Engineering Physics - Waves

Introduction to Waves

• Major Aim: Understanding the classification of waves based on physical properties (mechanical, water, EM, sound, and matter waves).
• Discussion Criteria: Longitudinal, transverse, and mixed waves.
• Dimensionality: 1-D, 2-D, and 3-D waves.
• Topics: Waves on a string, wave equations, harmonic waves, reflection and transmission at boundaries, standing waves and eigenfrequencies.
• Reference: Chapter 5, Section 5.2 from 'The Physics of Vibrations and Waves’, 1st Edition, H J Pain and P Rankin, John Wiley and Sons Ltd, 2015.

Waves: The Concept

• Definition: A wave is a disturbance that propagates in a medium and transfers energy without a net transfer of medium particles.
• Forms: Elastic deformation, pressure variation, electric or magnetic intensity, electric potential, or temperature.
• Examples: Sound waves, water waves, seismic waves, radio waves, light, and microwaves.
• Crowd Wave: An example of a wave.

Characteristics of Waves

• Crest: The highest point of a wave.
• Trough: The lowest point of a wave.
• Time Period (T): Time taken for the wave to complete one oscillation.
• Velocity (v):
  $v = λ / T$

Classification of Waves

• Based on Medium:
  • Mechanical Waves
  • Non-mechanical Waves (Electromagnetic Waves)
  • Matter Waves
• Based on Vibrations:
  • Transverse Waves
  • Longitudinal Waves
  • Mixed Waves
• Based on Propagation Dimension:
  • 1D Waves
  • 2D Waves
  • 3D Waves

Classification by Medium

• Mechanical Waves: Require a medium to travel; cannot travel in a vacuum.

• Electromagnetic Waves: Do not require a medium; can travel in a vacuum. Examples include microwaves, X-rays, radio waves, ultraviolet waves, and light.

• Matter Waves: Associated with each particle; described by the De-Broglie wavelength.

  \λ = h / p

  • Cannot be understood by classical theories; require Quantum Physics.

Classification by Vibration

• Transverse Waves: Vibrations are perpendicular to the direction of propagation. Examples include EM waves and water waves.
• Longitudinal Waves: Vibrations are along the direction of propagation. Example: Sound waves.
  *Note: Electromagnetic waves can only be transverse waves.

Sound Waves: Longitudinal Waves

• Description: A wave of compression and rarefaction that propagates in an elastic medium such as air.

EM Waves: Transverse Waves

• Propagation: Propagate by varying electric and magnetic fields perpendicular to each other.

Surface Waves

• Characteristics: Exhibit properties of both longitudinal and transverse waves.
• Motion: Particles move up and down as well as back and forth, resulting in an overall circular motion.

1D, 2D & 3D Waves

• Classification: Based on dimensionality.

Mathematical Interpretation of Waves

• Description: Representing a disturbance moving with speed $v$ in the forward direction.
• Equation:
  • At $t = 0$, $y(x, 0) = f(x)$
  • At time $t = t<em>1$, $y(x, t</em>1) = y(x - vt, 0) = f(x - vt)$
  • General form: $y(x, t) = f(x ± vt)$

Vector Resolution

• Definition: Splitting a single vector into two or more vectors in different directions which together produce a similar effect as is produced by a single vector itself.
• Component Vectors: The vectors formed after splitting.

Concept of Partial Derivative

• For any function of two independent variables, $g(x,y)$
• First order partial derivative
  $\frac{{\partial g}}{{\partial x}} = \lim_{\Delta x \to 0} \frac{{g(x+\Delta x,y) - g(x,y)}}{{\Delta x}}$
• Second order partial derivative
  $\frac{{\partial^2 g}}{{\partial x^2}} = \lim_{\Delta x \to 0} \frac{{\frac{{\partial g}}{{\partial x}}(x+\Delta x, y) - \frac{{\partial g}}{{\partial x}}(x, y)}}{{\Delta x}}$

Wave Equation: 1D Wave on a String

• Assumptions:

  1. The string is perfectly flexible and offers no resistance to bending; tension is tangential.
  2. Points move only vertically; no horizontal motion.
  3. Gravitational forces are negligible.

