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20'</span> </ol> <article id="main-article"> <p><img alt="" src="../../waves/thomas_young__scientist_.jpg" style="width: 197px; float: left; height: 250px;">In Standard Level Waves, you learnt the basics of everything we will observe here, but with a few notable extensions.</p> <p>What is the Doppler effect? What are the effects of a moving source, observer and medium? What are the equations for simple harmonic motion? How is the diffraction pattern formed for light passing through a single slit? What affects the interference patterns for two or more slits? What is thin film interference?</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-turquoise panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Doppler effect</p> </div> </div> <div class="panel-body"> <div> <p>The Doppler effect is the change in frequency in observed sound due to relative motion between source and observer. </p> <p>For a moving source and stationary observer, the observed frequency can be calculated using the following equation:</p> <p style="text-align: center;"><span class="math-tex">\(f'=f({v\over v\pm u_s})\)</span></p> <ul> <li><span class="math-tex">\(f'\)</span> is the observed frequency</li> <li><span class="math-tex">\(f\)</span> is the frequency emitted by the source</li> <li><span class="math-tex">\(v\)</span> is the speed of sound (ms<sup>-1</sup>)</li> <li><span class="math-tex">\(u_s\)</span> is the speed of the source (ms<sup>-1</sup>), where <span class="math-tex">\(u_s>0\)</span> for the source moving towards the observer</li> </ul> <p>If the source is stationary and the observer moves, we adapt the equation:</p> <p style="text-align: center;"><span class="math-tex">\(f'=f({v\pm u_o \over v})\)</span></p> <ul> <li><span class="math-tex">\(f'\)</span> is the observed frequency</li> <li><span class="math-tex">\(f_0\)</span> is the frequency emitted by the source</li> <li><span class="math-tex">\(v\)</span> is the speed of sound (ms<sup>-1</sup>)</li> <li><span class="math-tex">\(u_o\)</span> is the speed of the observer (ms<sup>-1</sup>), where <span class="math-tex">\(u_o>0\)</span> for the source moving towards the observer</li> </ul> <p>A sonic boom is produced when the relative speed of the source exceeds the speed of the wave in the medium.</p> <p>The Doppler effect affects light waves and can be used to determine the relative speeds of objects in the universe. The light of receding objects is red shifted. The light of approaching objects is blue shifted.</p> <p style="text-align: center;"><span class="math-tex">\({\Delta f \over f}={\Delta \lambda \over \lambda} \approx{v\over c}\)</span></p> <ul> <li><span class="math-tex">\(\Delta f\)</span> is the change in observed frequency (Hz)</li> <li><span class="math-tex">\(f\)</span> is the frequency emitted by the source (Hz)</li> <li><span class="math-tex">\(\Delta \lambda\)</span> is the change in the oberved wavelength (m)</li> <li><span class="math-tex">\(\lambda\)</span> is the wavelength emitted by the source (m)</li> <li><span class="math-tex">\(v\)</span> is the speed of the source (ms<sup>-1</sup>)</li> <li><span class="math-tex">\(c\)</span> is the speed of light (ms<sup>-1</sup>)</li> </ul> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-turquoise panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Simple harmonic motion</p> </div> </div> <div class="panel-body"> <div> <p>Simple harmonic motion is defined as oscillations where the acceleration is directly proportional to the displacement from a fixed point and always acts towards that point.</p> <p style="text-align: center;"><span class="math-tex">\(a = -ω^2x\)</span></p> <ul> <li><span class="math-tex">\(a\)</span> is acceleration (ms<sup>-2</sup>)</li> <li><span class="math-tex">\(\omega\)</span> is angular frequency: <span class="math-tex">\(\omega=2\pi f={2\pi\over T}\)</span> (Hz)</li> <li><span class="math-tex">\(x\)</span> is displacement (m)</li> </ul> <p>We can also calculate the displacement and velocity.