Questions and Answers
What is a History Graph?
Description of wave at a single point in space throughout all time
What is a Snapshot Graph?
Description of wave at one instant of time throughout all space
What is the general form of the Wave Equation?
y(x,t) = f(x - vt) + g(x + vt)
What does the Principle of Superposition state?
Signup and view all the answers
What are Sinusoidal waves generated by?
Signup and view all the answers
What does K represent in wave terminology?
Signup and view all the answers
What is the equation for K?
Signup and view all the answers
What is ω in wave terminology?
Signup and view all the answers
What is the equation for Phase Velocity?
Signup and view all the answers
What is the definition of Group Velocity?
Signup and view all the answers
What does the Transverse Velocity represent?
Signup and view all the answers
What assumptions are made with waves on a string?
Signup and view all the answers
What occurs when there is a Density Change from high to low μ?
Signup and view all the answers
What is a Normal Mode?
Signup and view all the answers
What does a Fourier Series represent?
Signup and view all the answers
What is characteristic of an odd function in Fourier Series?
Signup and view all the answers
What defines an even function in Fourier Series?
Signup and view all the answers
What happens at a discontinuity in a Fourier Series?
Signup and view all the answers
What are Nodes in a standing wave?
Signup and view all the answers
What are Anti-nodes?
Signup and view all the answers
Study Notes
Wave Graph Types
- History Graph: Displays wave behavior at a single point over time (y vs. t at a fixed x).
- Snapshot Graph: Illustrates the wave shape across space at a particular instant (y vs. x at a fixed t).
Wave Fundamentals
- Wave Equation: Describes wave motion; solutions take the form y(x,t) = f(x - vt) + g(x + vt) indicating both right and left traveling waves.
- Principle of Superposition: States that the total displacement at a point from multiple waves is the sum of individual displacements.
Sinusoidal Waves
- Characteristics: Generated from a source oscillating in simple harmonic motion (SHM) described by y(x,t) = Acos(kw - ωt).
- Relationship: Velocity relates to wavelength and frequency via v = λf = λ/T.
Wavenumber and Angular Frequency
- Wavenumber (k): Defined as the number of wave cycles in a unit distance, k = 2π/λ, representing spatial wave density.
- Angular Frequency (ω): Indicates how rapidly the wave oscillates over time, showing the phase shift at specific points in time.
Wave Representation
- Complex Notation: Represents waves using complex numbers; the real part describes the wave's displacement y(x,t) = Re( Ae^(i(kx - ωt + Φ₀)).
- Phase: Given by y(x,t) = Acos(Φ); it defines the position in the wave cycle.
Phase Properties
- Phase Difference: Quantified as ΔΦ = 2π(Δx/λ) = kΔx, which measures the difference in phase between points in space.
- Phase Velocity: The velocity at which a point of constant phase travels, expressed as Vph = ω/k = fλ.
Velocities in Waves
- Group Velocity: Indicates the speed of the wave's overall envelope, which represents energy transfer.
- Transverse Velocity (Vy): Velocity of a point on a string in the y-direction, defined mathematically by ∂y/∂t = ∂/∂t(Acos(kx - ωt)).
Wave Behavior on Strings
- Assumptions: For waves on a string, consider tension, linear density, neglect of gravity, and no acceleration in the x-direction. Resulting behaviors affect transmitted and reflected waves.
Reflection and Transmission
- Density Change: Waves transitioning from high to low density (μ2 < μ1) result in a faster transmitted wave (v2 > v1) and a positive reflection coefficient.
- Boundary Behavior: Reflection of a wave from a boundary results in inversion, with the frequency remaining constant, while wave characteristics like velocity and wavelength may change.
Energy Conservation
- Energy Balance: The power incident at a boundary equals the sum of reflected and transmitted power |Pinst,i| = |Pinst,r| + |Pinst,t|.
Standing Waves
- Formation: Created by two sinusoidal waves of equal amplitude and frequency traveling in opposite directions, resulting in nodes (zero amplitude) and anti-nodes (maximum amplitude).
- Normal Modes: Frequencies allowed for standing waves, with the first harmonic corresponding to wavelength λ = 2L.
Fourier Series
- Representation: Almost any periodic function can be expressed as a sum of cosine and sine terms, providing a powerful method for analyzing waveforms.
-
Types of Functions:
- Odd Functions: Only sine terms, originating at the f(x) axis.
- Even Functions: Composed solely of cosine terms, symmetric about the f(x) axis.
- Neither: A mix of sine and cosine terms, not symmetric.
- Discontinuities: Square waves, which are discontinuous, will appear smoother in Fourier representation but will never have sharp corners regardless of the terms used.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Explore the different types of wave graphs, including history and snapshot graphs. Understand wave fundamentals such as the wave equation, principle of superposition, and characteristics of sinusoidal waves. Delve into key concepts like wavenumber and angular frequency to grasp wave behavior more effectively.