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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"> Standing waves on a string <a href="#" class="mark-page-favorite pull-right" data-pid="281" 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="../278/waves.html">Waves</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Standing waves on a string</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 10 minutes"><i class="fa fa-clock-o"></i> 10'</span> </ol> <article id="main-article"> <p><img alt="" src="../../waves/6-cello.jpg" style="float: left; width: 167px; height: 250px;">It is easy to observe the wave motion in a string but it doesn´t tell the whole story of wave properties. A string wave can't diffract or refract.</p> <p>Sometimes simple explanations cause misunderstandings. Here a string wave is polarised by a narrow slit. Light, however, is polarised by a special Polaroid filter and not a slit. </p> <hr class="hidden-separator"> <div class="panel panel-has-colored-body panel-purple"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <h4>Wave speed in a string</h4> <p>The velocity of a wave in a string is related to the tension and mass per unit length of the string:</p> <p style="text-align: center;"><span class="math-tex">\(v=\sqrt{T\over \mu}\)</span></p> <div class="smart-object center" data-id="172" style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="480" mozallowfullscreen="" src="https://player.vimeo.com/video/170354191" webkitallowfullscreen="" width="640"></iframe></div> <p>Different guitar strings have different diameters and so different values of <em>μ</em>, mass per unit length. Therefore, the speed in each string will be different and they will vibrate with different frequencies.</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/171054524"></iframe></div> </div> </div> <div class="panel-footer"> <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><i>Use quizzes to practise application of theory.</i> </p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#af3b1cd8"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="af3b1cd8"> <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%;"> Waves in strings <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="6-89-281" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>A wave in a string passes from a thin string into a thicker one.</p><p style="text-align: center;"><img alt="" height="217" src="../../stringq1.jpg" width="267"></p><p>Which diagram correctly represents the wave ?</p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> C</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> D</label></p></div><div class="q-explanation">Wave slows down so wavelength is shorter. Also takes more energy to give same displacement so amplitude less.</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 wave of wavelength 2m travels along a rope. The rope is attached to a second one which has the same diameter but 4 times the density.</p><p>What is the wavelength of the wave in the second rope?</p><p style="text-align: center;">v = √(T/m/L)</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 32 m</label></p><p><label class="radio"><input class="c" type="radio"> 1 m</label></p><p><label class="radio"><input type="radio"> 8 m</label></p><p><label class="radio"><input type="radio"> 0.5 m</label></p></div><div class="q-explanation"><p>The mass per unit length is 4 times bigger.<br>The velocity is proportional to 1/√(m/L) so if m/L is x4 velocity is x 1/2<br>Wavelength is proportional to velocity<br> </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 velocity of a wave in a string is 2 ms<sup>-1</sup>. What will the velocity be if the tension in the string is changed from 20 N to 80 N</p><p style="text-align: center;">v = √(T/m/L)</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 8 ms<sup>-1</sup></label></p><p><label class="radio"><input type="radio"> 0.5 ms<sup>-1</sup></label></p><p><label class="radio"><input class="c" type="radio"> 4 ms<sup>-1</sup></label></p><p><label class="radio"><input type="radio"> 1 ms<sup>-1</sup></label></p></div><div class="q-explanation"><p>The velocity is proportional to √T</p><p>T is x4 so velocity is x2</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 axis below are velocity and Tension.</p><p style="text-align: center;"><img alt="" height="283" src="../../stringwaveq2.jpg" width="298"></p><p>Which line is the correct shape for the velocity of a wave in a string vs the tension of the string?</p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> D</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> C</label></p></div><div class="q-explanation"><p>A is 1/5x<br>B is y = x<sup>2</sup><br>C is y = x<br>D is y = √x</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 pulse moves towards the end of a string.</p><p style="text-align: center;"><img alt="" src="../../stringwaveq4.jpg" style="width: 340px; height: 175px;"></p><p>Here are diagram representing the string a short time later.</p><p style="text-align: center;"><img alt="" src="../../stringwaveq5.jpg" style="width: 291px; height: 485px;"></p><p>Which diagram represents the following situation</p></div><div class="q-answer"><p>No end <input type="text" style="height: auto;" data-c="C"> <span class="review"></span></p><p>Fixed end <input type="text" style="height: auto;" data-c="B"> <span class="review"></span></p><p>Loose end <input type="text" style="height: auto;" data-c="A"> <span class="review"></span></p><p>Connected to a thinner string <input type="text" style="height: auto;" data-c="D"> <span class="review"></span></p><p>Connected to a thicker string <input type="text" style="height: auto;" data-c="E"> <span class="review"></span></p></div><div class="q-explanation"></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 diagram represents a pulse on a 1m long string fixed at both ends.</p><p style="text-align: center;"><img alt="" height="122" src="../../stringwave7.jpg" width="362"></p><p>The following represent the string a short time later.