Demonstrations of two-dimensional standing waves using Chladni plates are certainly impressive and can help instill a sense of awe and wonder in our students. We want to expose our students to things they can’t yet explain. We want to spark enough curiosity to get them to ask how something like this is even possible. But we don’t want to just leave them wondering. We want to give our students the conceptual tools to make sense of their observations. This is what the discipline of science is all about!
In this blog post, I will walk you through a digital resource I created to give students a conceptual understanding of “standing waves”, the phenomena behind the patterns created on this vibrating plate. All of the content is conceptual and can be used for the middle school level, up through introductory physics at the college level. This Standing Waves Digital Resource outlines the basic progression of concepts needed for students to understand what a standing wave is, and how it can be created. The Standing Waves Digital Resource includes animations, links to free simulations, suggested demonstrations, and videos that can be used in your classroom.
All the equipment used in the suggested standing wave demonstrations come from the “Mechanical Wave Complete Bundle”.
Developing the Standing Wave Concept
Step #1 (Mechanical Wave Introduction)
The first step in helping students build a conceptual understanding of standing waves involves a basic understanding of mechanical waves. Slinkies and coiled springs are a great hands-on way to introduce the basics of mechanical waves to students. Arbor Scientific has many options including their “helical spring”, “spring wave”, and “super spring”. Students can also investigate the basics of mechanical waves using an interactive simulation, like the one available from the PhET Interactive Simulations website titled “Waves on a String”.
Any introduction to mechanical waves must give students an understanding that mechanical waves are disturbances that move through a solid, liquid or gas when the material or medium is disturbed, travel outward from the initial disturbance in all available directions of the material, and a “wave” is moving energy, not moving matter. It helps to show or discuss several different examples of mechanical waves moving through solids, liquids and gasses.
I have students use the Waves on a String simulation to further investigate the properties and behavior of moving disturbances or waves. The investigation gives students an understanding of how to describe a wave’s amplitude, pulse width and speed. The investigation also introduces the concept of damping, reflected wave behavior, and the effect tension has on wave speed. You can find links to the simulation and corresponding student worksheet in the Standing Waves Digital Resource.
Through the Waves on a String activity students should be able to identify characteristics of a wave like amplitude and pulse width, recognize that the wave speed is only changed by the tension in the string, wave pulses are reflected differently off of open or fixed ends, and damping refers to a decrease in wave height or amplitude over time.
Step #2 (Periodic Waves)
Next, students need to be given a basic understanding of “periodic waves”. Students need to understand that periodic waves are created by periodic or repeated disturbances, and have a quantifiable amplitude, frequency, wavelength and wave speed.
The Waves on a String simulation is a great tool to introduce periodic waves because the simulation allows you to easily change the frequency of disturbance and see its effect on the periodic wave’s wavelength. The simulation can also be paused or stepped through frame by frame to show how amplitude and wavelength can be measured. At the high school and college level students need to also be introduced to the wave-speed equation showing the quantitative relationship between a periodic wave’s frequency, wavelength and speed.
Step #3 (Wave Interference)
Once students understand the basic properties of waves and periodic waves, the students need to be introduced to how waves interact. If you have two waves moving in opposite directions through the same slinky or string, what happens when they meet? Is it like two cars that collide in a head-on collision? Do the waves collide and stop? Do the waves bounce off each other? Does anything change about each wave pulse after they interact?
Even with careful observation, the results of wave interactions are very difficult to observe using slinkies or strings, unless you are using high speed video. For this reason it is again helpful to use the Waves on a String simulation to investigate how waves interact. Use the simulation to create different scenarios where two pulses meet while traveling in opposite directions. It is helpful to show the students examples of pulses with different widths, and examples where the pulses meet on the same side of the string and opposite sides.
The standing waves digital resource has four different scenarios that will help students draw conclusions about how waves interact (see below). You want students to see that the two waves pass through each other unaffected by the interaction. Each wave looks the same and is traveling in the same direction, at the same speed as it was before the interaction. The really interesting thing happens when the two waves are in the same location at the same time. Remind students that the string can only do one thing. The string can’t be in two locations at the same time. The result of the wave interaction is known as wave interference or superposition. When the pulses are on the same side of the string there is “constructive interference” where the total displacement of the string in the area of overlap is LARGER than it would be due to either of the individual pulses or waves. When the pulses are on the opposite side of the string there is “destructive interference” where the total displacement of the string in the area of overlap is SMALLER than it would be due to either of the individual pulses or waves.