</p> <p style="text-align: center;"> <span class="math-tex">\(x=x_0\cos{\omega t} \Rightarrow v=-\omega x_0\sin \omega t \)</span></p> <p style="text-align: center;"><span class="math-tex">\(v=\pm \omega\sqrt{{x_0}^2-x^2}\)</span></p> <ul> <li><span class="math-tex">\(x_0\)</span> is amplitude (m)</li> <li><span class="math-tex">\(t\)</span> is time (s)</li> <li><span class="math-tex">\(v\)</span> is velocity (ms<sup>-1</sup>)</li> </ul> <p>When an oscillating system is modelled as a mass on a spring, we can calculate the time period (and therefore frequency) of the oscillations:</p> <p style="text-align: center;"><span class="math-tex">\(T=2\pi \sqrt{m\over k}\)</span></p> <ul> <li><span class="math-tex">\(T\)</span> is time period (s)</li> <li><span class="math-tex">\(m\)</span> is mass (kg)</li> <li><span class="math-tex">\(k\)</span> is spring constant (N m<sup>-1</sup>)</li> </ul> <p>Simple pendula have a similar equation, but with different variable quantities:</p> <p style="text-align: center;"><span class="math-tex">\(T=2\pi\sqrt{l\over g}\)</span></p> <ul> <li><span class="math-tex">\(T\)</span> is time period (s)</li> <li><span class="math-tex">\(l\)</span> is the length of the string</li> <li><span class="math-tex">\(g\)</span> is gravitational field strength (N kg<sup>-1</sup>)</li> </ul> <p>The total energy of an oscillating system is the sum of the kinetic and potential energies. Potential energy falls to zero when kinetic energy is maxium, and vice versa.</p> <p style="text-align: center;"><span class="math-tex">\(E_k={1\over 2}m\omega^2({x_0}^2-x^2)\)</span></p> <p style="text-align: center;"><span class="math-tex">\(E_T={1\over 2}m\omega^2{x_0}^2\)</span></p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-turquoise panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Single-slit diffraction</p> </div> </div> <div class="panel-body"> <div> <p>Diffraction is the spreading out of a wave when passing through a gap in a boundary.</p> <p style="text-align: center;"><img alt="" src="../../waves/diffraction.jpg"></p> <p>We can determine the position of the first minimum using the approximation equation:</p> <p style="text-align: center;"><span class="math-tex">\(\theta = {\lambda \over a}={y\over D}\)</span></p> <ul> <li><span class="math-tex">\(\theta\)</span> is the angle subtended at the slit by the first minimum and the centre (rad)</li> <li><span class="math-tex">\(\lambda\)</span> is the wavelength of the light (m)</li> <li><span class="math-tex">\(a\)</span> is the width of the slit (m)</li> <li><span class="math-tex">\(y\)</span> is the distance between the centre and the first minimum (on either side since symmetrical) (m)</li> <li><span class="math-tex">\(D\)</span> is the distance from the slit to the screen (m)</li> </ul> <p>The larger the wavelength, the more spread the pattern becomes. Diffraction separates the colours of white light in the outer maxima, with the red light subtending larger angles than blue.</p> <p style="text-align: center;"><img alt="" src="../../waves/white-light-only.png" style="height: 68px; width: 400px;"></p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <p> </p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-has-border panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Resolution</p> </div> </div> <div class="panel-body"> <div> <p>Two sources of light become resolved (distinguishable as separate) when the Rayleigh criterion applies: the principal maximum of one diffraction pattern coincides with the first minimum of the other.</p> <p style="text-align: center;"><img alt="" src="../../waves/justresolved.png"></p> <p>In the instance that the Rayleigh criterion applies, we can calculate the separation of the sources:</p> <p style="text-align: center;"><span class="math-tex">\(\theta=1.22{\lambda\over b} ={\text{separation}\over D}\)</span></p> <ul> <li><span class="math-tex">\(\theta\)</span> is the angle subtended by the centre and the first minimum (rad)</li> <li><span class="math-tex">\(\lambda\)</span> is the wavelength of light (m)</li> <li><span class="math-tex">\(b\)</span> is the diameter of the aperture (m)</li> <li><span class="math-tex">\(D\)</span> is the perpendicular distance from the centre of the two objects to the slit (m)</li> </ul> <p>Diffraction gratings can be used to resolve two wavelengths of light.</p> <p style="text-align: center;"><span class="math-tex">\(R={\lambda\over \Delta \lambda}=mN\)</span></p> <ul> <li><span class="math-tex">\(R\)</span> is the resolvance of the diffraction grating</li> <li><span class="math-tex">\(\lambda\)</span> is the mean wavelength (m)</li> <li><span class="math-tex">\(\Delta \lambda\)</span> is the smallest resolvable wavelength difference (m)</li> <li><span class="math-tex">\(m\)</span> is the order of the diffraction</li> <li><span class="math-tex">\(N\)</span> is the total number of slits illuminated</li> </ul> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Multiple-slit interference</p> </div> </div> <div class="panel-body"> <div> <p>Light interferes when it passes through two or more slits in a boundary. The effect is observable for coherent light beams (constant phase difference) with the same amplitude.