - \omega t + \phi)$. Which term represents the phase constant?

<p>$\phi$ (D)</p> Signup and view all the answers

In the context of wave motion, what is the physical significance of the 'crest'?

<p>The point of maximum positive displacement in a wave. (D)</p> Signup and view all the answers

If the angular frequency of a wave is doubled, what happens to its period if all other factors remain constant?

<p>The period is halved. (A)</p> Signup and view all the answers

A wave has a frequency of 4 Hz. What is its period?

<p>0.25 s (A)</p> Signup and view all the answers

How does the speed of a wave on a stretched string change if the tension in the string is quadrupled, assuming the linear mass density remains constant?

<p>The speed is doubled. (A)</p> Signup and view all the answers

What does the Laplace correction address concerning Newton's formula for the speed of sound in a gas?

<p>It corrects for the isothermal conditions assumed by Newton, considering adiabatic conditions instead. (D)</p> Signup and view all the answers

When two waves overlap in a medium, what principle is used to determine the resultant displacement?

<p>Superposition principle (D)</p> Signup and view all the answers

During the reflection of a wave at a rigid boundary, what phase change occurs in the reflected wave?

<p>A phase change of $\pi$ radians (D)</p> Signup and view all the answers

In a standing wave, what are the positions of zero amplitude called?

<p>Nodes (D)</p> Signup and view all the answers

For a standing wave in a string fixed at both ends, what is the condition for the formation of nodes?

<p>$x = \frac{n\lambda}{2}$, where n = 0, 1, 2, 3,... (B)</p> Signup and view all the answers

What is the relationship between the fundamental frequency ($v_1$) and the length (L) of a string fixed at both ends?

<p>$v_1 = \frac{v}{2L}$ (D)</p> Signup and view all the answers

In a closed pipe (closed at one end), which harmonics are present?

<p>Only odd harmonics (C)</p> Signup and view all the answers

What is the condition for the antinode in a closed pipe?

<p>$L = (n + \frac{1}{2}) \frac{\lambda}{4}$ (A)</p> Signup and view all the answers

What is the relationship between the frequencies of successive harmonics in an open pipe?

<p>They are integer multiples of the fundamental frequency. (A)</p> Signup and view all the answers

When two waves with slightly different frequencies interfere, what phenomenon is observed?

<p>Beats (C)</p> Signup and view all the answers

If two sound waves of frequencies 250 Hz and 255 Hz are superimposed, what is the beat frequency?

<p>5 Hz (D)</p> Signup and view all the answers

Flashcards

What are Waves?

Patterns that transfer energy without physical material transfer.

What are Mechanical Waves?

Waves requiring a material medium for propagation and governed by Newton's laws.

What are Electromagnetic Waves?

Waves that do not require a medium for propagation; travels through vacuum at speed of light.

What are Matter Waves?

Waves associated with moving subatomic particles like electrons and protons.

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What are Transverse Waves?

Waves where particles oscillate perpendicular to wave propagation direction.

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What are Longitudinal Waves?

Waves where particles oscillate parallel to the direction of wave propagation.

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What is a Crest?

Point of maximum positive displacement in a wave.

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What is a Trough?

Point of maximum negative displacement in a wave.

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What is Amplitude?

Maximum displacement of elements from their equilibrium position.

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What is Phase?

Describes the state of motion as wave sweeps through a string element

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What is Phase Constant?

The initial phase angle in wave equations.

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What is Wavelength?

Minimum distance between two consecutive troughs or crests.

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What is Angular Wave Number?

Parameter defined as 2π / wavelength.

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What is Period?

Time taken for any element to complete one oscillation.

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What is Frequency?

Number of oscillations per unit time made by an element.

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Wave speed on a string

The speed of a wave on a string depends on tension and linear mass density.

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What are Beats?

Phenomenon of periodic variations in sound intensity due to superposition.

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What are Nodes?

Positions of zero amplitude in a standing wave.

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What are Antinodes?

Positions of maximum amplitude in a standing wave.

