Podcast
Questions and Answers
Which of the following best distinguishes matter waves from mechanical waves?
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:
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?
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?
What is the primary restoring force for capillary waves?
A wave is described by the equation $y(x, t) = a \sin(kx - \omega t + \phi)$. Which term represents the phase constant?
A wave is described by the equation $y(x, t) = a \sin(kx - \omega t + \phi)$. Which term represents the phase constant?
In the context of wave motion, what is the physical significance of the 'crest'?
In the context of wave motion, what is the physical significance of the 'crest'?
If the angular frequency of a wave is doubled, what happens to its period if all other factors remain constant?
If the angular frequency of a wave is doubled, what happens to its period if all other factors remain constant?
A wave has a frequency of 4 Hz. What is its period?
A wave has a frequency of 4 Hz. What is its period?
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?
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?
What does the Laplace correction address concerning Newton's formula for the speed of sound in a gas?
What does the Laplace correction address concerning Newton's formula for the speed of sound in a gas?
When two waves overlap in a medium, what principle is used to determine the resultant displacement?
When two waves overlap in a medium, what principle is used to determine the resultant displacement?
During the reflection of a wave at a rigid boundary, what phase change occurs in the reflected wave?
During the reflection of a wave at a rigid boundary, what phase change occurs in the reflected wave?
In a standing wave, what are the positions of zero amplitude called?
In a standing wave, what are the positions of zero amplitude called?
For a standing wave in a string fixed at both ends, what is the condition for the formation of nodes?
For a standing wave in a string fixed at both ends, what is the condition for the formation of nodes?
What is the relationship between the fundamental frequency ($v_1$) and the length (L) of a string fixed at both ends?
What is the relationship between the fundamental frequency ($v_1$) and the length (L) of a string fixed at both ends?
In a closed pipe (closed at one end), which harmonics are present?
In a closed pipe (closed at one end), which harmonics are present?
What is the condition for the antinode in a closed pipe?
What is the condition for the antinode in a closed pipe?
What is the relationship between the frequencies of successive harmonics in an open pipe?
What is the relationship between the frequencies of successive harmonics in an open pipe?
When two waves with slightly different frequencies interfere, what phenomenon is observed?
When two waves with slightly different frequencies interfere, what phenomenon is observed?
If two sound waves of frequencies 250 Hz and 255 Hz are superimposed, what is the beat frequency?
If two sound waves of frequencies 250 Hz and 255 Hz are superimposed, what is the beat frequency?
Flashcards
What are Waves?
What are Waves?
Patterns that transfer energy without physical material transfer.
What are Mechanical Waves?
What are Mechanical Waves?
Waves requiring a material medium for propagation and governed by Newton's laws.
What are Electromagnetic Waves?
What are Electromagnetic Waves?
Waves that do not require a medium for propagation; travels through vacuum at speed of light.
What are Matter Waves?
What are Matter Waves?
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What are Transverse Waves?
What are Transverse Waves?
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What are Longitudinal Waves?
What are Longitudinal Waves?
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What is a Crest?
What is a Crest?
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What is a Trough?
What is a Trough?
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What is Amplitude?
What is Amplitude?
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What is Phase?
What is Phase?
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What is Phase Constant?
What is Phase Constant?
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What is Wavelength?
What is Wavelength?
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What is Angular Wave Number?
What is Angular Wave Number?
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What is Period?
What is Period?
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What is Frequency?
What is Frequency?
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Wave speed on a string
Wave speed on a string
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What are Beats?
What are Beats?
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What are Nodes?
What are Nodes?
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What are Antinodes?
What are Antinodes?
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Superposition Principle
Superposition Principle
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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.