</p> <p>Young (pictured above) observed the interference pattern for a double slit.</p> <p style="text-align: center;"><img alt="" src="../../waves/double-6.png"></p> <p>The fringe spacing can be calculated as follows:</p> <p style="text-align: center;"><span class="math-tex">\(s={D\lambda\over d}\)</span></p> <ul> <li><span class="math-tex">\(s\)</span> is the distance between consecutive maxima (m)</li> <li><span class="math-tex">\(D\)</span> is the distance from the slits to the screen (m)</li> <li><span class="math-tex">\(\lambda\)</span> is the wavelength of light (m)</li> <li><span class="math-tex">\(d\)</span> is the distance separating the centres of the two slits (m)</li> </ul> <p>The two-slit <strong>interference</strong> pattern is modulated by the one-slit <strong>diffraction</strong> effect.</p> <p style="text-align: center;"><img alt="" src="../../waves/double-5.png"></p> <p>When light shines through multiple slits, an interference pattern forms similar to that through a double slit.</p> <p style="text-align: center;"><span class="math-tex">\(n\lambda = d \sin \theta\)</span></p> <ul> <li><span class="math-tex">\(n\)</span> is the number (order) of the maximum, where the central maximum is order 0</li> <li><span class="math-tex">\(\lambda\)</span> is the wavelength of the light (m)</li> <li><span class="math-tex">\(d\)</span> is the distance separating the slits (m)</li> <li><span class="math-tex">\(\theta\)</span> is the angle subtended by the maximum and the centre</li> </ul> <p>The more slits added, the more narrow and intense the peaks become. A diffraction grating is an array of identical, equally-spaced slits.</p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Thin film interference</p> </div> </div> <div class="panel-body"> <div> <p>Thin film interference is the effect observed on soap bubbles, oily puddles and peacock feathers.</p> <p>When a wave reflects off a loose boundary, it experiences no phase change. However, when a wave reflects off a fixed boundary, it experiences a phase change of <span class="math-tex">\(\pi\)</span> radians.</p> <p>Thin film interference is a result of the combination of the different wavelengths of light and the different distances travelled through the film.</p> <p>The condition for constructive interference is <span class="math-tex">\(2dn=(m+{1\over 2})\lambda\)</span>:</p> <ul> <li><span class="math-tex">\(d\)</span> is the thickness of the film (m)</li> <li><span class="math-tex">\(n\)</span> is the refractive index of the film</li> <li><span class="math-tex">\(m\)</span> is an integer value</li> <li><span class="math-tex">\(\lambda\)</span> is the wavelength of light (m)</li> </ul> <p>The condition for destructive interference is <span class="math-tex">\(2dn=m\lambda\)</span>.</p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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</textarea><a href="#" class="btn btn-default btn-sm btn-block text-center save-notes" title="Save personal notes for this flashcard" style="margin-bottom: 10px;"><i class="fa fa-pencil"></i> Save notes</a> </div><div class="fc-viewer-actions-sm hidden-md hidden-lg"><button type="button" class="btn btn-sm btn-xs-block reset-stats"><i class="fa fa-fw fa-undo"></i>Reset</button> <button type="button" class="btn btn-sm btn-xs-block" data-toggle="modal" data-target="#card-notes-sm-modal"><i class="fa fa-fw fa-pencil"></i>Notes</button><button type="button" class="btn btn-sm btn-xs-block" data-toggle="modal" data-target="#report-problem-sm-modal"><i class="fa fa-fw fa-exclamation-circle"></i>Report error</button></div> </div></div><div class="fc-viewer-controls-bottom hidden-xs hidden-sm"><div class="fc-move"> <button class="btn bg-turquoise text-center disabled" rel="prev" title="Previous flashcard"><i class="fa fa-fw fa-chevron-left"></i>Prev</button><button class="btn bg-turquoise text-center disabled" rel="next" title="Next flashcard">Next<i class="fa fa-fw fa-chevron-right"></i></button></div><div class="fc-actions"><button class="btn btn-default btn-block text-center shuffle"><i class="fa fa-random"></i> Shuffle</button><div class="fc-progress"><div class="progress"><div class="progress-bar"></div></div><div class="text-right"><span class="fc-viewer-results-counter"></span></div></div></div></div><div class="fc-close-sm hidden-md hidden-lg"><button