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24:[["$","$L33",null,{"feature":"flashcards"}],"$L34",["$","$L35",null,{"href":"/a-level/physics/aqa/progressive-waves/flashcards","label":"Progressive waves","subjectCode":"a_level_physics__aqa","surface":"flashcards"}],["$","$L36",null,{"runner":["$","div",null,{"className":"min-h-0 flex-1 overflow-hidden bg-background","children":["$","$L37",null,{"decks":[{"id":"84db061c-572a-446f-ad5a-294e2c47d405","title":"Progressive waves Flashcards","topicId":"90032f68-85ea-57fa-b3c8-bbafda7d1936","topicSlug":"progressive-waves","cards":[{"id":"18bf7313-4191-4b88-8e0a-02bac83aea30","cardType":"basic","points":1,"front":"What does a progressive wave transfer from one position to another?","back":"Energy (without transferring matter).","mcqOptions":[{"text":"Energy (without transferring matter).","correct":true,"explanation":"A progressive wave carries energy between positions while particles of the medium only oscillate around equilibrium, so there is no net transfer of matter with the wave."},{"text":"Matter (without transferring energy).","correct":false,"explanation":"This reverses the defining behaviour of a progressive wave. The wave transfers energy, while the particles of the medium oscillate without being permanently transported."},{"text":"Particles (without transferring energy).","correct":false,"explanation":"Particles may oscillate as the wave passes, but they are not carried from one position to another. It is energy that is transferred through the medium."},{"text":"The medium (without transferring energy).","correct":false,"explanation":"The medium supports the wave but does not travel along with it. Its particles oscillate locally while the progressive wave transfers energy between positions."}]},{"id":"742c671b-eb42-475a-8c14-584eb6c2ab04","cardType":"cloze","points":null,"front":"In a progressive wave, particles oscillate around their [ ] position rather than moving permanently with the wave.","back":"In a progressive wave, particles oscillate around their equilibrium (rest) position rather than moving permanently with the wave.","mcqOptions":[{"text":"equilibrium (rest)","correct":true,"explanation":"The equilibrium position is the particle’s undisturbed central position. During wave motion, the particle moves repeatedly to either side of this position."},{"text":"maximum (crest)","correct":false,"explanation":"A crest represents maximum positive displacement, not the central position about which a particle oscillates. Particles oscillate around their equilibrium or rest position."},{"text":"minimum (trough)","correct":false,"explanation":"A trough represents maximum negative displacement rather than the centre of oscillation. The particle repeatedly returns through and oscillates around its equilibrium or rest position."},{"text":"instantaneous (current)","correct":false,"explanation":"The instantaneous position changes throughout the oscillation, so it cannot be the fixed centre of motion. That fixed centre is the equilibrium or rest position."}]},{"id":"aa05e5cb-11d0-4373-b418-4327f11bd5aa","cardType":"cloze","points":null,"front":"In a progressive wave, particles oscillate around their equilibrium (rest) position rather than moving [ ] with the wave.","back":"In a progressive wave, particles oscillate around their equilibrium (rest) position rather than moving permanently with the wave.","mcqOptions":[{"text":"permanently","correct":true,"explanation":"Particles oscillate locally and return repeatedly towards equilibrium, so they do not undergo permanent displacement in the direction in which the progressive wave travels."},{"text":"uniformly","correct":false,"explanation":"Uniform motion is not the key distinction because oscillating particles continually change velocity. The important point is that particles do not move permanently with the wave."