It’s worth pausing here just to highlight how big of a difference there is between how waves interact compared to how objects made from matter interact. When two cars collide, they push on each other, and their speeds will change, and either both or one could change its direction of travel. If two cars interacted like waves, they would pass through one another without changing speed, and where they overlap, the part of the car at that location would either be bigger or smaller than either of the two individual cars.
Step #4 (Interference of Periodic Waves)
Now that students understand how two individual wave pulses interact, you can help students think about how two periodic waves would interact when passing through each other in the same material, moving in opposite directions.
We know the periodic waves will pass through each other, unchanged in wavelength, amplitude and wave speed, but what about the material the waves are traveling through? Each part of the material will be equal to the sum of the individual waves moving through that location at that specific time. There will either be constructive or destructive interference happening at each location along the length of the material. It gets really complicated to think this through without using an animation. The Standing Waves Digital Resource includes animations to help students visualize how two periodic waves interfere with each other while moving through the same material.
Imagine we have two students shaking each end of a slinky or string, each creating periodic waves that will pass through each other. The red and blue lines represent the individual periodic waves, but remember, each part of the material can only be in one place at one time. The purple line shows the sum of the red and blue periodic waves at each point along the length of the material. The purple line represents the actual behavior of the material. There are locations that do not move from equilibrium, and other locations that are being displaced farther from equilibrium than the amplitude of either periodic wave. These locations along the material’s length do not change position over time. They appear to stand still. This is why this behavior is called a “standing wave”, and it is created by two periodic waves moving through the same material in opposite directions.
Highlight several locations along the material and have students try to explain why it behaves that way. The locations along the material’s length that stay at equilibrium are known as “nodes''. These are locations of complete destructive interference between the two period waves. Every time a crest from one periodic wave passes through that location, a trough from the other periodic wave cancels it out, and vice-versa. The locations along the material’s length that have the largest displacement from equilibrium are known as “antinodes”. This is a result of maximum constructive interference between the two periodic waves. Maximum constructive interference happens at locations where the crests from each periodic wave meet, or the troughs from each periodic wave meet. This is a good time to help students see the connection between the wavelength of the periodic waves moving through the material, and the distance between the nodes or antinodes. The animation should help students see that the distance between nodes is equal to one half of the wavelength of the periodic waves moving through the material.
Martin Kirby, another physics teacher, has created a great simulation for students to explore the interference patterns created by many different periodic waves called “Wave Add”. There is a link for this simulation included in the Standing Waves Digital Resource. One last thing to note is that a standing wave can be created with only one source of periodic disturbances. As already discussed, standing waves are created by two periodic waves moving through the same material in opposite directions, but the second source of periodic waves could just be the reflected waves from a fixed end. An animation from Dr. Dan Russell from Penn State University illustrates this really well.
Step #5 (Standing Waves in Strings)
Now you’re ready to show students some demonstrations of standing waves to help them further investigate the phenomena. Use the Mechanical Wave Driver and Sine Wave Generator to attempt to create standing waves on a stretched string. Start with a low frequency and gradually increase the frequency until the first standing wave is created. Have students count the number of nodes and antinodes in the first standing wave and determine how much of a wavelength fits in the length of the string. For the first possible standing wave, half of a wavelength fits in the string’s length since each end was a node and there was a single antinode in the middle.
Now gradually increase the frequency until the next possible standing wave is found, and have students count the number of nodes and antinodes. Again, ask the students how much of a wavelength fits in the length of the string. The students will quickly identify a pattern: standing waves are created in a string with two fixed ends ONLY when the string’s length is equal to one half the periodic wave’s wavelength, or two halves of a wavelength, or three halves of a wavelength, etc… So the Mechanical Wave Driver will only create a standing wave if the driving frequency creates periodic waves with certain wavelengths.
