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14px"><i class="fa fa-fw"></i><a href="../1126/multiple-slit-interference.html">Multiple slit interference</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../1124/thin-film-interference.html">Thin film interference</a></label></li></ul></li></ul></div> <div class="hidden-xs hidden-sm"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> Doppler effect <a href="#" class="mark-page-favorite pull-right" data-pid="1122" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../physics.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../444/oscillations-and-waves.html">Oscillations and waves</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1128/ahl-waves.html">AHL Waves</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Doppler effect</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../waves/doppler-s.jpg" style="float: left; width: 250px; height: 165px;"></p> <p>The Doppler effect is the first wave phenomenon that we consider in this Additional Higher Level topic. It is a change in frequency in observed sound caused by relative motion between the source, the observer or the medium. Here we will learn how to calculate these changes in frequency according to the type of relative motion, explain the impact of a sonic boom and discuss the equivalent effect for light.</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> <p>The Doppler effect is the change in frequency in observed sound due to relative motion between source and observer. </p> <p>This is often observed when a car drives past sounding its horn. The frequency is increased when the car approaches because the car catches up with the waves, causing them to be squashed. The result is a reduction in wavelength, which leads to a higher frequency as a result of the wave equation:</p> <p style="text-align: center;"><span class="math-tex">\(f={c\over \lambda}\)</span></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/369635810"></iframe></div> <p>This diagram shows how you might sketch the Doppler effect.</p> <p style="text-align: center;"><img alt="" src="../../astro/doppler.png" style="width: 300px; height: 243px;"></p> </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>Equation</p> </div> </div> <div class="panel-body"> <div> <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>Ahead of the source, the waves have been squashed into the distance <span class="math-tex">\(vt-u_st\)</span> (equal to the distance traveled by the sound less the distance caught up by the source). To find the observed wavelength, divide this distance by the number of waves produced in the same time, <span class="math-tex">\(ft\)</span>: </p> <p style="text-align: center;"><span class="math-tex">\(\lambda'={vt-u_st \over ft}={v-u_s\over f}\)</span></p> <p>Substituting into the wave equation for the observed wave: <span class="math-tex">\(f'={c\over \lambda'}=f({v\over v-u_s})\)</span></p> <p>The converse is true for an observer behind the source.</p> <p>Notice that, the greater the speed of the source, the more the observed frequency is changed. However, the speed of the source relative to an observer within the source itself is constant.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/369636306"></iframe></div> </div> </div> <div class="panel-footer"> <div> </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>Calculating velocity</p> </div> </div> <div class="panel-body"> <div> <p>We can use determined values of the frequency of the sound emitted by a source car in the video above as it approaches and recedes and passes the observer to find the velocity of the source.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/369636122"></iframe></div> </div> </div> <div class="panel-footer"> <div> </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>Measuring the Doppler effect</p> </div> </div> <div class="panel-body"> <div> <ol> <li>Determine the frequency of the source by measuring the time for the formation of, for example, 10 waves: <span class="math-tex">\(f={1\over T}\)</span></li> <li>Determine the frequency ahead of the source by measuring the time for 10 waves to pass a chosen point or line.</li> <li>Using the “set scale” tool set the width of the video window to 1m.</li> <li>Measure the velocity of the source and waves from the gradient of displacement-time graphs.</li> <li>Use the Doppler equation to calculate the change in frequency.</li> </ol> </div> </div> <div class="panel-footer"> <div> </div> </div> </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>Moving observer</p> </div> </div> <div class="panel-body"> <div> <p>Doppler shift is also experienced when an observer moves towards or away from a source that is stationary relative to the medium. This effect is simply due to the relative velocity between the observer and source.</p> <p>The equation to calculate the observed frequency is amended as follows:</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>This equation comes from use of the relative velocity of the sound as it approaches the observer, now <span class="math-tex">\(v+u_o\)</span> and using <span class="math-tex">\(\lambda={v\over f}\)</span>:</p> <p style="text-align: center;"><span class="math-tex">\(f'={v+u_o\over \lambda}\Rightarrow f'=f{(v+u_o)\over v}\)</span></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/369636063"></iframe></div> </div> </div> <div class="panel-footer"> <div> </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>Moving medium</p> </div> </div> <div class="panel-body"> <div> <p>Sound travels through media containing particles. The medium may too have a speed.