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Types of Mechanical Waves

Level 1 - Physics topic page in Mechanical Waves.

Principle

Mechanical waves are organized by how the medium moves compared with how the disturbance travels.

The central idea is to keep two motions separate: local oscillation of the medium and transport of the wave pattern.

Notation

\(\vec \xi\)
displacement of a small part of the medium
\(\mathrm{m}\)
\(\vec v_w\)
wave-propagation velocity
\(\mathrm{m\,s^{-1}}\)
\(\lambda\)
wavelength
\(\mathrm{m}\)
\(T\)
period
\(\mathrm{s}\)
\(f\)
frequency
\(\mathrm{Hz}\)

Method

Derivation 1: Separate medium motion from wave motion

A wave is a moving pattern. A small piece of the medium usually oscillates about equilibrium instead of traveling with the pattern.

Local displacement
\[\vec \xi(t)\]
Pattern speed
\[\vec v_w=\frac{\Delta \vec r_{\text{same phase}}}{\Delta t}\]
Follow a crest, compression, or any other fixed phase point.

Derivation 2: Classify the wave by direction

Once those two directions are separated, the wave type is set by geometry.

Transverse case
\[\vec \xi\perp \vec v_w\]
The medium moves across the direction the pattern travels.
Longitudinal case
\[\vec \xi\parallel \vec v_w\]
The medium moves along the same line as the wave propagation.

For the transverse snapshot below, the plotted curve shows the shape of the medium at one instant. The particle displacement is vertical, while the wave pattern advances horizontally.

0π/2π3π/22π-1.2-0.600.61.2xyparticlevw

The same logic classifies sound in air as longitudinal: air parcels oscillate back and forth along the same line that the compression pattern travels.

Derivation 3: Recover the standard wave-speed relation

A periodic wave repeats after one period. During that time, a fixed phase point moves forward by one wavelength.

One-period shift
\[v_w=\frac{\lambda}{T}\]
Use frequency
\[f=\frac{1}{T}\]
Wave-speed form
\[v_w=f\lambda\]

Rules

These are the compact results from the method above.

Transverse condition
\[\vec \xi\perp \vec v_w\]
Longitudinal condition
\[\vec \xi\parallel \vec v_w\]
Wave speed
\[v_w=\frac{\lambda}{T}=f\lambda\]

Examples

Question
A wave on a string has wavelength
\[0.80\,\mathrm{m}\]
and frequency
\[12\,\mathrm{Hz}\]
Find its speed and identify the wave type.
Answer
The speed is
\[v_w=f\lambda=(12)(0.80)=9.6\,\mathrm{m\,s^{-1}}\]
A string wave is transverse because the string elements move across the direction of propagation.

Checks

  • A material point in the medium is not usually carried forward with the wave.
  • The classification uses the direction of local displacement, not the shape of the graph.
  • A wave can be periodic and still be either transverse or longitudinal.
  • Use the speed of the pattern, not the speed of an individual particle in the medium.

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Periodic Wave Patterns