You can use the animations in the Standing Waves Digital Resource to show the first four possible standing waves for a string with two fixed ends. Even though the oscillator attached to one end of the string is moving up and down, it is effectively a fixed end since its displacement from equilibrium is small relative to the displacement of the standing wave at the antinodes. If you know the length of the string, you can then calculate the specific wavelengths that would create each specific standing wave, and at the high school and college levels you could then use the wave-speed equation to calculate the driving frequencies needed to generate those required wavelengths.
Step #6 (More Standing Wave Demonstrations)
The Mechanical Wave Complete Bundle has ways to demonstrate other types of standing waves including metal strips with one fixed end and one free end, longitudinal standing waves in stretched springs, circular standing waves, and of course two-dimensional standing waves using the included Chladni Plates.
Now when students see the patterns of sand on the vibrating metal plate, they are ready to understand what creates those patterns. Just like a stretched string, only specific frequencies of disturbance will create standing waves in the metal surface. If standing waves are created in the two-dimensional surface, there should be antinodes, locations in the material that are displaced up and down more than the displacement of the Mechanical Wave Driver. These would be locations of maximum constructive interference where crests are overlapping with other crests, and troughs are overlapping with other troughs. When a standing wave is present the salt will move away from these antinode locations to parts of the material with less up and down displacement, like the node locations. There is no up and down displacement at the node locations because there is complete destructive interference between the periodic waves moving through the material. So when a standing wave is present at a particular frequency, the sand will move away from antinodes and settle at the nodes.
The complexity of the standing wave patterns increase, as you increase the frequency of the Mechanical Wave Driver. This is due to the fact that a higher frequency creates smaller wavelengths and the resulting standing waves will have higher numbers of both nodes and antinodes.
I hope this article has helped give you an idea of what it could look like to help your students understand the basics of standing waves. Remember to check out the animations, simulations and related videos in the Standing Waves Digital Resource.
Indian Hill High School
Aaron Debbink is a physics teacher with 16 years of classroom experience who has an undergraduate engineering degree and a masters degree in physics. He is passionate about building a classroom culture that values exploration and a drive for understanding. Aaron uses Modeling Instruction in his introductory and AP physics classes and has been a Modeling Instruction workshop leader since 2011. Aaron is a Knowles Teaching Fellow and the recipient of a 2019 Yale Educator Award.