},{"text":"steadily","correct":false,"explanation":"Particles do not travel steadily with a progressive wave because their motion reverses during each oscillation. They remain near equilibrium rather than moving permanently with it."},{"text":"irreversibly","correct":false,"explanation":"Wave-driven particle motion is repeatedly reversed, but the missing term specifically contrasts oscillation with permanent transport. Particles do not move permanently with the progressive wave."}]},{"id":"bb78d31e-dc54-4c6d-b742-6ce078ca0aab","cardType":"basic","points":2,"front":"How are displacement and amplitude related?","back":"Amplitude is the maximum displacement from the equilibrium position.","mcqOptions":[{"text":"Amplitude is the maximum displacement from the equilibrium position.","correct":true,"explanation":"Amplitude measures the greatest magnitude of displacement reached on either side of equilibrium. It is therefore the maximum displacement from the equilibrium position."},{"text":"Amplitude is the instantaneous displacement from the equilibrium position.","correct":false,"explanation":"Instantaneous displacement varies throughout an oscillation and may have any value between the extremes. Amplitude is the maximum displacement from the equilibrium position."},{"text":"Amplitude is the average displacement from the equilibrium position.","correct":false,"explanation":"Over a complete symmetric oscillation, the average displacement is zero. Amplitude instead measures the maximum displacement reached from the equilibrium position."},{"text":"Amplitude is the total displacement from crest to trough.","correct":false,"explanation":"The crest-to-trough displacement is twice the amplitude because it spans both sides of equilibrium. Amplitude is the maximum displacement from equilibrium to one extreme."}]},{"id":"aba0f6a6-8e8d-49b9-ae9e-ef19faa35978","cardType":"basic","points":1,"front":"What is the definition of wavelength ($\\lambda$)?","back":"The least distance between two adjacent particles oscillating in phase.","mcqOptions":[{"text":"The least distance between two adjacent particles oscillating in phase.","correct":true,"explanation":"Particles separated by one wavelength are at identical points in their oscillations. Wavelength is the shortest distance between adjacent particles that are in phase."},{"text":"The greatest distance between two adjacent particles oscillating in phase.","correct":false,"explanation":"Many particles separated by whole-number multiples of wavelength can be in phase, so there is no defining greatest distance. Wavelength is the least such distance."},{"text":"The least distance between two adjacent particles oscillating in antiphase.","correct":false,"explanation":"Particles in antiphase differ by half a cycle, so their least separation is half a wavelength. A wavelength separates adjacent particles oscillating in phase."},{"text":"The least distance between a crest and its adjacent trough oscillating in phase.","correct":false,"explanation":"An adjacent crest and trough are separated by half a wavelength and represent opposite phases. Wavelength is the least distance between adjacent points in phase."}]},{"id":"1c40741a-e6ca-4595-b637-f4f948c6cb70","cardType":"cloze","points":null,"front":"To find the wavelength of a wave from a graph, the horizontal axis must represent [ ].","back":"To find the wavelength of a wave from a graph, the horizontal axis must represent distance.","mcqOptions":[{"text":"distance","correct":true,"explanation":"Wavelength is a spatial separation, so it is measured from a displacement-distance graph. The horizontal distance between adjacent in-phase points gives one wavelength."},{"text":"time","correct":false,"explanation":"A displacement-time graph shows how one point oscillates and allows the time period to be measured. Wavelength requires distance on the horizontal axis."