# Diffraction

- Page ID
- 3353

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This little document was written as a Supplementary Course Note for the course PHY132F - *Introduction to Physics II*.

## Course Specifics

All section and equation numbers refer to Randall D. Knight **Physics for Scientists and Engineers**, 2^{nd} ed. (Pearson Addison-Wesley, 2008).

We have dropped the details of *diffraction * from the syllabus. The dropped material appears in the text in §22.4 and §22.5. However, there are some qualitative aspects of the phenomenon of diffraction that we will wish to know, and this document discusses them.

## Review of Two Slit Interference

You will want to recall that for two slit *interference *, the angle between successive maxima is approximately given by:

\[\theta_m=m\frac{\lambda}{d} ~~,m=0,1,2,3,... \tag{Eqn 22.4}\]

As discussed in class:

- As the distance between the slits
*d*is decreased, the angles of the maxima increase. Small distances correspond to a spread out interference pattern; large distances correspond to an interference pattern that gets squished. - The position of the maxima scales as the wavelength divided by the distance between the slits. This means that the in this context the phrases small distances and large distances used above mean in comparison to the wavelength of the waves.

## Diffraction

The photograph to the right shows a water wave propagating to the right and passing through a gap in a sea wall. As you can see, the wave spreads out after it has passed through the slit This phenomenon is called *diffraction*, and occurs for all waves.

You can see in the photograph that the wavelength of the water waves is about the size of the width of the slit.

You can also see that the amplitude of the diffracted wave is largest in the center, and decreases as the magnitude of the angle from the horizontal increases.

It turns out that, similar to the case for interference, as the width of the slit is decreased the diffraction pattern spreads out more, and as the width of the slit is increased the pattern gets squished. In fact, if the width or the slit is much much larger than the wavelength of the wave, essentially no diffraction occurs and the wave continues propagating as a plane wave.

These facts are used in the design of loudspeakers.

Speakers that produce bass notes are called "woofers". They need to move a lot of air, so need to be big. Diameters of 30 cm or more are common.

However, a typical high treble note has a frequency of 5000 Hz or so, which corresponds to a wavelength of 0.07 m. This is much less than the diameter required for a woofer, so if we try to use the woofer to generate a high frequency note the sound wave will beam straight ahead without significant diffraction. Thus you won't hear those notes unless you are right in front of the speaker.

Thus speakers to produce high frequency notes, "tweeters", must have a much smaller diameter than required for the woofer. In the speaker to the right the woofer is above and the tweeter is below. A simple electric circuit inside the enclosure directs low frequencies to the woofer and high frequencies to the