</p><p style="text-align: center;"><img alt="" height="460" src="../../stringwave6.jpg" width="362"></p><p>If the speed of the wave is 1ms<sup>-1</sup>, which diagram represents the string after the following times</p></div><div class="q-answer"><p>1s <input type="text" style="height: auto;" data-c="D"> <span class="review"></span></p><p>2s <input type="text" style="height: auto;" data-c="A"> <span class="review"></span></p><p>3s <input type="text" style="height: auto;" data-c="D"> <span class="review"></span> </p><p>4s <input type="text" style="height: auto;" data-c="A"> <span class="review"></span></p></div><div class="q-explanation"><p style="text-align: center;"><img alt="" class="gifffer" data-gifffer="/media/physics/strinqwavea7.gif" style="width: 304px; height: 86px;"></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 diagram represents a pulse in a 1m string with one fixed end and one loose end.</p><p style="text-align: center;"><img alt="" height="123" src="../../stringwave9.jpg" width="330"></p><p>The following represent the string a short time later.</p><p style="text-align: center;"><img alt="" height="475" src="../../tringwaveq8.jpg" width="327"></p><p>If the speed of the wave is 1ms<sup>-1</sup>, which diagram represents the string after the following times</p></div><div class="q-answer"><p>1s <input type="text" style="height: auto;" data-c="B"> <span class="review"></span></p><p>2s <input type="text" style="height: auto;" data-c="C"> <span class="review"></span></p><p>3s <input type="text" style="height: auto;" data-c="D"> <span class="review"></span></p><p>4s <input type="text" style="height: auto;" data-c="A"> <span class="review"></span></p></div><div class="q-explanation"><p style="text-align: center;"><img alt="" class="gifffer" data-gifffer="/media/physics/stringwavea9.gif"></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 pulses travel towards each other as shown.</p><p style="text-align: center;"><img alt="" height="119" src="../../stringwaveq10.jpg" width="291"></p><p>The following represent the string a short time later.</p><p style="text-align: center;"><img alt="" src="../../stringwaveq11.jpg" style="width: 209px; height: 361px;"></p><p>If the speed of the wave is 1ms<sup>-1</sup> which image represents the wave after:</p></div><div class="q-answer"><p>0.5 s <input type="text" style="height: auto;" data-c="C"> <span class="review"></span></p><p>1 s <input type="text" style="height: auto;" data-c="A"> <span class="review"></span></p><p>1.5 s <input type="text" style="height: auto;" data-c="C"> <span class="review"></span></p><p>2 s <input type="text" style="height: auto;" data-c="D"> <span class="review"></span></p><p>5 s <input type="text" style="height: auto;" data-c="B"> <span class="review"></span></p></div><div class="q-explanation"><p style="text-align: center;"><img alt="" class="gifffer" data-gifffer="/media/physics/stringave11.gif"></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 pulses travel towards each other on a clamped string as shown.</p><p style="text-align: center;"><img alt="" src="../../stringwave12.jpg" style="width: 377px; height: 137px;"></p><p>The diagram below shows positions of the string some time later.</p><p style="text-align: center;"><img alt="" src="../../stringwaveq13.jpg" style="width: 212px; height: 203px;"></p><p>If the wave speed is 1 ms<sup>-1</sup> which image represents the wave after:</p></div><div class="q-answer"><p>1 s <input type="text" style="height: auto;" data-c="A"> <span class="review"></span></p><p>5 s <input type="text" style="height: auto;" data-c="B"> <span class="review"></span></p><p>9 s <input type="text" style="height: auto;" data-c="A"> <span class="review"></span></p><p>13 s <input type="text" style="height: auto;" data-c="B"> <span class="review"></span></p></div><div class="q-explanation"><p style="text-align: center;"><img alt="" class="gifffer" data-gifffer="/media/physics/stringwave12.gif"></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 pulses travel towards each other on a string that is clamped at one end and loose at the other.</p><p style="text-align: center;"><img alt="" src="../../stringwave14.jpg" style="width: 330px; height: 127px;"></p><p>The diagram below shows the string some time later</p><p style="text-align: center;"><img alt="" src="../../stringwave15.jpg" style="width: 216px; height: 269px;"></p><p>If the wave speed is 1 ms<sup>-1</sup> which image represents the wave after:</p></div><div class="q-answer"><p>1 s <input type="text" style="height: auto;" data-c="A"> <span class="review"></span></p><p>5 s <input type="text" style="height: auto;" data-c="C"> <span class="review"></span></p><p>9 s <input type="text" style="height: auto;" data-c="B"> <span class="review"></span></p><p>13 s <input type="text" style="height: auto;" data-c="C"> <span class="review"></span></p><p>18 s <input type="text" style="height: auto;" data-c="A"> <span class="review"></span></p></div><div class="q-explanation"><p style="text-align: center;"><img alt="" class="gifffer" data-gifffer="/media/physics/stringwavea14.gif"></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>Tow aves travelling towards each other cancel out as shown.</p><p style="text-align: center;"><img alt="" height="192" src="../../stringwaveq15.jpg" width="280"></p><p>What form does the energy take when the waves cancel?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> there is none</label></p><p><label class="radio"><input type="radio"> gravitational PE</label></p><p><label class="radio"><input class="c" type="radio"> KE</label></p><p><label class="radio"><input type="radio"> elastic PE</label></p></div><div class="q-explanation"><p>The string isn´t stretched or lifted but is moving even though it doesn´t look like it in the still image.</p><p style="text-align: center;"><img alt="" class="gifffer" data-gifffer="/media/physics/stringwaveq15.gif"></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>This diagram shows a wave approaching a narrow slit.