Maonyesho ya mawimbi ya kusimama ya pande mbili kwa kutumia sahani za Chladni ni ya kuvutia na yanaweza kusaidia kuingiza hisia ya hofu na ajabu katika wanafunzi wetu. Tunataka kuwafunulia wanafunzi wetu mambo ambayo bado hawawezi kueleza. Tunataka kuwasha udadisi wa kutosha kuwafanya waulize jinsi kitu kama hiki kinavyowezekana. Lakini hatutaki kuwaacha wakijiuliza tu. Tunataka kuwapa wanafunzi wetu zana za dhana ya kufanya maana ya uchunguzi wao. Hili ndilo somo la sayansi! Katika chapisho hili la blogi, nitakuongoza kupitia rasilimali ya dijiti niliyounda ili kuwapa wanafunzi uelewa wa dhana ya "mafuriko ya kusimama", matukio nyuma ya mifumo iliyoundwa kwenye sahani hii ya kutetemeka. Mafunzo yote ni ya dhana na yanaweza kutumika kwa kiwango cha shule ya kati, hadi fizikia ya utangulizi katika ngazi ya chuo kikuu. "Kifaa hiki cha Digital Standing Waves kinaelezea maendeleo ya msingi ya dhana zinazohitajika kwa wanafunzi kuelewa nini ""wimbi la kusimama"" na jinsi linavyoweza kuundwa." Rasilimali ya Dijiti ya Mawimbi ya kusimama inajumuisha uhuishaji, viungo vya uigizaji wa bure, maonyesho yaliyopendekezwa, na video ambazo zinaweza kutumika katika darasa lako. Vifaa vyote kutumika katika mapendekezo ya kusimama wimbi maonyesho kuja kutoka <unk>Mechanical Wave Complete Bundle<unk>. Kuendeleza Wimbi la Kuegemea Hatua ya 1 (Mwongozo wa Wimbi la Mechanical) Hatua ya kwanza katika kuwasaidia wanafunzi kujenga uelewa wa dhana ya mawimbi ya kusimama inahusisha uelewa wa msingi wa mawimbi ya mitambo. Slinkies na springs coiled ni njia kubwa ya mikono juu ya kuanzisha misingi ya mawimbi ya mitambo kwa wanafunzi. Arbor Scientific ina chaguzi nyingi ikiwa ni pamoja na <unk> helical spring<unk>, <unk> spring wave<unk>, na <unk>super spring<unk>. Wanafunzi wanaweza pia kuchunguza misingi ya mawimbi ya mitambo kwa kutumia simulation maingiliano, kama moja inapatikana kutoka PhET Interactive Simulations tovuti yenye jina <unk>Waves juu ya String<unk>. "Kama ilivyoelezwa hapo juu, mawimbi ya mitambo ni ""mchanganyiko ambao husafiri kupitia kitu kigumu, kioevu au gesi wakati nyenzo au kati inaharibiwa, kusafiri nje kutoka kwa usumbufu wa awali katika mwelekeo wote unaopatikana wa nyenzo, na ""wimbi"" ni nishati inayohama, sio suala linalohama." Inasaidia kuonyesha au kujadili mifano kadhaa tofauti ya mawimbi ya mitambo kusonga kupitia vimeng'enya, kioevu na gesi. Wanafunzi wanaweza kutumia Waves on a String simulation ili kuchunguza zaidi mali na tabia ya kusonga usumbufu au mawimbi. Uchunguzi hutoa wanafunzi uelewa wa jinsi ya kuelezea wimbi amplitude, msukumo upana na kasi. Uchunguzi pia huanzisha dhana ya damping, mwenendo wa wimbi iliyoonyeshwa, na athari ya mvutano juu ya kasi ya wimbi. Unaweza kupata viungo kwa simulation na sambamba mwanafunzi worksheet katika Standing Waves Digital Resource. Kwa kutumia Waves on a String, wanafunzi wanaweza kutambua sifa za wimbi kama vile amplitude na upana wa msukumo, kutambua kwamba kasi ya wimbi inabadilika tu na mvutano katika kamba, msukumo wa wimbi unaonyeshwa tofauti kutoka kwa mwisho ulio wazi au uliowekwa, na kupungua kwa urefu wa wimbi au amplitude kwa muda. Hatua ya pili: Kujifunza kuhusu mawimbi ya mara kwa mara (Periodic Waves) Wanafunzi wanahitaji kuelewa kwamba mawimbi ya mara kwa mara ni kuundwa na usumbufu wa mara kwa mara au mara kwa mara, na kuwa na amplitude quantifiable, mzunguko, urefu wa wimbi na kasi ya wimbi. Waves juu ya String simulation ni chombo kubwa ya kuanzisha mawimbi ya mara kwa mara kwa sababu simulation inaruhusu wewe kwa urahisi kubadilisha mzunguko wa usumbufu na kuona athari yake juu ya mawimbi ya mawimbi ya mara kwa mara. Uigaji pia inaweza kusimamishwa au hatua kwa njia ya fremu kwa fremu kuonyesha jinsi amplitude na urefu wa wimbi inaweza kupimwa. Katika shule