5: Interference
- Page ID
- 134582
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Explain how the principle of superposition applies to waves
- Define interference, constructive interference, destructive interference, beats, beat frequency, difference tone
- Apply the principle of superposition
- Explain these phenomena using the principle of superposition:
- beats
- standing waves
- reflection
- nodal (and antinodal) lines
- Determine whether two sources will constructively (or destructively interfere) at any given location
Experience with sound suggests that waves don’t collide like particles do. When two people talk simultaneously during a conversation, each person hears both people- not just themselves. This simple observation suggests two important properties of not just sound, but all waves. First, waves (sound or otherwise) from different sources can (and do) pass through each other. Second, waves from multiple sources can co-exist at the same location at the same time.
This chapter explores what happens when waves overlap. When waves overlap, the individual waves add together to form a single wave. The math is addition. The central idea is called the principle of superposition:
At all locations where two (or more) waves (or pulses) overlap, the displacement of the resulting wave (or pulse) is equal to the sum of the displacements each wave (or pulse) would have if it was at the location alone.
Most of the time, superposition leads to the results you expect. When two people are talking at you at the same time, what you hear is both people at once. When you hear two instruments play two notes together, you usually hear two notes playing together.
However, the results of superposition can sometimes be very surprising. The height of a wave at any location can be either “positive” or “negative,” so when waves occupy the same location, they can up to zero! In other words, sounds can “cancel each other out” and at any moment and at any location. Most of the time, the cancellation is temporary and has no practical effect on the sound. However, depending on the types of sounds and how the sound waves line up, superposition can lead to interesting and noticeable effects.
- 5.1: Principle of superposition
- This page covers wave interference and the principle of superposition, describing how overlapping waves combine to create a new wave based on their individual displacements. It differentiates between constructive interference (aligned crests) and destructive interference (crest meeting trough), applicable universally but often simplified in practice. The page also includes problem-solving strategies for analyzing wave interactions using the principle of superposition.
- 5.2: Sine waves, phase and interference
- This page explains phase difference in sine waves, detailing how they can be in phase (0 degrees) or out of phase (180 degrees). The amplitude of the resulting wave is affected by this phase relationship: constructive interference occurs with in-phase waves, while destructive interference happens with out-of-phase waves. When neither condition is met, the resulting amplitude falls between the sum and difference of the original wave amplitudes.
- 5.3: How to make standing waves
- This page details the formation of standing waves via superposition, illustrating how two identical, oppositely traveling waves create nodes with no motion. It uses the example of sound waves from two speakers to demonstrate standing waves and acoustic levitation. Additionally, it explains the relationship between the wavelength of standing waves and the distance between nodes, and briefly discusses the effects of waves with varying amplitudes or traveling in the same direction.
- 5.4: Beats
- This page explores beats, a sound phenomenon resulting from the interference of two nearly identical pitches, producing a fluctuating volume. The beat frequency is determined by the frequency difference, with smaller differences yielding lower frequencies of beats. As the frequency difference increases, listeners transition from perceiving a single pitch to recognizing distinct sounds. Additionally, varying amplitude levels can influence how beats are perceived.
- 5.5: Interference in two dimensions
- This page covers two-dimensional interference patterns created by point sources of waves, detailing how sound waves from identical sources can interfere constructively or destructively based on travel distances. It explores factors such as source spacing, phase, and frequency in determining patterns, including nodal and antinodal lines.
- 5.6: Interference- Review and homework
- This page explains wave interference concepts such as constructive and destructive interference, phase, beats, and difference tones. It includes review questions about wave interactions, noise canceling technology, and sound perception affected by phase. Additionally, it provides numerical problems on sound basics and interference patterns, highlighting nodal and antinodal lines with practical examples. Lastly, it suggests using Audacity for hands-on sound manipulation experience.
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