</p><p style="text-align: center;"><img alt="" src="../../polarisedstringq1.jpg" style="width: 223px; height: 515px;"></p><p>Which image shows the wave after passing through the slit?</p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> C</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> D</label></p></div><div class="q-explanation"><p>Only the vertical component will pass.</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 nodes in a standing wave are separated by:</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> λ/2</label></p><p><label class="radio"><input type="radio"> 2λ</label></p><p><label class="radio"><input type="radio"> λ</label></p><p><label class="radio"><input type="radio"> λ/4</label></p></div><div class="q-explanation"><p><label class="radio"><img alt="" src="../../standingwaveq1.jpg" style="width: 326px; height: 132px;"></label></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 diagram represents a standing wave in a string.</p><p style="text-align: center;"><img alt="" src="../../standuingwaveq2.jpg" style="width: 286px; height: 132px;"></p><p>Points A and B are</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> in phase with the same amplitude</label></p><p><label class="radio"><input type="radio"> π out of phase with different amplitude</label></p><p><label class="radio"><input type="radio"> in phase with different amplitude</label></p><p><label class="radio"><input class="c" type="radio"> π out of phase with the same amplitude</label></p></div><div class="q-explanation"><p style="text-align: center;"><img alt="" class="gifffer" data-gifffer="/media/physics/standingq3.gif" style="width: 235px; height: 105px;"></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 diagram represents a standing wave in a string.</p><p style="text-align: center;"><img alt="" src="../../standingstringq2.jpg" style="width: 291px; height: 132px;"></p><p>Points A and B are</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> in phase with different amplitude</label></p><p><label class="radio"><input type="radio"> π out of phase with different amplitude</label></p><p><label class="radio"><input class="c" type="radio"> in phase with the same amplitude</label></p><p><label class="radio"><input type="radio"> π out of phase with the same amplitude</label></p></div><div class="q-explanation"><p style="text-align: center;"><img alt="" class="gifffer" data-gifffer="/media/physics/standansq2.gif" style="width: 235px; height: 105px;"></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 diagram represents a standing wave in a string.</p><p style="text-align: center;"><img alt="" src="../../standstring4.jpg" style="width: 291px; height: 132px;"></p><p>Points A and B are</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> in phase with the different amplitude</label></p><p><label class="radio"><input type="radio"> in phase with the same amplitude</label></p><p><label class="radio"><input type="radio"> out of phase with different amplitude</label></p><p><label class="radio"><input type="radio"> out of phase with same amplitude</label></p></div><div class="q-explanation"><p style="text-align: center;"><img alt="" class="gifffer" data-gifffer="/media/physics/standstringa5.gif" style="width: 235px; height: 105px;"></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 diagram represents a standing wave in a string.</p><p style="text-align: center;"><img alt="" src="../../standuingwaveq2.jpg" style="width: 286px; height: 132px;"></p><p>If the wavelength is 3m the distance between points A and B is:</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 1.5 m</label></p><p><label class="radio"><input type="radio"> 3 m</label></p><p><label class="radio"><input type="radio"> 1 m</label></p><p><label class="radio"><input type="radio"> 2 m</label></p></div><div class="q-explanation"><p>The distance between antinodes is λ/2</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 diagram represents a standing wave in a string</p><p style="text-align: center;"><img alt="" src="../../standstringq6.jpg" style="width: 291px; height: 132px;"></p><p>If the frequency of this wave is 12 Hz what is the frequency if the next harmonic?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 18 Hz</label></p><p><label class="radio"><input type="radio"> 24 Hz</label></p><p><label class="radio"><input class="c" type="radio"> 16 Hz</label></p><p><label class="radio"><input type="radio"> 4 Hz</label></p></div><div class="q-explanation"><p>This is the 3rd harmonic.</p><p>The frequency of the 1st harmonic = 4 Hz</p><p>The frequency of the 4th harmonic = 4 x 4 = 16Hz</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 graph represents a frequency spectrum for a stretched string</p><p style="text-align: center;"><img alt="" height="262" src="../../standstringq7.jpg" width="268"></p><p>The frequency of the 8th harmonic would be</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 800 Hz</label></p><p><label class="radio"><input type="radio"> 600 Hz</label></p><p><label class="radio"><input type="radio"> The 8th harmonic is not possible</label></p><p><label class="radio"><input class="c" type="radio"> 400 Hz</label></p></div><div class="q-explanation"><p>1st harmonic = 50 Hz</p><p>8th Harmonic = 8 x 50 Hz</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 graph represents the frequency spectrum for the string shown.</p><p style="text-align: center;"><img alt="" height="258" src="../../standstringq8.jpg" width="262"></p><p>Át which point is the string being touched?</p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> C</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> D</label></p></div><div class="q-explanation"><p>The 3rd harmonic is being played so there must be a node at 1/3 of the length</p><p style="text-align: center;"><img alt="" src="../../standstringq6.jpg" style="width: 291px; height: 132px;"></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> 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