},{"text":"displacement","correct":false,"explanation":"Displacement normally appears on the vertical axis and shows particle position relative to equilibrium. The horizontal axis must show distance to measure wavelength."},{"text":"phase","correct":false,"explanation":"Phase can identify matching stages of oscillation, but wavelength is a physical separation. The horizontal axis must represent distance so that separation can be measured."}]},{"id":"5e5dee3a-6c4d-4805-89bc-10911e9276ba","cardType":"basic","points":1,"front":"What property of a wave is found by measuring the horizontal interval between peaks on a displacement-time graph?","back":"Time period ($T$).","mcqOptions":[{"text":"Time period ($T$).","correct":true,"explanation":"On a displacement-time graph, successive peaks occur one complete oscillation apart. Their horizontal time interval is therefore the time period, $T$."},{"text":"Frequency ($f$).","correct":false,"explanation":"The interval between successive peaks gives the time for one oscillation, not the number per second. It is the time period, with $f=1/T$."},{"text":"Wavelength ($\\lambda$).","correct":false,"explanation":"Wavelength is measured between peaks on a displacement-distance graph. On a displacement-time graph, the horizontal peak-to-peak interval gives the time period, $T$."},{"text":"Phase difference ($\\phi$).","correct":false,"explanation":"Phase difference compares positions within an oscillation rather than directly naming the peak-to-peak time interval. That interval is one complete time period, $T$."}]},{"id":"db63eb03-180b-4bb9-8b18-48f911e7c7f5","cardType":"basic","points":1,"front":"State the relationship between frequency ($f$) and time period ($T$).","back":"$$f = \\frac{1}{T}$","mcqOptions":[{"text":"$$f = \\frac{1}{T}$","correct":true,"explanation":"Frequency is the number of complete oscillations per second, while period is the time for one oscillation. They are reciprocals, so $f=1/T$."},{"text":"$$f = T$","correct":false,"explanation":"This treats frequency and period as the same quantity despite their different units. They are reciprocals: frequency in hertz is given by $f=1/T$."},{"text":"$$f = \\frac{1}{2T}$","correct":false,"explanation":"This incorrectly counts one oscillation as lasting two periods, halving the calculated frequency. One complete oscillation takes time $T$, so $f=1/T$."},{"text":"$$f = \\frac{2}{T}$","correct":false,"explanation":"This incorrectly counts two complete oscillations during each period, doubling the frequency. By definition, one period contains one oscillation, so $f=1/T$."}]},{"id":"59b881ad-39dd-4f8e-8342-1befb6a0a8a8","cardType":"basic","points":1,"front":"State the wave equation linking speed ($c$), frequency ($f$), and wavelength ($\\lambda$).","back":"$$c = f\\lambda$","mcqOptions":[{"text":"$$c = f\\lambda$","correct":true,"explanation":"Wave speed equals the number of cycles passing per second multiplied by the distance covered in each cycle, so speed is frequency multiplied by wavelength."},{"text":"$$c = \\frac{f}{\\lambda}$","correct":false,"explanation":"Dividing frequency by wavelength does not produce units of speed. The correct relationship is $c = f\\lambda$, since frequency multiplied by wavelength gives metres per second."},{"text":"$$c = \\frac{\\lambda}{f}$","correct":false,"explanation":"Dividing wavelength by frequency incorrectly multiplies wavelength by the period. Wave speed is instead found using $c = f\\lambda$."},{"text":"$$c = \\frac{1}{f\\lambda}$","correct":false,"explanation":"Taking the reciprocal of the product gives inverse speed, not speed. The wave equation is $c = f\\lambda$ because each cycle advances one wavelength."}]},{"id":"fe170e71-0292-4642-9302-02d06685a107","cardType":"basic","points":1,"front":"What is the standard unit for frequency?","back":"Hertz ($\\text{Hz}$).","mcqOptions":[{"text":"Hertz ($\\text{Hz}$).","correct":true,"explanation":"Frequency measures the number of complete cycles occurring each second, so its standard unit is the hertz, where $1\\text{ Hz}=1\\text{ s}^{-1}$."