• Parameters:

  • Linear mass density: $\rho$
  • Tension: $T$

• Derivation:

  • Net force along x direction = 0

    $F<em>1 \cos(\theta</em>1) = F<em>2 \cos(\theta</em>2) = T$

  • Net force along y direction:

    $\Delta F = F<em>1 \sin(\theta</em>1) - F<em>2 \sin(\theta</em>2) = \rho \Delta x a_y$

  • Approximation:

    $\Delta F = T[\tan(\theta<em>1) - \tan(\theta</em>2)] = T [\frac{\partial y}{\partial x}<em>{x+ \Delta x} - \frac{\partial y}{\partial x}</em>{x}]$
    $\Delta F = T \frac{\partial^2 y}{\partial x^2} \Delta x = \rho \Delta x \frac{\partial^2 y}{\partial t^2}$

  • Wave Equation:

    $\frac{\partial^2 y}{\partial x^2} = \frac{\rho}{T} \frac{\partial^2 y}{\partial t^2}$

  • Wave Velocity:

    $v = \sqrt{\frac{T}{\rho}}$

  • Final Wave Equation:

    $\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}$

• Dimensional Analysis:

  • $[v] = [LT^{-1}]$

Wave Equation: General

• Description: A second-order linear partial differential equation.
• Origin: Newton’s second law of motion, $F = ma$
• Application: Vibrations of a String.
• Derivation Steps: Using a 1-D infinite string.

Examples

• Problem: A copper wire is pulled by a tension $T = 0.98$ N. The density is $\rho = 9.86 g/cm$. Compute the speed of the wave.

• Solution:

  $v = \sqrt{\frac{T}{\rho}} = 1.01 m/s$

Verifying Wave Equation Solutions

• Problem: Show that $f(x, t) = A \sin[B(x - vt)]$ satisfies the wave equation.
• Problem: Show that the function $y(x,t) = Ae^{-b (x-vt)^2}$ satisfies the wave equation.
• Problem: Show that the function $y(x,t) = \frac{A}{1+B(x+vt)}$ satisfies the wave equation. What is the direction of the wave in this case?

Simple Harmonic Motions

• Definition: A motion where the restoring force is directly proportional to the displacement from the mean position.

  $F = -ky$

• Harmonic waves are sinusoidal.

Harmonic Wave - Displacement

• Model: Amplitude modeled by a sine function.
• Propagation: Propagates one wavelength ($\lambda$) along the x-axis during one time period (T).
• Speed: Constant, $v = \frac{\lambda}{T}$
• Amplitude Range: -A to +A.
• Mathematical Representation:
  • $\omega = \frac{2\pi}{T} \implies T = \frac{2\pi}{\omega}$
  • $k = \frac{2\pi}{\lambda}$
  • $y = A \sin(\frac{2\pi}{\lambda} x + \phi) = A \sin(kx + \phi)$

Harmonic Wave - General Solution and Direction

• General Wave Equation:

  $y(x,t) = A \sin(k x \pm \omega t + \phi)$

• Relating to General Solution:

  $y(x,t) = A \sin(2\pi(\frac{x}{\lambda} \pm \frac{t}{T}))$
  $y(x,t) = A \sin(k x \pm \omega t)$
  Here, v = \frac{\omega}{k} => \omega = kv

• Direction of Propagation:

  • $+$ sign: negative x-direction.
  • $-$ sign: positive x-direction.
  • Note: $k = \frac{\omega}{v} = \omega*\frac{1}{v}$, $[\frac{\omega}{v}] = [\frac{\omega}{x}]$ and $\omega$ = [rad/sec]

Concept of Phase

• General Form:

  $y(x, t) = A \sin(kx \pm \omega t + \phi_0)$

  • $A$ = Amplitude
  • $k$ = Wavenumber
  • $\omega$ = Angular frequency
  • $\phi_0$ = Initial phase

• Temporal Term $\omega t$: Phase of the wave with respect to the reference oscillator.

• Spatial Term $kx$: Displacement of the oscillator.

Properties of SH Waves

1. Amplitude (A): Maximum displacement from the undisturbed position.
2. Wavelength ($\lambda$) / Wavenumber (k):
   • Wavelength is the distance between successive crests/troughs.
   • Wavenumber: $k = \frac{2\pi}{\lambda}$
3. Period (T) and Frequency (f):
   • Period is the time between arrivals of two successive crests/troughs.
   • Frequency is the number of crests/troughs passing a point per unit time.
   • Relationships:
     • $f = \frac{1}{T}$
     • $\omega = \frac{2\pi}{T} = 2\pi f$
4. Velocity (v):
   • Relationship: $v = \frac{\lambda}{T} = \lambda f = \frac{\omega}{k}$

Simple Harmonic Wave Representation

• Problem: For a given wave $y(x, t) = 0.02 \sin(\frac{x}{0.01} + \frac{t}{0.05})$, find amplitude, wavelength, frequency, and velocity.