ya sekondari na wanafunzi wa ngazi ya chuo kikuu pia wanahitaji kuanzishwa kwa equation ya kasi ya wimbi inayoonyesha uhusiano wa kiasi kati ya mzunguko wa wimbi la mara kwa mara, urefu wa wimbi na kasi. Hatua ya 3 - Kuingiliana kwa Mawimbi: Mara baada ya wanafunzi kuelewa mali za msingi za mawimbi na mawimbi ya mara kwa mara, wanafunzi wanahitaji kuanzishwa kwa jinsi mawimbi yanavyoingiliana. Ikiwa kuna mawimbi mawili yanayoenda katika mwelekeo tofauti kupitia kamba moja, nini kinatokea wanapokutana? Ni kama magari mawili yanayogongana? Je, mawimbi hayo hukutana na kusimamishwa? Je, mawimbi hayo hupindukia? Je, kuna badiliko lolote kuhusu kila msukumo wa mawimbi baada ya kuingiliana? Hata kwa uchunguzi wa makini, matokeo ya mwingiliano wa wimbi ni vigumu sana kuchunguza kwa kutumia slinkies au kamba, isipokuwa wewe ni kutumia video ya kasi. Kwa sababu hii ni tena muhimu kutumia Waves juu ya String simulation kuchunguza jinsi mawimbi kuingiliana. Tumia simulation kuunda matukio tofauti ambapo pulses mbili kukutana wakati wa kusafiri katika mwelekeo kinyume. Ni muhimu kuwaonyesha wanafunzi mifano ya pulses na upana tofauti, na mifano ambapo pulses kukutana upande huo wa kamba na pande kinyume. Wimbi la kusimama la digital lina matukio manne tofauti ambayo yatasaidia wanafunzi kutoa hitimisho kuhusu jinsi mawimbi yanavyoingiliana (tazama hapa chini). Unataka wanafunzi waone kwamba mawimbi hayo mawili hupita bila kuathiriwa na mwingiliano. Kila wimbi linaonekana sawa na husafiri katika mwelekeo uleule, kwa kasi ileile kama ilivyokuwa kabla ya mwingiliano. Jambo la kuvutia sana hutokea wakati mawimbi hayo mawili yanapokuwa mahali pamoja kwa wakati mmoja. Wakumbushe wanafunzi kwamba kamba inaweza kufanya jambo moja tu. kamba haiwezi kuwa katika maeneo mawili kwa wakati mmoja. Matokeo ya mwingiliano wa mawimbi yanajulikana kama kuingiliana kwa mawimbi au superposition. Wakati pulses ni upande huo wa kamba kuna <unk> constructive kuingiliana<unk> ambapo jumla ya uhamisho wa kamba katika eneo la overlap ni kubwa kuliko ingekuwa kutokana na yoyote ya pulses binafsi au mawimbi. Wakati pulses ni upande kinyume cha kamba kuna <unk>haribu kuingiliana<unk> ambapo jumla ya uhamisho wa kamba katika eneo la overlap ni ndogo kuliko ingekuwa kutokana na yoyote ya pulses mtu binafsi au mawimbi. Ni thamani ya kusitisha hapa tu ili kuonyesha jinsi kubwa ya tofauti kuna kati ya jinsi mawimbi kuingiliana ikilinganishwa na jinsi vitu kufanywa kutoka suala kuingiliana. Magari mawili yanapogongana, yanabadilika mwendo na mwelekeo wa mwendo. Ikiwa magari mawili yanashirikiana kama mawimbi, yatapita bila kubadilisha kasi, na mahali wanapojipinda, sehemu ya gari katika eneo hilo itakuwa kubwa au ndogo kuliko yoyote ya magari mawili. Hatua ya 4 (Interference ya Waves ya Kipindi) Sasa kwamba wanafunzi wanaelewa jinsi pulses mbili za wimbi za mtu binafsi zinavyoingiliana, unaweza kuwasaidia wanafunzi kufikiria jinsi mawimbi mawili ya mara kwa mara yanavyoshiriki wakati wa kupita kupitia kila mmoja katika nyenzo moja, wakisonga katika mwelekeo tofauti. Wimbi la mzunguko hupita kwa njia tofauti, bila mabadiliko ya urefu wa wimbi, amplitude na kasi ya wimbi, lakini vipi kuhusu nyenzo ambayo mawimbi yanapita? Kila sehemu ya nyenzo hiyo itakuwa sawa na jumla ya mawimbi ya mtu binafsi yanayopita katika eneo hilo kwa wakati huo hususa. Kutakuwa ama constructive au uharibifu kuingiliana kutokea katika kila eneo kando ya urefu wa nyenzo. Ni