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Superposition Principle

Algebraic sum of individual wave displacements when two or more waves overlap

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Study Notes

  • Waves are patterns that move without the physical transfer or flow of matter
  • The three main types of waves are mechanical waves, electromagnetic waves, and matter waves

Mechanical Waves

  • Mechanical waves require a material medium such as water, air, or rock for propagation
  • They're governed by Newton's laws
  • Examples include water waves, sound waves, and seismic waves

Electromagnetic Waves

  • Electromagnetic waves do not need a medium to propagate
  • All electromagnetic waves travel through a vacuum at the speed of light (c), which equals 3 x 10^8 m/s
  • Examples include visible light, ultraviolet light, radio waves, microwaves, and x-rays

Matter Waves

  • Matter waves are associated with moving electrons, protons, neutrons, other fundamental particles, atoms, and molecules
  • Matter waves associated with electrons are used in electron microscopes

Transverse and Longitudinal Waves

  • Mechanical waves can be transverse or longitudinal
  • This depends on the relationship between the direction of vibrations of particles in the medium and the propagation of the wave

Transverse Waves

  • The medium's constituents oscillate perpendicular to the direction of wave propagation
  • They travel in the form of crests and troughs
  • Transverse waves can only propagate in solids and strings, not fluids
  • Waves on a stretched string are an example

Longitudinal Waves

  • The constituents of the medium oscillate along the direction of wave propagation
  • They travel as compressions and rarefactions
  • Longitudinal waves can propagate through solids, liquids, and gases
  • Sound waves and vibrations in a spring are examples

Capillary and Gravity Waves

  • Capillary and gravity waves are types of waves on the surface of water

Capillary Waves

  • Capillary waves are ripples with short wavelengths, not more than a few centimeters
  • Surface tension of water is the restoring force

Gravity Waves

  • Gravity waves have wavelengths ranging from meters to hundreds of meters
  • Gravity is their restoring force, working to keep the water surface at its lowest level

Traveling (Progressive) Waves

  • Traveling or progressive waves are transverse or longitudinal
  • These waves travel from one point of a medium to another

Displacement Relation (Progressive Wave Along a String)

  • A progressive wave traveling along the positive x-axis can be represented as y(x, t) = a sin(kx – ωt + φ)
  • A progressive wave traveling along the negative x-axis can be represented as y(x, t) = a sin(kx + ωt + φ)

Wave Crest

  • A crest corresponds to a point of maximum positive displacement in a wave

Wave Trough

  • A trough corresponds to a point of maximum negative displacement in a wave

Amplitude

  • The amplitude 'a' represents the magnitude of the maximum displacement of elements from their equilibrium positions
  • Amplitude is a positive quantity

Phase

  • Phase describes the state of motion as the wave sweeps through a string element at a particular position x
  • Phase is the argument (kx – ωt + φ) of the oscillatory term sine
  • It changes linearly with time t

Phase Constant

  • The constant ϕ is called the initial phase angle
  • Its value is determined by the initial displacement (at t=0) and velocity of the element (at x=0)

Wavelength

  • Wavelength (λ) is the minimum distance between two consecutive troughs or crests
  • It's also the distance between two consecutive points in the same phase of wave motion

Propagation Constant (Angular Wave Number)

  • Propagation constant or angular wave number (k) = 2π / λ
  • The SI unit of angular wave number is radians per meter (rad m⁻¹)

Period

  • The period (T) is the time taken by an element to finish one complete oscillation

Angular Frequency

  • Angular frequency (ω) = 2π / T and its SI unit is radians per second (rad s⁻¹)
  • T can be calculated; T = 2π / ω

Frequency

  • Frequency (𝜈) is the number of oscillations per unit time made by an element as the wave passes
  • Frequency (𝜈) = 1 / T or 𝜈 = ω / 2π and is usually measured in hertz

Traveling Wave Speed

  • For a wave propagating in the positive x direction with initial phase ϕ = 0, the wave can be expressed as y(x, t) = a sin(kx - ωt)
  • As the wave moves, each point on the waveform retains its displacement y
  • The argument (kx - ωt) remains constant
  • wave speed (v) = ω / k

