Summary

General Principles

The Wave Model

This model is based on the idea of a traveling wave, which is an organized disturbance traveling at a well-defined wave speed v.

In transverse waves the displacement is perpendicular to the direction in which the wave travels.
In longitudinal waves the particles of the medium are displaced parallel to the direction in which the wave travels.

A wave transfers energy, but no material or substance is transferred outward from the source.

The figure illustrates a transverse wave above a longitudinal wave one.

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Two basic classes of waves:

Mechanical waves travel through a material medium such as water or air.
Electromagnetic waves require no material medium and can travel through a vacuum.

For mechanical waves, such as sound waves and waves on strings, the speed of the wave is a property of the medium. Speed does not depend on the size or shape of the wave.

Important Concepts

The displacement D of a wave is a function of both position (where) and time (when).

A snapshot graph shows the wave’s displacement as a function of position at a single instant of time.

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A history graph shows the wave’s displacement as a function of time at a single point in space.

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For a transverse wave on a string, the snapshot graph is a picture of the wave. The displacement of a longitudinal wave is parallel to the motion; thus the snapshot graph of a longitudinal sound wave is not a picture of the wave.

Sinusoidal waves are periodic in both time (period T) and space (wavelength $λ$ ):

\begin{matrix} D (x, t) & = & A sin [2 π (x / λ - t / T) + ϕ_{0}] \\ = & A sin (k x - ω t + ϕ_{0}) \end{matrix}

where A is the amplitude, $k = 2 π / λ$ is the wave number, $ω = 2 π f = 2 π / T$ is the angular frequency, and $ϕ_{0}$ is the phase constant that describes initial conditions.

The fundamental relationship for any sinusoidal wave is $v = λ f$ .

Applications

String (transverse): $v = \sqrt{T_{s} / μ}$
Sound (longitudinal): $v = \sqrt{B / ρ} = 343 m / s$ in $20 ° C$ air
Light (transverse): $v = c / n$ , where $c = 3.00 \times 10^{8} m / s$ is the speed of light in a vacuum and n is the material’s index of refraction

The wave intensity is the power-to-area ratio: $I = P / a$

For a circular or spherical wave: $I = P_{source} / 4 π r^{2}$

The sound intensity level is

β = (10 dB) \log_{10} (I / 1.0 \times 10^{- 12} W / m^{2})

The Doppler effect occurs when a wave source and detector are moving with respect to each other: the frequency detected differs from the frequency $f_{0}$ emitted.

Approaching source	Observer approaching a source
$f_{+} = \frac{f_{0}}{1 - v_{s} / v}$	$f_{+} = (1 + v_{o} / v) f_{0}$
Receding source	Observer receding from a source
$f_{-} = \frac{f_{0}}{1 + v_{s} / v}$	$f_{-} = (1 - v_{o} / v) f_{0}$

The Doppler effect for light uses a result derived from the theory of relativity.