},{"text":"Seconds ($\\text{s}$).","correct":false,"explanation":"Seconds measure period, the time taken for one cycle, rather than frequency. Frequency is measured in hertz ($\\text{Hz}$), equivalent to inverse seconds."},{"text":"Metres ($\\text{m}$).","correct":false,"explanation":"Metres measure distance or wavelength, not the rate of oscillation. Frequency counts cycles per second and is measured in hertz ($\\text{Hz}$)."},{"text":"Radians per second ($\\text{rad s}^{-1}$).","correct":false,"explanation":"Radians per second measure angular frequency, usually denoted $\\omega$. Ordinary frequency $f$ counts complete cycles per second and is measured in hertz ($\\text{Hz}$)."}]},{"id":"1184567b-397b-4af8-a969-20f1d030d582","cardType":"cloze","points":null,"front":"One full cycle of an oscillation is equal to $\\text{[...]}$ or $2\\pi$ radians.","back":"One full cycle of an oscillation is equal to $\\boxed{360^\\circ}$ or $2\\pi$ radians.","mcqOptions":[{"text":"$$360^\\circ$","correct":true,"explanation":"A complete rotation or oscillation cycle corresponds to $360^\\circ$. This is equivalent to $2\\pi$ radians because $180^\\circ$ equals $\\pi$ radians."},{"text":"$$180^\\circ$","correct":false,"explanation":"An angle of $180^\\circ$ is only half a cycle and equals $\\pi$ radians. One full cycle is $360^\\circ$, equivalent to $2\\pi$ radians."},{"text":"$$90^\\circ$","correct":false,"explanation":"An angle of $90^\\circ$ represents one quarter of a cycle and equals $\\pi/2$ radians. A full cycle is $360^\\circ$."},{"text":"$$720^\\circ$","correct":false,"explanation":"An angle of $720^\\circ$ represents two complete cycles, not one. A single full cycle is $360^\\circ$, which equals $2\\pi$ radians."}]},{"id":"546b6152-7c1b-42ca-82a8-421b4c36e68d","cardType":"cloze","points":null,"front":"One full cycle of an oscillation is equal to $360^\\circ$ or $\\text{[...]}$ radians.","back":"One full cycle of an oscillation is equal to $360^\\circ$ or $\\boxed{2\\pi}$ radians.","mcqOptions":[{"text":"$$2\\pi$","correct":true,"explanation":"One full cycle is $360^\\circ$, and converting degrees to radians gives $360\\times\\pi/180=2\\pi$ radians."},{"text":"$$\\pi$","correct":false,"explanation":"An angle of $\\pi$ radians equals $180^\\circ$, so it represents only half a cycle. One complete cycle is $2\\pi$ radians."},{"text":"$$\\frac{\\pi}{2}$","correct":false,"explanation":"An angle of $\\pi/2$ radians equals $90^\\circ$, which is one quarter of a cycle. A full cycle corresponds to $2\\pi$ radians."},{"text":"$$4\\pi$","correct":false,"explanation":"An angle of $4\\pi$ radians corresponds to $720^\\circ$, so it represents two complete cycles. One full cycle is $2\\pi$ radians."}]},{"id":"901062ed-7cfb-4ce6-b0de-44f771ab6899","cardType":"basic","points":1,"front":"What does phase difference measure?","back":"How far out of step two particles or waves are.","mcqOptions":[{"text":"How far out of step two particles or waves are.","correct":true,"explanation":"Phase difference compares positions within an oscillation cycle, indicating how far the motions of two particles or waves are out of step."},{"text":"How far apart in space two particles or waves are.","correct":false,"explanation":"Physical separation is a distance and does not by itself define phase difference. Phase difference measures how far out of step two oscillations are within their cycles."},{"text":"How different in amplitude two particles or waves are.","correct":false,"explanation":"Amplitude difference compares maximum displacements, not positions within the oscillation cycle. Phase difference measures how far out of step two particles or waves are."},{"text":"How much energy two particles or waves transfer.","correct":false,"explanation":"Energy transfer depends on factors such as amplitude and power, rather than defining phase. Phase difference measures how far out of step two particles or waves are."}]},{"id":"ca27b818-59aa-4c12-b1f1-303a2cdd6ea1","cardType":"basic","points":2,"front":"State two possible phase differences (in radians) for two particles oscillating in phase.","back":"