General Relation for Progressive Waves

  • The speed of a wave is related to its wavelength and frequency
  • Although it is determined by the properties of the medium
  • wave speed (v) = 𝜈λ

Transverse Wave Speed on a Stretched String

  • Wave speed (v) = √(T/μ), where T is tension and μ is linear mass density

Longitudinal Wave Speed (Speed of Sound)

  • Longitudinal waves travel through a compressions and rarefactions or changes in medium density (ρ)

Speed of Propagation in a Fluid

  • Speed (v) = √(B/ρ), where B is the bulk modulus of the fluid

Speed of Longitudinal Wave in Solid Bar

  • Speed (v) = √(Y/ρ), where Y is Young's modulus

Speed of a Longitudinal Wave (Ideal Gas)

  • Using Newton's Formula, assuming isothermal conditions during sound propagation, Speed (v) = √(B/ρ) where B=P
  • Using Laplace correction to Newton's formula assuming adiabatic conditions
  • Speed (v) = √(γP/ρ)

Laplace Correction

  • Laplace stated pressure variations during sound propagation occur so rapidly that there is no time for heat transfer to maintain constant temperature
  • The Laplace correction leads to v = √(γP/ρ), where γ = Cp/Cv

Speed of Sound in Air

  • The speed of sound in air at STP(Standard Temperature and Pressure) is 331.3 m/s

Superposition Principle

  • When two or more waves overlap, the resultant displacement is the algebraic sum of individual displacements
  • If y₁(x, t) and y₂(x, t) are individual waves, then the resultant displacement y(x, t) = y₁(x, t) + y₂(x, t)
  • For two waves with same amplitude, angular frequency, and wavenumber traveling in positive x direction but differing in initial phase (ϕ), the resultant displacement y(x, t) = [2a cos(Φ/2)] sin(kx-ωt + Φ/2)

Resultant Wave Analysis

  • The resultant wave is sinusoidal
  • The initial phase of resultant wave = Φ/2
  • The amplitude of resultant wave is A= 2a cos(Φ/2)

Waves in Phase

  • If Φ = 0, the two waves are in phase
  • The resultant displacement y(x,t) = 2a sin(kx – ωt)
  • The amplitude A = 2a, the largest possible value

Reflected Waves- Rigid Boundary

  • The reflected wave experiences a phase reversal (phase difference of π radians or 180°)
  • There is no displacement at the boundary as the string is fixed

Incident Wave (Rigid Boundary)

  • yi(x, t) = a sin(kx – ωt)

Reflected Wave (Rigid Boundary)

  • yr(x, t) = a sin(kx + ωt + π) or -a sin(kx + ωt)

Reflected Waves- Open Boundary

  • The reflected wave has the same sign (no phase reversal) and amplitude as the incident wave
  • There is maximum displacement at the boundary
  • Twice the amplitude of either of the pulses

Incident Wave (Open Boundary)

  • yi(x, t) = a sin(kx – ωt)

Reflected Wave (Open Boundary)

  • yr(x, t) = a sin(kx + ωt)

Standing Waves and Normal Modes

  • Standing waves are produced by the interference of two identical waves moving in opposite directions

Incident Wave

  • Wave travelling in the positive direction of x-axis
  • y1(x, t) = a sin(kx – ωt)

Reflected Wave

  • Wave travelling in the negative direction of x-axis
  • y2(x, t) = a sin(kx + ωt)

Standing Wave Equation

  • By the principle of superposition
  • y(x, t) = (2a sin kx) cos ωt
  • The waveform does not move
  • The amplitude of the wave = 2a sin kx

Nodes

  • Positions of zero amplitude in a standing wave

Antinodes

  • Positions of maximum amplitude in a standing wave

Condition for Nodes

  • wave amplitude = 0
  • 2a sin kx=0
  • kx= nπ, for n = 0, 1, 2, 3,..
  • Nodes are at x=n(λ/2); n=0, 1, 2, 3,..

