Summary

General Principles

Principle of Superposition

The displacement of a medium when more than one wave is present is the sum at each point of the displacements due to each individual wave.

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Important Concepts

Standing Waves

Standing waves are due to the superposition of two traveling waves moving in opposite directions.

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The amplitude at position x is

A (x) = 2 a sin k x

where a is the amplitude of each wave.

The boundary conditions determine which standing-wave frequencies and wavelengths are allowed. The allowed standing waves are modes of the system.

The figure illustrates two standing waves labeled m equals 1, m equals 2, and m equals 3.

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Solving Interference Problems

Maximum constructive interference occurs where crests are aligned with crests and troughs with troughs. The waves are in phase.

Maximum destructive interference occurs where crests are aligned with troughs. The waves are out of phase.

Model Model the wave as linear, circular, or spherical.

Visualize Find distances to the sources.

Solve Interference depends on the phase difference $Δ ϕ$ between the waves:

\begin{array}{l} Constructive : & Δ ϕ = 2 π \frac{Δ r}{λ} + Δ ϕ_{0} = m \cdot 2 π \\ Destructive : & Δ ϕ = 2 π \frac{Δ r}{λ} + Δ ϕ_{0} = (m + \frac{1}{2}) \cdot 2 π \end{array}

$Δ r$ is the path-length difference of the two waves, and $Δ ϕ_{0}$ is any phase difference between the sources. For identical (in-phase) sources:

Constructive : Δ r = m λ Destructive : Δ r = (m + \frac{1}{2}) λ

Review Is the result reasonable?

The figure illustrates nodal and antinode points on overlapping circular fronts.

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Applications

Boundary Conditions

Strings, electromagnetic waves, and sound displacement waves in closed-closed tubes must have nodes at both ends:

λ_{m} = \frac{2 L}{m} f_{m} = m \frac{v}{2 L} = m f_{1} m = 1, 2, 3, \dots

The frequencies and wavelengths are the same for a sound wave in an open-open tube, which has antinodes at both ends.

A sound displacement wave in an open-closed tube must have a node at the closed end but an antinode at the open end. This leads to

λ_{m} = \frac{4 L}{m} f_{m} = m \frac{v}{4 L} = m f_{1} m = 1, 3, 5, 7, \dots

Beats (loud-soft-loud-soft modulations of intensity) occur when two waves of slightly different frequency are superimposed.

The figure illustrates a graphical representation of beats.

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The beat frequency between waves of frequencies $f_{1}$ and $f_{2}$ is

f_{beat} = | f_{1} - f_{2} |