$0$
$2\\pi$

","mcqOptions":[{"text":"

$0$
$2\\pi$

","correct":true,"explanation":"In-phase particles occupy the same point in their oscillation cycles, so their phase difference is zero or any whole multiple of $2\\pi$ radians."},{"text":"

$\\pi$
$3\\pi$

","correct":false,"explanation":"Odd multiples of $\\pi$ describe anti-phase oscillations, where the particles are half a cycle apart. In-phase differences include $0$ and $2\\pi$ radians."},{"text":"

$\\frac{\\pi}{2}$
$\\frac{5\\pi}{2}$

","correct":false,"explanation":"These values are odd quarter-cycle differences, so the particles are neither in phase nor anti-phase. In-phase differences include $0$ and $2\\pi$ radians."},{"text":"

$0$
$\\pi$

","correct":false,"explanation":"Although $0$ represents particles in phase, $\\pi$ places them in anti-phase. Two valid in-phase differences are $0$ and $2\\pi$ radians."}]},{"id":"79a5efc8-91f9-468f-b63c-4a5fdada5c66","cardType":"basic","points":2,"front":"Describe the motion of two particles in anti-phase.","back":"One reaches maximum positive displacement while the other reaches maximum negative displacement.","mcqOptions":[{"text":"One reaches maximum positive displacement while the other reaches maximum negative displacement.","correct":true,"explanation":"Anti-phase particles differ in phase by $\\pi$ radians, so their displacements are equal in magnitude and opposite in direction at corresponding instants."},{"text":"One reaches maximum positive displacement while the other also reaches maximum positive displacement.","correct":false,"explanation":"Reaching maximum positive displacement together describes particles in phase, not anti-phase. Anti-phase particles reach equal maximum displacements on opposite sides of equilibrium."},{"text":"One reaches maximum positive displacement while the other reaches the equilibrium position.","correct":false,"explanation":"A maximum and an equilibrium position are separated by a quarter cycle, giving a phase difference of $\\pi/2$, not the $\\pi$ difference required for anti-phase."},{"text":"Both pass through the equilibrium position while moving in the same direction.","correct":false,"explanation":"Particles passing equilibrium in the same direction are in phase. Anti-phase particles can pass equilibrium together only while moving in opposite directions."}]},{"id":"bc42bb8a-a21e-4195-80a0-045317c3754a","cardType":"cloze","points":null,"front":"On a single wave, particles in anti-phase are separated by a distance of $\\text{[...]}$.","back":"On a single wave, particles in anti-phase are separated by a distance of $\\boxed{(n + 0.5)\\lambda}$.","mcqOptions":[{"text":"$$(n + 0.5)\\lambda$","correct":true,"explanation":"Anti-phase points are separated by an odd number of half-wavelengths. This is expressed as $(n+0.5)\\lambda$ for integer values of $n$."},{"text":"$$n\\lambda$","correct":false,"explanation":"Separations of whole wavelengths place particles at identical points in the wave cycle, so they are in phase. Anti-phase separation is $(n+0.5)\\lambda$."},{"text":"$$(n + 0.25)\\lambda$","correct":false,"explanation":"A quarter-wavelength offset corresponds to a phase difference of $\\pi/2$, not anti-phase. Anti-phase requires an odd half-wavelength separation, $(n+0.5)\\lambda$."},{"text":"$$(n + 1)\\lambda$","correct":false,"explanation":"Adding one whole wavelength gives another integer-wavelength separation, so the particles remain in phase. Anti-phase points are separated by $(n+0.5)\\lambda$."