id="fc-xs-close" type="button" class="btn btn-default btn-block text-center" data-dismiss="modal" aria-label="Close">Close</button> <hr class="thin"> <div class="fc-actions" style="width: 100%;"> <button class="btn btn-default btn-block text-center shuffle"> <i class="fa fa-random"></i> Shuffle </button><div class="fc-progress"><div class="progress"><div class="progress-bar"></div></div><div class="text-right"><span class="fc-viewer-results-counter hidden-md hidden-lg"></span></div></div> </div></div></div></div></div><div class="modal-footer"><div id="report-error-container" class="report-problem"><div class="row"><div class="col-md-8"><div class="msg"></div><div class="form-group"><label>Report a problem for flashcard <span class="fc-viewer-id"></span></label><textarea class="form-control issue-description" cols="4"></textarea></div><button type="button" class="btn btn-primary pull-right send-report disabled"><i class="fa fa-paper-plane"></i>Send</button><button id="close-send-report" type="button" class="btn btn-default" style="margin-left: 0;">Close</button></div></div></div></div></div></div></div><div id="card-notes-sm-modal" class="modal fade overlay-modal" tabindex="-1" role="dialog"><div class="modal-dialog modal-sm" role="document"><div class="modal-content"><div class="modal-header p-xs-3"><h4 class="modal-title text-center my-xs-2">Notes</h4></div><div class="modal-body p-xs-3"><textarea class="form-control notes fc-back card-notes p-xs-3" rows="8" style="resize: none;">
</textarea></div><div class="modal-footer p-xs-3"><button type="button" class="btn btn-sm btn-primary btn-xs-block save-notes"><i class="fa fa-fw fa-pencil"></i>Save</button><button type="button" class="btn btn-sm btn-default btn-xs-block mx-xs-0" data-dismiss="modal" aria-label="Close">Close</button></div></div></div></div><div id="report-problem-sm-modal" class="modal fade overlay-modal report-problem" tabindex="-1" role="dialog"><div class="modal-dialog modal-sm" role="document"><div class="modal-content"><div class="modal-header p-xs-3"><h4 class="modal-title text-center my-xs-2">Report a problem for flashcard <span class="fc-viewer-id"></span></h4></div><div class="modal-body p-xs-3"><div class="msg"></div><div class="form-group mb-xs-0"><textarea class="form-control issue-description p-xs-3" rows="8"></textarea></div></div><div class="modal-footer p-xs-3"><button type="button" class="btn btn-sm btn-primary text-center btn-xs-block mb-xs-3 disabled send-report"><i class="fa fa-fw fa-paper-plane"></i>Send</button><button type="button" class="btn btn-sm btn-default text-center btn-xs-block mx-xs-0" data-dismiss="modal" aria-label="Close">Close</button></div></div></div></div> <div id="fcgame-viewer" class="modal fade modal-flashcard" tabindex="-1" role="dialog"> <div class="modal-dialog modal-lg modal-dialog-fcgame" role="document"> <div class="modal-content mc-flashcard modal-content-fcgame"> <div class="modal-header" style="background-color: #fafafa;"> <div class="row text-center" style="width: 100%;"> <div class="col-md-12"> <div style="display: flex; justify-content: space-between; align-items: center;"> <div class="text-left"> Your time: <span class="chronometer c-mobile" id="chronometer">00:00:</span> <span class="chronometer msec c-mobile" id="chronometer-msec">000</span> <br> Your best time: <span class="chronometer c-mobile" id="best-record"></span> </div> <button type="button" class="close" data-dismiss="modal" aria-label="Close"> <span aria-hidden="true">×</span> </button> </div> </div> </div> </div> <div id="fcgame-body" class=" modal-body modal-body-fcgame"> <div class="row-fluid"> <div class="col-lg-12 col-md-12 col-sm-12 fc-content" id="fc-content" style="padding: 15px;"></div> </div> </div> <div id="end-game" style="display: none;"> <div class="col-lg-10 col-lg-offset-1 col-md-10 col-md-offset-1 col-sm-10 col-sm-offset-1 result shadow-bottom"> <div id="end-game-dismiss"></div> <div id="end-game-results"></div> <div id="show-percent" style="margin-bottom: 30px;font-size: 16px;"></div> </div> <div id="end-game-fc" class="col-lg-12"></div> </div> <div id="start-game" class="col-lg-12 col-md-12 col-sm-12 centered"> <div style="width: 200px;"> <button class="btn btn-success btn-xs-block" id="play-game" style="display: block; width: 100%; margin: 10px 0;">Start</button> <button class="btn btn-default btn-xs-block" id="cancel-game" style="display: block; width: 100%; margin: 10px 0;">Cancel</button> </div> </div> <div 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