Mechanical Waves: Properties, Types, and Wave Functions

Mechanical Waves

Definition and Examples

Mechanical waves are disturbances that propagate through a medium at a well-defined velocity. These waves transfer energy and momentum but do not transport mass from one place to another.

• Examples: Water waves, sound waves, stadium waves.

Types of Waves

Classification by Particle Motion

• Transverse Waves: The medium's particles move perpendicular to the direction of wave propagation. Example: Waves on a string.

• Longitudinal Waves: The medium's particles move parallel to the direction of wave propagation. Example: Sound waves.

• Torsional Waves: The medium undergoes twisting motion that propagates through the medium.

Wave Pulses

Localized Disturbances

A wave pulse is a single, localized disturbance that travels through a medium at the medium's wave velocity. Unlike periodic waves, pulses do not repeat.

Periodic Waves

Repetitive Structure

Periodic waves have a repeating structure, with each point in the medium undergoing repetitive motion. A snapshot of a periodic wave shows the displacement of the medium from equilibrium at a given instant.

Periodic-Wave Properties

Key Quantities

• Amplitude (A): Maximum displacement of a point on the wave.

• Period (T): Time for a point on the wave to complete one oscillation. $f = \frac{1}{T}$ $f = \frac{\omega}{2\pi}$

• Wavelength (\lambda): Distance between identical points (e.g., crest to crest) on the wave.

Sinusoidal Waves

Simple Harmonic Motion (SHM)

Sinusoidal waves are a special type of periodic wave where each point in the medium undergoes simple harmonic motion (SHM).

• SHM Solution: $x(t) = A \cos(\omega t + \phi)$ Where:

  • $A$ = amplitude

  • $\omega$ = angular frequency ($\omega = \sqrt{\frac{k}{m}}$)

  • $\phi$ = phase constant

• Velocity in SHM: $v(t) = -A\omega \sin(\omega t + \phi)$ The velocity is maximum when displacement is zero, and zero when displacement is maximum.

Wave Function

Mathematical Description

The wave function describes the displacement of the wave at every point in space ($x$) and time ($t$).

• For a sinusoidal wave: $y(x, t) = A \cos(\omega t - kx)$ Where $k = \frac{2\pi}{\lambda}$ is the wave number.

• For a wave moving in the $+x$ direction: $y(x, t) = A \cos(\omega t - kx)$

• For a wave moving in the $-x$ direction: $y(x, t) = A \cos(\omega t + kx)$

Wave Speed

Dependence on Medium

• The speed of a wave ($v$) depends only on the properties of the medium, not on amplitude, wavelength, or period.

• Relationship between speed, wavelength, and period: $v = \frac{\lambda}{T}$ $v = \lambda f$

• For a wave on a string: $v = \sqrt{\frac{F}{\mu}}$ Where $F$ is the tension and $\mu$ is the mass per unit length.

• Increasing tension increases wave speed; increasing mass per unit length decreases wave speed.

Principle of Superposition

Wave Interference

When two or more waves meet at a point, their displacements add algebraically. This is known as the principle of superposition.

• Constructive Interference: When displacements add to produce a larger amplitude.

• Destructive Interference: When displacements partially or completely cancel each other.

Standing Waves on a String

Formation and Properties

Standing waves are formed by the superposition of two waves traveling in opposite directions. For a string fixed at both ends, only certain wavelengths are allowed, resulting in nodes (points of zero displacement) and antinodes (points of maximum displacement).

• Wave function for standing wave: $y_{tot} = 2A \sin\left(\frac{2\pi x}{\lambda}\right) \sin\left(\frac{2\pi t}{T}\right)$

• Node positions: $x = n \frac{\lambda}{2}$, where $n = 0, 1, 2, ...$

• Allowed wavelengths for string of length $L$: $L = n \frac{\lambda}{2}$

• Fundamental mode (1st harmonic): $\lambda = 2L$

• Frequency of nth harmonic: $f_n = n \frac{v}{2L}$

Summary Table: Key Properties of Mechanical Waves

Property	Symbol	Definition
Amplitude	A	Maximum displacement from equilibrium
Period	T	Time for one complete oscillation
Frequency	f	Oscillations per second ($f = 1/T$)
Wavelength	$\lambda$	Distance between identical points on wave
Wave speed	v	Speed at which wave propagates ($v = \lambda f$)

Example Problem

Boat and Wave Crests

A boat is moored and waves make it move up and down. If the spacing between wave crests is 20 meters and the speed of the waves is 5 m/s, the time to go from the top of a crest to the bottom of a trough is:

• Wavelength $\lambda = 20$ m

• Speed $v = 5$ m/s

• Period $T = \lambda / v = 20 / 5 = 4$ s

• Time from crest to trough = $T/2 = 2$ s

Answer: 2 seconds. Additional info: This example illustrates the relationship between wave speed, wavelength, and period.