}]},{"id":"e26c72b2-232c-43a3-9e1f-ba28dd3ff947","cardType":"basic","points":1,"front":"State the formula for phase difference ($\\Delta \\phi$) in radians between two points separated by distance $\\Delta x$.","back":"$$\\Delta \\phi = \\frac{2\\pi \\Delta x}{\\lambda}$","mcqOptions":[{"text":"$$\\Delta \\phi = \\frac{2\\pi \\Delta x}{\\lambda}$","correct":true,"explanation":"A complete wavelength corresponds to a phase change of $2\\pi$ radians, so a separation $\\Delta x$ corresponds to the fraction $\\Delta x/\\lambda$ of $2\\pi$."},{"text":"$$\\Delta \\phi = \\frac{\\Delta x}{\\lambda}$","correct":false,"explanation":"This omits the factor of $2\\pi$, giving the separation as a fraction of a wavelength rather than as an angle. The correct formula is $\\Delta \\phi = \\frac{2\\pi \\Delta x}{\\lambda}$."},{"text":"$$\\Delta \\phi = \\frac{2\\pi \\lambda}{\\Delta x}$","correct":false,"explanation":"This reverses the wavelength fraction: phase difference increases with separation and decreases with wavelength. The correct relationship is $\\Delta \\phi = \\frac{2\\pi \\Delta x}{\\lambda}$."},{"text":"$$\\Delta \\phi = \\frac{\\pi \\Delta x}{\\lambda}$","correct":false,"explanation":"Using $\\pi$ treats one wavelength as only half a cycle. One complete wavelength represents $2\\pi$ radians, so $\\Delta \\phi = \\frac{2\\pi \\Delta x}{\\lambda}$."}]},{"id":"df21b95b-416a-4998-b242-76d635ac8211","cardType":"basic","points":1,"front":"If two points on a wave are separated by $1.5\\lambda$, what is their phase relationship?","back":"Anti-phase.","mcqOptions":[{"text":"Anti-phase.","correct":true,"explanation":"A separation of $1.5\\lambda$ gives a phase difference of $3\\pi$ radians, equivalent to $\\pi$ radians modulo $2\\pi$. The points are therefore in anti-phase."},{"text":"In phase.","correct":false,"explanation":"Points in phase must be separated by a whole number of wavelengths. A separation of $1.5\\lambda$ leaves an extra half-wavelength, so the points are in anti-phase."},{"text":"Quarter-phase.","correct":false,"explanation":"Quarter-phase corresponds to an odd multiple of $\\lambda/4$, not three half-wavelengths. Since $1.5\\lambda$ produces a phase difference equivalent to $\\pi$, the points are in anti-phase."},{"text":"Three-quarter-phase.","correct":false,"explanation":"A three-quarter-cycle phase relationship would correspond to a phase difference of $3\\pi/2$ modulo $2\\pi$. Here $1.5\\lambda$ gives $3\\pi$, equivalent to anti-phase."}]},{"id":"68aeeecc-7835-41c7-9169-249763701d4d","cardType":"basic","points":2,"front":"A wave has $T = 0.40 \\text{ s}$ and $\\lambda = 1.2 \\text{ m}$. Calculate its speed ($c$).","back":"$$3.0 \\text{ m s}^{-1}$","mcqOptions":[{"text":"$$3.0 \\text{ m s}^{-1}$","correct":true,"explanation":"Wave speed is $c=\\lambda/T$. Substituting $\\lambda=1.2\\text{ m}$ and $T=0.40\\text{ s}$ gives $c=1.2/0.40=3.0\\text{ m s}^{-1}$."},{"text":"$$0.48 \\text{ m s}^{-1}$","correct":false,"explanation":"This comes from multiplying wavelength by period instead of dividing by it. Wave speed is $c=\\lambda/T=1.2/0.40=3.0\\text{ m s}^{-1}$."},{"text":"$$0.33 \\text{ m s}^{-1}$","correct":false,"explanation":"This reverses the required division by calculating $T/\\lambda$. Speed is distance per unit time, so $c=\\lambda/T=1.2/0.40=3.0\\text{ m s}^{-1}$."},{"text":"$$1.5 \\text{ m s}^{-1}$","correct":false,"explanation":"This effectively divides by twice the period, halving the speed. One wavelength is travelled in one period, so $c=1.2/0.40=3.0\\text{ m s}^{-1}$."}]},{"id":"dd0d36d4-1ea5-436b-8c84-0cc9b960079c","cardType":"basic","points":1,"front":"Why should you use fractions of $\\pi$ when calculating phase difference in A-Level Physics?","back":"It is the standard convention for radians and avoids rounding errors from long decimals.","mcqOptions":[{"text":"It is the standard convention for radians and avoids rounding errors from long decimals.","correct":true,"explanation":"Radians are conventionally expressed as exact multiples or fractions of $\\pi$. Keeping $\\pi$ in the result preserves exact values and prevents accumulated rounding errors in later calculations."