</p> <p>This diagram shows the effect when a steady drip of water falls from a bridge into the river below.</p> <p style="text-align: center;"><img src="../../waves/doppler.png" style="height: 178px; width: 300px;"></p> </div> </div> <div class="panel-footer"> <div> </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>Sonic boom</p> </div> </div> <div class="panel-body"> <div> <p>A sonic boom is generated when the speed of the source exceeds the speed of sound, creating a geometrical cone behind the object. This loud *BANG* from the shock waves can awaken sleeping people nearby and damage delicate objects.</p> <p>Sonic booms can more easily be achieved by aircraft at height. The speed of sound decreases as the atmosphere becomes less dense, so the speed required to exceed the speed of sound also decreases.</p> <p style="text-align: center;"><img alt="" src="../../waves/sonic-boom.jpg" style="width: 200px; height: 250px;"></p> <p style="text-align: center;"><iframe frameborder="0" height="440" scrolling="no" src="https://tube.geogebra.org/material/iframe/id/464311/width/577/height/440/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/true/ld/false/sri/true/at/auto" width="577"></iframe></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/369636811"></iframe></div> </div> </div> <div class="panel-footer"> <div> </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>Electromagnetic waves</p> </div> </div> <div class="panel-body"> <div> <h4>Equation</h4> <p>We must use an approximate equation for all electromagnetic waves, in which we assume that the speed of the source does not approach the speed of light. Thus, there are no relativistic effects:</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> <p>For a source moving away from the observer, the observed wavelength is higher than that at source, but the observed frequency is lower than that at source. A galaxy emitting visible light will have this red-shifted if it moves away from the observed. Red shift is evidence for the expansion of the universe, as the light from all distant galaxies is red-shifted. Thus, all galaxies are receding and so the universe must be expanding in all directions.</p> <p style="text-align: center;"><img alt="" src="../../astro/redshift.jpg" style="width: 291px; height: 173px;"></p> <h4>Uses</h4> <p>The Doppler effect in electromagnetic waves can also also be used for more 'earthly' means.</p> <p>A Doppler radar produces velocity data about objects at a distance. Speed cameras are a good example. They emit a beam of radiation (microwaves or infrared) at an approaching vehicle where it is reflected. The Doppler effect occurs twice: once as the observer approaches the radiation and again as the reflection is emitted from the new moving source. The speed of the vehicle can be calculated from the change in frequency. Doppler radars can also be found in aviation, sounding satellites, Major League Baseball, meteorology, radar guns, and radiology and medicine.</p> <p>In meteorology, the direction, speed and type of objects of precipitation may be detected. Storms may be analysed in this way to assess structure and likely severity.</p> <p>In medicine, the direction and speed of blood flow in arteries and veins are determined. This technique is used in echocardiograms and medical ultrasonography and is an effective tool in diagnosis of vascular problems.</p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <div> <p><em>Use quizzes to practise application of theory.</em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#871734ea"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="871734ea"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Doppler <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="6-343-1122" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The image is a still from a GeoGebra simulation.</p><p style="text-align: center;"><img alt="" height="289" src="../../screenshot-2019-09-07-at-09.19.38.png" width="376"></p><p>The wavelength of the wave in front of the source is:</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 4.00 m</label></p><p><label class="radio"><input class="c" type="radio"> 0.25 m</label></p><p><label class="radio"><input type="radio"> 1.00 m</label></p><p><label class="radio"><input type="radio"> 0.50 m</label></p></div><div class="q-explanation"><p>There are 16 waves produced in 4s. These fit in the space between the source and the first wavefront.</p><p>The wave has progressed 3 x 4 = 12 m</p><p>The source has moved 2 x 4 = 8 m</p><p>The difference between the source and wavefront distance = 4 m</p><p><span class="math-tex">\(\lambda'={vt-u_st \over ft}={4\over 16}=0.25\text{ m}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The image is a still from a GeoGebra simulation.</p><p style="text-align: center;"><img alt="" height="289" src="../../screenshot-2019-09-07-at-09.19.38.png" width="376"></p><p>The wavelength of the wave behind the source is:</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 1.00 m</label></p><p><label class="radio"><input class="c" type="radio"> 1.25 m</label></p><p><label class="radio"><input type="radio"> 4.00 m</label></p><p><label class="radio"><input type="radio"> 0.50 m</label></p></div><div class="q-explanation"><p>There are 16 waves produced in 4s. These fit in the space between the source and the first wavefront.