Fundamentals of Waves

🎯 Learning Objectives

Understand the types and properties of waves, Learn mathematical representations of traveling and standing waves, Explore the relationship between wavelength, frequency, and velocity, Analyze wave energy, power, and intensity, Examine the principle of superposition, interference, and diffraction

Introduction to Waves:

A wave is a disturbance that travels through a medium (or space) transferring energy without transporting matter. Waves are ubiquitous in nature:

Mechanical waves: Require a medium (e.g., sound waves in air, water waves, seismic waves).
Electromagnetic waves: Do not require a medium (e.g., light, radio waves).
Matter waves: Quantum mechanical wave functions representing probabilities.

Signal domain illustration — **Figure 1.** Signal domains: physical, analog, and digital.

Key Properties of Waves:

Waves are characterized by:

Property	Symbol	Description
Wavelength	$λ$	Distance between consecutive peaks or troughs
Frequency	$f$	Number of oscillations per second (Hz)
Angular frequency	$ω$	$ω = 2πf$ (rad/s)
Wave number	$k$	$k = 2π/λ$ (rad/m)
Velocity	$v$	Speed of wave propagation; $v = fλ$
Amplitude	$A$	Maximum displacement from equilibrium

Traveling Waves:

A one-dimensional traveling wave can be expressed as:

y(x,t) = A \, \sin(kx - \omega t + \phi)

Where:

$A$ is the amplitude
$k = \frac{2\pi}{\lambda}$ is the wavenumber
$\omega = 2\pi f$ is the angular frequency
$\phi$ is the phase constant

Wave velocity is given by:

v = \frac{\omega}{k} = f \lambda

Note: The negative sign in the argument indicates propagation in the $+x$ direction. For propagation in $-x$ , use $kx + ωt$ .

Harmonic Waves

The general sinusoidal wave form:

y(x,t) = A \, \cos(kx - \omega t)

The energy of a mechanical wave on a string of linear density μ and tension T:

E = \frac{1}{2} \mu \omega^2 A^2 \lambda

The power transmitted by the wave:

P = \frac{1}{2} \mu \omega^2 A^2 v

Wave Equation:

The classical wave equation in one dimension:

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

This equation governs all linear, non-dispersive waves.
For three-dimensional waves, the Laplacian generalizes the spatial derivative:

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

Superposition Principle:

The principle of superposition states:

If two or more waves coexist in the same medium, the resultant displacement is the algebraic sum of individual displacements.

y_{total} = y_1 + y_2 + \dots + y_n

This leads to interference:

Constructive interference: Peaks align → amplitude increases
Destructive interference: Peaks align with troughs → amplitude decreases

Standing Waves

Formed by two traveling waves in opposite directions:

y(x,t) = 2A \, \sin(kx) \, \cos(\omega t)

Nodes: Points of zero amplitude, (kx = n\pi)
Antinodes: Points of maximum amplitude, (kx = (n + 0.5)\pi)

Energy and Intensity:

For a mechanical wave:

I = \frac{1}{2} \rho v \omega^2 A^2

Where:

$I$ = intensity (power per unit area)
$rho$ = medium density
$v$ = wave speed
$A$ = amplitude

For sound waves, intensity is related to decibels (dB):

L = 10 \log_{10} \frac{I}{I_0}

Wave Types:

Transverse waves: Oscillation perpendicular to propagation (light, water surface waves)
Longitudinal waves: Oscillation parallel to propagation (sound in air)
Surface waves: Combination of longitudinal and transverse (ocean waves)

Transverse Wave

Longitudinal Wave

One-DOF Mass-Spring System

A single-degree-of-freedom mass-spring system has one natural mode of oscillation.

One mass connected to one spring oscillates back and forth at the frequency $ω=(\frac{s}{m})^2$

One mass, connected to two springs in parallel, oscillates back and forth at the slightly higher frequency $ω=(2\frac{s}{m})^2$

Two-DOF Mass-Spring System

A two degree-of-freedom system (consisting of two identical masses connected by three identical springs) has two natural modes, each with a separate resonance frequency.

The second natural mode of oscillation occurs at a frequency of $ω=(3\frac{s}{m})^2$ At this frequency, the two masses move with the same amplitude but in opposite directions so that the coupling spring between them alternately stretched and compressed. The center of the coupling spring does not move; this location is called a node.

Three-DOF Mass-Spring System

N-DOF Mass-Spring System

Dispersion and Phase Velocity:

Phase velocity: $v_p = \frac{\omega}{k}$
Group velocity: $v_g = \frac{d\omega}{dk}$
Dispersion occurs when $v_p \neq v_g$

Boundary Conditions:

Reflection at fixed or free boundaries
Transmission into a new medium
Standing waves in finite strings or tubes:
Open-open, open-closed, closed-closed tubes
Resonance frequencies: $f_n = n \frac{v}{2L}$ (strings or open tubes)

Wave Packets and Fourier Analysis:

Real-world signals are superpositions of sinusoids
Any periodic signal $s(t)$ can be expressed using Fourier series:

s(t) = \sum_{n=1}^{\infty} [a_n \cos(n \omega_0 t) + b_n \sin(n \omega_0 t)]

Non-periodic signals use Fourier transform:

S(f) = \int_{-\infty}^{\infty} s(t) e^{-j 2 \pi f t} dt

🧪 Interactive Deoms

Interactive 3D Wave Simulation

Wave Type: travelingFrequency: 1.0 HzAmplitude: 1.0Wave Length: 10.0

Frequency: 5.0 Hz

Amplitude: 1.0

Source Distance: 2.0 m

Wave Speed: 0.50

Wave Type Visualizer

Select Wave Type:

Particles oscillate ⬆️⬇️ perpendicular to wave motion — e.g., light, water surface ripples.

📝 Key Takeaways

🧠 Quick Quiz

Test your understanding - select and submit an answer.

1) What is the relationship between wave speed, frequency, and wavelength?

A) v = f / λ
B) v = λ / f
C) v = f × λ
D) v = λ² × f

2) What type of wave has particle motion perpendicular to wave propagation?

A) Longitudinal
B) Transverse
C) Surface
D) Sound

3) The wave equation in 1D is:

A) ∂²y/∂x² = v² ∂²y/∂t²
B) ∂²y/∂x² = (1/v²) ∂²y/∂t²
C) ∂y/∂x = (1/v²) ∂y/∂t
D) ∂²y/∂t² = v ∂²y/∂x²

4) RMS voltage is particularly useful for:

A) Peak amplitude
B) Power calculations
C) Frequency analysis
D) Wavelength

5) For two waves with same amplitude, perfectly in phase, what happens?

A) Destructive interference
B) Constructive interference
C) No change
D) Reflection

6) The first node in a standing wave on a string fixed at both ends occurs at:

A) x = 0
B) x = λ/4
C) x = λ/2
D) x = λ

7) The group velocity equals the phase velocity when:

A) Wave is dispersive
B) Wave is non-dispersive
C) Wavelength is zero
D) Amplitude is constant

8) A square wave with peak voltage 5 V has RMS voltage:

A) 5 V
B) 3.54 V
C) 2.5 V
D) 7.07 V

9) Which type of wave can travel without a medium?

A) Sound
B) Seismic
C) Electromagnetic
D) Water wave

10) The Fourier transform decomposes a signal into:

A) Time-domain components
B) Frequency-domain components
C) Energy-domain components
D) Phase-domain components

Chapter - 2

Key Words: Wave properties, Sound waves, Mechanical waves