inakuwa ngumu sana kufikiri hii kupitia bila kutumia uhuishaji. The Standing Waves Digital Resource inajumuisha uhuishaji kusaidia wanafunzi kuona jinsi mawimbi mawili ya mara kwa mara yanavyoingiliana wakati wa kusonga kupitia nyenzo moja. Fikiria wanafunzi wawili wakitikisa kila ncha ya kamba au kamba, kila mmoja akitoa mawimbi ya mara kwa mara ambayo yatapita kwa kila mmoja. Mstari wa bluu na nyekundu unawakilisha mawimbi ya mzunguko, lakini kumbuka, kila sehemu ya nyenzo inaweza kuwa mahali moja tu kwa wakati mmoja. Mstari wa zambarau unaonyesha jumla ya mawimbi ya mara kwa mara ya nyekundu na bluu katika kila hatua kando ya urefu wa nyenzo. Mstari wa zambarau unawakilisha tabia halisi ya nyenzo hiyo. Kuna maeneo ambayo hayaondoki katika usawa, na maeneo mengine ambayo yanaondolewa mbali zaidi na usawa kuliko amplitude ya mawimbi ya mara kwa mara. Maeneo haya kando ya urefu wa nyenzo hayaibadili nafasi kwa muda. Wanaonekana wamesimama kimya. Hii ndiyo sababu tabia hii inaitwa "wimbi la kusimama", na huundwa na mawimbi mawili ya mara kwa mara yanayohama kupitia nyenzo ileile katika mwelekeo tofauti. Weka alama kwenye sehemu kadhaa za habari hiyo na uwaombe wanafunzi wajaribu kueleza ni kwa nini inafanya hivyo. Maeneo kando ya urefu wa nyenzo ambayo kubaki katika usawa ni inayojulikana kama <unk>nodes<unk>. Hizi ni maeneo ya kuingiliana kabisa uharibifu kati ya mawimbi mawili ya kipindi. Kila wakati kilele kutoka wimbi moja la mara kwa mara hupita kupitia eneo hilo, shimo kutoka wimbi jingine la mara kwa mara hufuta, na kinyume chake. Maeneo kando ya urefu wa nyenzo ambayo yana uhamisho mkubwa kutoka usawa yanajulikana kama <unk>antinodes<unk>. Hii ni matokeo ya usumbufu mkubwa wa ujenzi kati ya mawimbi mawili ya mara kwa mara. Kuingiliana kwa ujenzi hufanyika mahali ambapo vilele vya kila wimbi la mara kwa mara hukutana, au mitaro ya kila wimbi la mara kwa mara hukutana. Hii ni wakati mzuri wa kuwasaidia wanafunzi kuona uhusiano kati ya urefu wa wimbi la mawimbi ya mara kwa mara kusonga kupitia nyenzo, na umbali kati ya nodes au antinodes. Picha hiyo inapaswa kuwasaidia wanafunzi kuona kwamba umbali kati ya node ni sawa na nusu ya urefu wa wimbi la mawimbi ya mara kwa mara kusonga kupitia nyenzo. Martin Kirby, mwalimu mwingine wa fizikia, ameunda simulation kubwa kwa wanafunzi kuchunguza mifumo ya kuingiliana iliyoundwa na mawimbi mengi tofauti ya mara kwa mara aitwaye <unk>Wave Add<unk>. Kuna kiungo kwa simulation hii ni pamoja na katika Standing Waves Digital Resource. Jambo moja la mwisho la kukumbuka ni kwamba wimbi la kusimama linaweza kuundwa kwa chanzo kimoja tu cha usumbufu wa mara kwa mara. Kama ilivyoelezwa hapo awali, mawimbi ya kusimama hutengenezwa na mawimbi mawili ya mara kwa mara ambayo husafiri kupitia nyenzo moja kwa mwelekeo tofauti, lakini chanzo cha pili cha mawimbi ya mara kwa mara inaweza kuwa mawimbi yaliyoonyeshwa kutoka mwisho wa kudumu. Picha ya mkurugenzi wa utafiti wa Dr. Dan Russell kutoka Chuo Kikuu cha Penn State inaonyesha jinsi ya kufanya hivyo. Hatua ya 5 (Standing Waves in Strings) Sasa uko tayari kuwaonyesha wanafunzi baadhi ya maonyesho ya mawimbi ya kusimama ili kuwasaidia kuchunguza zaidi matukio. Tumia Mechanical Wave Driver na Sine Wave Generator kujaribu kuunda mawimbi ya kusimama kwenye kamba iliyoinuliwa. Anza na mzunguko wa chini na hatua kwa hatua kuongeza mzunguko mpaka wimbi la kwanza kusimama ni kuundwa. Wanafunzi wanahesabu idadi ya nodes na antinodes