Simple Harmonic Wave: Mathematical Representation

• Equivalent Expressions:
  • $y(x, t) = f(x ± vt)$
  • $y(x, t) = A \sin(kx ± \omega t + \phi)$
  • $y(x, t) = A \sin(kx ± \omega t)$
  • $y(x, t) = A \sin(\omega(\frac{x}{v} ± t))$
  • $y(x, t) = A \sin(2\pi(\frac{x}{\lambda} ± \nu t))$
  • $y(x, t) = A \sin(\frac{2\pi}{\lambda} (x ± vt))$

Simple Harmonic Wave: Wave Velocity & Particle Velocity

• Right-Moving Wave:

  $\frac{\partial y}{\partial t} = -v \frac{\partial y}{\partial x}$

• Left-Moving Wave:

  $\frac{\partial y}{\partial t} = v \frac{\partial y}{\partial x}$

• Particle Velocity: direction × wave velocity × slope

Simple Harmonic Wave: Particle Motion

• Determining which particles move up/down based on the wave's shape.

Characteristic Impedance of a String

• Impedance: Opposition to wave motion offered by a medium.

• Characteristic Impedance (Z): Impedance offered by the string to transverse waves.

  $Z = \frac{Transverse Force}{Transverse Velocity} = \frac{F<em>y}{v</em>y} = \frac{F_y}{\frac{\partial y}{\partial t}}$

Characteristic Impedance Details

• Transverse Force: $F<em>y = T \sin(\theta</em>1)$

• For small angles: $F_y \approx -T \tan(\theta) = -T(\frac{\partial y}{\partial x})$

• Impedance:

  $Z = \frac{F<em>y}{v</em>y} = \ - \frac{-T \frac{\partial y}{\partial x}}{-v \frac{\partial y}{\partial x}} = \frac{T}{v} = \sqrt{T\rho}$

Wave Travel in Different Medium

• Understanding wave properties when traveling between two different mediums in terms of impedance.

Boundary Conditions

• At the boundary, $x = 0$: Displacement and Tension must be continuous.

• Displacement Continuous:

  $y<em>i(x, t) + y</em>r(x, t) = y_t(x, t) \implies A + B = C$

• Tension Continuous:

  $T(\frac{\partial y}{\partial x})<em>i + T(\frac{\partial y}{\partial x})</em>r = T(\frac{\partial y}{\partial x})<em>t \implies A - B = \frac{k</em>2}{k_1} C$
  Where:

• Medium 1: $Z<em>1, \rho</em>1, v<em>1, k</em>1, \lambda_1$

• Medium 2: $Z<em>2, \rho</em>2, v<em>2, k</em>2, \lambda_2$

Reflection and Transmission Coefficients

• Using Boundary Conditions:

  • $A - B = \frac{k<em>2}{k</em>1} C$
  • $A + B = C$

• Reflection Coefficient:

  $\frac{B}{A} = \frac{k<em>1 - k</em>2}{k<em>1 + k</em>2} = \frac{Z<em>1 - Z</em>2}{Z<em>1 + Z</em>2} = \frac{v<em>2 - v</em>1}{v<em>2 + v</em>1}$

  • Independent of $\omega$ ($k<em>1 = \frac{\omega}{v</em>1}$ and $k<em>2 = \frac{\omega}{v</em>2}$)

• Transmission Coefficient:

  $\frac{C}{A} = \frac{2k<em>1}{k</em>1 + k<em>2} = \frac{2Z</em>1}{Z<em>1 + Z</em>2} = \frac{2v<em>2}{v</em>1 + v_2}$

Wave Travel in Different Medium: Rigid End

• Infinite Impedance: $Z_2 \rightarrow \infty$

• Transmission Coefficient:

  $\frac{C}{A} = 0$

• Reflection Coefficient:

  $\frac{B}{A} = -1$

  • Phase shift of $\pi$ (180 degrees).

• Qualitatively transmitted amplitude approaches zero and wave reflects with a 180-degree phase shift.

Wave Travel in Different Medium: Free End

• Low Impedance: $Z_2 \rightarrow 0$

• Transmission Coefficient:

  $\frac{C}{A} = 2$

  • The coefficient of the transmitted wave will be as large as it can possibly be: TWICE the size of the incident wave

• Reflection Coefficient:

  $\frac{B}{A} = 1$

Boundary with Same Medium

• Equal Impedance: $Z<em>1 = Z</em>2$

• Transmission Coefficient:

  $\frac{C}{A} = 1$

• Reflection Coefficient:

  $\frac{B}{A} = 0$

• No boundary exists; the incident wave keeps traveling.

Reflection and Transmission of Wave in String: Free End

Superposition Principle