},{"text":"It is the standard conversion to degrees and avoids using radian measures in calculations.","correct":false,"explanation":"Fractions of $\\pi$ do not convert radians into degrees; they remain radian measures. They are used because they are the standard exact form and avoid decimal rounding errors."},{"text":"It is the standard method for cancelling units and makes every calculated quantity dimensionless.","correct":false,"explanation":"Using fractions of $\\pi$ does not remove units from every physical quantity. It expresses phase angles exactly in the standard radian convention and avoids rounding long decimal approximations."},{"text":"It is the preferred calculator format and makes trigonometric functions evaluate more quickly.","correct":false,"explanation":"Fractions of $\\pi$ are not inherently easier for calculators, which can evaluate decimals too. Their advantage is that they follow radian convention and preserve exact values without rounding."}]},{"id":"e340d812-e640-4d64-a6d4-d70c62b10d8c","cardType":"cloze","points":null,"front":"When using the wave equation, prefixes like $\\text{M}$ (mega) or $\\text{n}$ (nano) must be converted to [ ].","back":"When using the wave equation, prefixes like $\\text{M}$ (mega) or $\\text{n}$ (nano) must be converted to standard powers of ten.","mcqOptions":[{"text":"standard powers of ten","correct":true,"explanation":"SI prefixes represent powers of ten, such as $\\text{M}=10^6$ and $\\text{n}=10^{-9}$. Converting them ensures all quantities are consistently substituted into the wave equation."},{"text":"positive powers of ten","correct":false,"explanation":"Prefixes must not automatically be converted to positive indices because some, including nano, represent negative powers. They should be converted to standard powers of ten with the correct sign."},{"text":"ordinary decimal fractions","correct":false,"explanation":"Decimal fractions can obscure very large or small scale factors and invite place-value errors. Prefixes should be replaced by standard powers of ten, such as $10^6$ or $10^{-9}$."},{"text":"unscaled SI base units","correct":false,"explanation":"Base units alone do not show how the prefix numerically scales the measurement. Each prefix must be converted to its standard power of ten before values are substituted."}]},{"id":"4f8c67eb-dd57-4c95-8026-e8ff6b21b00f","cardType":"basic","points":2,"front":"Calculate the phase difference in radians if distance $\\Delta x = \\frac{\\lambda}{4}$.","back":"$$\\frac{\\pi}{2}$ rad","mcqOptions":[{"text":"$$\\frac{\\pi}{2}$ rad","correct":true,"explanation":"A full wavelength corresponds to $2\\pi$ radians. A quarter-wavelength separation therefore gives $2\\pi\\times\\frac{1}{4}=\\frac{\\pi}{2}$ rad."},{"text":"$$\\pi$ rad","correct":false,"explanation":"This treats a quarter-wavelength as half a complete cycle. Since one wavelength is $2\\pi$ radians, a quarter-wavelength is $2\\pi/4=\\frac{\\pi}{2}$ rad."},{"text":"$$\\frac{1}{4}$ rad","correct":false,"explanation":"This treats the quarter-wavelength fraction itself as the angle and omits the factor of $2\\pi$. Multiplying by $2\\pi$ gives the correct phase difference, $\\frac{\\pi}{2}$ rad."},{"text":"$$\\frac{3\\pi}{2}$ rad","correct":false,"explanation":"This corresponds to three quarters of a wavelength rather than one quarter. Applying $\\Delta\\phi=2\\pi\\Delta x/\\lambda$ gives $2\\pi/4=\\frac{\\pi}{2}$ rad."}]}]}],"examCountdown":{"examDate":"2027-05-21","paperLabel":"Paper 1"},"heading":"Progressive waves","title":"Progressive waves Flashcards","topbar":{"backHref":"/a-level/physics/aqa/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"90032f68-85ea-57fa-b3c8-bbafda7d1936","walled":"$L39"}],"$L3a","$L3b"]