Condition for Antinodes

Positions of maximum amplitude in standing wave: kx = (n + ½), or (n = 0, 1, 2, 3, ...)

Nodes and Antinodes Separation

  • The antinodes are separated by λ/2
  • Located half way between pairs of nodes

Standing Waves (Stretched String Fixed at Both Ends)

  • For a stretched string with length L fixed at both ends that has two ends at x= 0 and x= L,
  • The condition for a node at L is L = n(λ/2), (n = 1, 2, 3, ...)

Fundamental Mode (First Harmonic)

  • The oscillation mode with n=1
  • The lowest frequency is called the fundamental mode or the first harmonic.
  • L = λ1/2, λ1 = 2L
  • Frequency is described as 𝝂𝟏 = v/2L

String Harmonics (2nd harmonic)

  • Second harmonic L=2(λ2/2)
  • Frequency is described as 𝝂2 = 2(v/2L)
  • Second harmonic (𝝂2) = to 2 times the first harmonic (𝟐𝝂𝟏)

String Harmonics (3rd harmonic)

  • Third harmonic L=3(λ3/2)
  • Frequency is described as 𝝂3 = 3(v/2L), or 3 (𝝂1)
  • The relationship between modes in a string is given as 𝝂𝟏 : 𝝂𝟐 : 𝝂𝟑 = 𝟏: 𝟐: 𝟑

Modes of Vibration (Closed Pipe)

  • System is closed at one end and open at the other

Calculating Condition for Antinode at L

  • L = (n+ ½)
  • Where n = 0, 1, 2, 3, ...
  • (λ/ 2)
  • The fundamental mode/ first harmonic is the oscillation mode where n=0

Closed Pipe Fundamental Mode or 1st Harmonic

  • L=(λ1)/4
  • Frequency, is calculated as; 𝝊𝟏 = v/4L

Third Harmonic in Closed Pipe

  • L=3(λ3)/4
  • The third harmonic is the oscillation mode where n = 1
  • Frequency, Is calculated as: 𝝊𝟑 = 3v/4L and 𝝊𝟑 = 3𝝂𝟏

Fifth Harmonic in Closed Pipe

  • L=5(λ4)/4
  • The fifth harmonic is the oscillation mode and n=2
  • Frequency, is calculated as 𝝊 = 5V/4L = =5 𝝂𝟏
  • The ratios can be written as: 𝝂𝟏 : 𝝂𝟑 : 𝝂𝟓 = 𝟏: 𝟑: 𝟓 and only odd harmonics are possible in a closed pipe

Modes of Vibration (Open Pipe)

  • This is a system open at both ends
  • For this type of length L, antinodes are formed at both ends.
  • L = n (λ/2)
  • Where, n = 1, 2, 3,...

Open Pipe Fundamental Mode/First Harmonic

  • L = a λ/2
  • The oscillation mode with n =1
  • Is also known as lowest frequency
  • Frequency, of first harmonic is found as 𝝊 = V/2L

Second Harmonic in Open Pipe

  • If L equals 2 λ/4
  • Frequency, will be described as 𝝊 =2(V2L) also equaling 2𝝂1

Third Harmonic in Open Pipe

  • L = 3(λ*3/2)
  • And can conclude frequency is, as is described; 𝝊 = 3(V/2L) , also: 3v1
  • Wave patterns can then be described as 𝝂𝟏 : 𝝂𝟐 : 𝝂𝟑 = 𝟏: 𝟐: 𝟑
  • All harmonics, thus, are possible in an open pipe, and are preferred to closed pipes in musical instruments.

Beats

  • These are periodic variations to the intensity of sounds such as a wavering traveling at similar amplitudes
  • Also know as periods of waxing and waning, beat frequencies can be expressed as 𝝂=𝝂-𝝂

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Description

Overview of mechanical, electromagnetic, and matter waves. Mechanical waves require a medium and follow Newton's laws. Electromagnetic waves do not require a medium and travel at the speed of light. Matter waves are associated with moving particles.

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