</p><p>The wave has progressed -3 x 4 = -12 m</p><p>The source has moved 2 x 4 = 8 m</p><p>The difference between the source and wavefront distance = 20 m</p><p><span class="math-tex">\(\lambda'={vt-u_st \over ft}={20\over 16}=1.25\text{ m}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The image is a still from a GeoGebra simulation.</p><p style="text-align: center;"><img alt="" height="289" src="../../screenshot-2019-09-07-at-09.19.38.png" width="376"></p><p>The frequency of the wave behind the source is:</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 2.0 Hz</label></p><p><label class="radio"><input type="radio"> 1.8 Hz</label></p><p><label class="radio"><input class="c" type="radio"> 2.4 Hz</label></p><p><label class="radio"><input type="radio"> 12.0 Hz</label></p></div><div class="q-explanation"><p>For a stationary observer behind a moving source:</p><p><span class="math-tex">\(f' =f{v\over (v+u_s)} = 4 \times {3\over 5} =2.4 \text{ Hz} \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The image is a still from a GeoGebra simulation.</p><p style="text-align: center;"><img alt="" height="289" src="../../screenshot-2019-09-07-at-09.19.38.png" width="376"></p><p>The frequency of the wave in front of the source is:</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 12 Hz</label></p><p><label class="radio"><input type="radio"> 18 Hz</label></p><p><label class="radio"><input type="radio"> 9 Hz</label></p><p><label class="radio"><input type="radio"> 20 Hz</label></p></div><div class="q-explanation"><p>For a stationary observer in front of a moving source:</p><p><span class="math-tex">\(f' =f{v\over (v-u_s)} = 4 \times {3\over 1} =12 \text{ Hz} \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A student tries to fool his teacher by changing the pitch of his whistle as he cycles past. The trick worked and his teacher heard no Doppler shift.</p><p>The student whistled with...</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>high pitch all the time</span></label> </p><p><label class="radio"> <input type="radio"> <span>high pitch changing to low pitch on passing</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>low pitch on approach changing to high pitch on passing</span></label> </p><p><label class="radio"> <input type="radio"> <span>low pitch all the time</span></label> </p></div><div class="q-explanation"><p>The Doppler shift increases the pitch on approach and decreases the pitch on recession. The student compensated by whisting a lower pitch on approach and higher pitch on recession.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Two observers listen to the whistle of an approaching train.</p><p style="text-align: center;"><img alt="" height="184" src="../../screenshot-2019-09-07-at-10.10.59.png" width="284"></p><p>Compared to A, the observer at B will hear a sound that is...</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>higher pitch and less loud</span></label> </p><p><label class="radio"> <input type="radio"> <span>lower pitch and louder</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>lower pitch and less loud</span></label> </p><p><label class="radio"> <input type="radio"> <span>same pitch and less loud</span></label> </p></div><div class="q-explanation"><p>B is further away so the sound will be less loud as its energy is dissipated.</p><p>The component of the train's velocity towards B is less than towards A so the Doppler shift will have a reduced effect.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Two observers listen to the whistle of an approaching train.</p><p style="text-align: center;"><img alt="" height="184" src="../../screenshot-2019-09-07-at-10.10.59.png" width="284"></p><p>As the train passes the pitch of the whistle drops. Compared to A the drop in pitch will... and the midpoint will take place...</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>...last longer ...earlier</span></label> </p><p><label class="radio"> <input type="radio"> <span>...last the same time ...at the same time</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>...last longer ...later</span></label> </p><p><label class="radio"> <input type="radio"> <span>...last the same time ...later</span></label> </p></div><div class="q-explanation"><p>The velocity at which the train approaches B decreases as the train gets nearer so the Doppler shift gradually reduces - this makes it last longer.</p><p>It takes more time for the sound to reach B as it is a greater distance from the track and so the drop in pitch will come later.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The image is a still from a <span class="scayt-misspell-word" data-scayt-word="GeoGebra" data-wsc-lang="en_US">GeoGebra</span> simulation.</p><p style="text-align: center;"><img alt="" height="251" src="../../screenshot-2019-09-07-at-11.08.12.png" width="334"></p><p>The frequency ahead of the source is 25 Hz. Calculate the velocity of the source.</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>7 ms<sup>-1</sup></span></label> </p><p><label class="radio"> <input type="radio"> <span>2 ms<sup>-1</sup></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>3 ms<sup>-1</sup></span></label> </p><p><label class="radio"> <input type="radio"> <span>4 ms<sup>-1</sup></span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(f' =f{v\over (v-u_s)} \Rightarrow v-u_s={f\over f'}v \)</span></p><p><span class="math-tex">\(u_s=v(1-{f\over f'})=5(1-{10\over 25})=3 \text{ ms}^{-1}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none 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