katika wimbi la kwanza la kusimama na kuamua ni kiasi gani cha urefu wa wimbi unafaa katika urefu wa kamba. Kwa wimbi la kwanza linalowezekana, nusu ya urefu wa wimbi inafaa katika urefu wa kamba kwani kila mwisho ulikuwa node na kulikuwa na antinode moja katikati. Sasa hatua kwa hatua kuongeza mzunguko mpaka ijayo uwezekano wa kusimama wimbi ni kupatikana, na kuwa na wanafunzi kuhesabu idadi ya nodes na antinodes. Tena, waulize wanafunzi ni kiasi gani cha urefu wa wimbi kinafaa katika urefu wa kamba. Mzunguko wa kusimama hutengenezwa katika kamba yenye ncha mbili za kudumu wakati tu urefu wa kamba ni sawa na nusu ya urefu wa mawimbi ya mara kwa mara, au nusu mbili za urefu wa mawimbi, au nusu tatu za urefu wa mawimbi, na kadhalika. Unaweza kutumia uhuishaji katika Waves Standing Digital Resource kuonyesha nne ya kwanza uwezekano wa mawimbi ya kusimama kwa kamba na mwisho mbili fasta. Ingawa oscillator iliyounganishwa na mwisho mmoja wa kamba ni kusonga juu na chini, ni kwa ufanisi mwisho fasta tangu displacement yake kutoka usawa ni ndogo ikilinganishwa na displacement ya wimbi kusimama katika antinodes. Ikiwa unajua urefu wa kamba, unaweza kuhesabu urefu wa mawimbi maalum ambayo itaunda kila wimbi la kusimama, na katika viwango vya shule ya sekondari na chuo kikuu unaweza kutumia usawa wa kasi ya wimbi kuhesabu masafa ya kuendesha yanayohitajika kuunda urefu huo wa wimbi. Mfano wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko wa Mzunguko. Sasa wanafunzi wanapoona michoro ya mchanga kwenye sahani ya chuma inayotetemeka, wako tayari kuelewa ni nini kinachotokeza michoro hiyo. Sawa na kamba iliyoinuliwa, masafa maalum tu ya usumbufu yatatengeneza mawimbi ya kusimama katika uso wa chuma. Kama mawimbi ya kusimama ni kuundwa katika uso wa pande mbili, lazima kuwe na antinodes, maeneo katika nyenzo ambayo ni kuhamishwa juu na chini zaidi ya kuhamishwa kwa Mechanical Wave Driver. Hizi zingekuwa maeneo ya kuingiliana kwa ujenzi wa juu ambapo vilele vinajipinda na vilele vingine, na mitaro inajipinda na mitaro mingine. Wakati wimbi la kusimama linapowepo, chumvi itaondoka kutoka kwa maeneo haya ya antinode hadi sehemu za nyenzo zenye kupinduka kidogo juu na chini, kama vile maeneo ya node. Hakuna juu na chini ya uhamisho katika maeneo ya node kwa sababu kuna kamili uharibifu kuingiliana kati ya mawimbi ya mara kwa mara kusonga kupitia nyenzo. Kwa hiyo wakati wimbi la kusimama linapokuwapo kwa mzunguko fulani, mchanga utahama kutoka kwa antinodes na kukaa kwenye nodes. Utata wa mifumo ya mawimbi ya kusimama kuongezeka, kama wewe kuongeza mzunguko wa Mechanical Wave Driver. Hii ni kwa sababu ya ukweli kwamba mzunguko wa juu huunda urefu mdogo wa wimbi na mawimbi ya kusimama yanayotokana yatakuwa na idadi kubwa ya nodes na antinodes. Natumaini makala hii imekusaidia kupata wazo la jinsi ya kusaidia wanafunzi wako kuelewa misingi ya mawimbi ya kusimama. Kumbuka kuangalia nje ya uhuishaji, simulations na video zinazohusiana katika Wimbi Standing Digital Rasilimali. Aaron Debbink ni mwalimu wa fizikia na uzoefu wa miaka 16 katika darasa, ana shahada ya uhandisi na shahada ya uzamili katika fizikia. Yeye ni shauku juu ya kujenga utamaduni darasani kwamba thamani ya uchunguzi na gari kwa ajili ya uelewa. Aaron ni mwanafunzi wa kwanza wa shule ya msingi ya Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi wa Ufundi. Aaron ni mwanafunzi wa Knowles Teaching Fellow na mpokeaji wa Tuzo ya Mwalimu wa Yale ya 2019.