Wave Dynamics and Graph Types

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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?

The displacement of the medium is the sum of the displacement due to each individual wave. Signup and view all the answers

What are Sinusoidal waves generated by?

A source oscillating with Simple Harmonic Motion (SHM) Signup and view all the answers

What does K represent in wave terminology?

Wavenumber Signup and view all the answers

What is the equation for K?

k = 2π/λ Signup and view all the answers

What is ω in wave terminology?

Angular frequency Signup and view all the answers

What is the equation for Phase Velocity?

Vph = ω/k = fλ Signup and view all the answers

What is the definition of Group Velocity?

How the overall 'envelope' of disruption moves Signup and view all the answers

What does the Transverse Velocity represent?

The velocity at which a point on the string is moving in the y-direction Signup and view all the answers

What assumptions are made with waves on a string?

String under tension T, linear density μ, zero stiffness, neglect Fg, no acceleration in x-direction Signup and view all the answers

What occurs when there is a Density Change from high to low μ?

Reflected wave has smaller amplitude than yi, transmitted wave is broader Signup and view all the answers

What is a Normal Mode?

Allowed frequency for standing wave starting at n = 1 Signup and view all the answers

What does a Fourier Series represent?

Almost any periodic function can be represented by a combination of cos and sin terms + a constant Signup and view all the answers

What is characteristic of an odd function in Fourier Series?

Sine terms only Signup and view all the answers

What defines an even function in Fourier Series?

Cos terms only Signup and view all the answers

What happens at a discontinuity in a Fourier Series?

The corners will get sharper and sharper but will never actually be pointed Signup and view all the answers

What are Nodes in a standing wave?

Points of zero amplitude on a standing wave Signup and view all the answers

What are Anti-nodes?

Points of maximum amplitude on a standing wave 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.

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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.