A sinusoidal transverse wave is travelling on a string. Any point on the string moves in
- (a)SHM with the same angular frequency as that of the wave
- (b)SHM with a different frequency than that of the wave
- (c)Uniform circular motion with the same angular speed as that of the wave
- (d)Uniform circular motion with a different angular speed than that of the wave
Answer
Why
Correct — A, (a) SHM with the same angular frequency as that of the wave.
THE QUESTION IS ABOUT THE MEDIUM, NOT ABOUT THE WAVE. A wave on a string travels; the string does not. Each little element of the string stays where it is along the length of the string and simply moves up and down about its own rest position. What travels along the string is the DISTURBANCE — the pattern, the phase and the energy — and it passes through element after element while every element stays at home.
WRITE THE WAVE DOWN AND THE ANSWER FALLS OUT. A sinusoidal transverse wave travelling along a string is
y(x, t) = A sin(omega t - k x)
where y is the sideways displacement, A the amplitude, omega the angular frequency and k the wave number. Now FIX x — that is, watch one particular point of the string. The term k x becomes a constant, and what is left is
y(t) = A sin(omega t - constant)
which is exactly the equation of SIMPLE HARMONIC MOTION: a sinusoidal oscillation about a mean position, with acceleration proportional to the displacement and directed towards that mean position. The angular frequency of that oscillation is omega — the SAME omega that appears in the wave. It could not be anything else, because omega came from the source that is shaking the string, and every element is being driven by its neighbour at that rate.
WHAT DIFFERS FROM POINT TO POINT IS THE PHASE, NOT THE FREQUENCY. Two elements separated by a distance x along the string are out of step by k x radians; that is why the string shows a travelling shape at all rather than moving as one block. Every element completes the same number of oscillations per second, and on an ideal string with no damping every element swings through the same amplitude.
THE ONE DISTINCTION THAT MATTERS MOST HERE — PARTICLE SPEED IS NOT WAVE SPEED. Differentiating y with respect to time gives the PARTICLE velocity, A omega cos(omega t - k x): it varies from moment to moment, is greatest, A omega, as the element crosses its mean position, and is zero at the two extremes. The WAVE speed is a different quantity altogether, v = omega / k, constant, and set by the medium — for a stretched string, v = square root of (T / mu), the tension divided by the mass per unit length. The element is transverse to the motion: it moves perpendicular to the direction in which the wave travels, which is what makes the wave transverse.
Why the others are wrong
- (b)SHM with a different frequency than that of the wave — Right about the kind of motion and wrong about its rate, and the error is worth naming because frequency is the one property of a wave that is fixed by the SOURCE and not by the medium. The hand or oscillator shaking the end of the string forces every element in turn to move at its own rate; an element vibrating at some other rate would fall out of step with its neighbours and the shape would not propagate unchanged, which is precisely what the wave does. That is also why frequency alone survives when a wave crosses from one medium into another: the speed changes and the wavelength changes with it, but the frequency stays exactly what the source imposed. In this item the point's angular frequency is the omega written in the wave itself. What genuinely differs between one element and another is not frequency but PHASE — how far through the cycle each one happens to be.
- (c)Uniform circular motion with the same angular speed as that of the wave — The angular rate named here is right and the path is wrong. An element of a stretched string moves along a STRAIGHT LINE at right angles to the string, up and down through its rest position; it has no second dimension to move in and it traces no circle. The reason this option is tempting is that a real and useful relation exists between the two motions: simple harmonic motion is the PROJECTION of uniform circular motion on to a diameter, which is why the reference-circle construction is used to picture phase and why angular frequency is measured in radians per second at all. But a projection is not the motion. The genuine case of near-circular particle paths is a surface water wave, where each parcel of water traces a small circle or ellipse because the disturbance there is part transverse and part longitudinal — a different situation from a wave on a string, and one worth keeping separate.
- (d)Uniform circular motion with a different angular speed than that of the wave — Wrong on both counts at once, and it is the option left over when the other three are built. The path is not circular — an element of the string oscillates along a straight line perpendicular to the string, which is the meaning of a TRANSVERSE wave — and the rate is not different, because every element is driven at the frequency imposed by the source and shares the wave's angular frequency omega. A useful discipline on option sets built this way is to test the two ideas separately rather than as a package: decide first what the path is, which eliminates both circular-motion choices at a stroke, and only then decide whether the rate matches. Notice also that the paper words the two pairs differently — the first two say 'frequency' and the last two say 'angular speed' — and neither difference of wording changes what is being asked.
Concept
A WAVE IS A DISTURBANCE THAT TRAVELS THROUGH A MEDIUM WITHOUT THE MEDIUM TRAVELLING WITH IT. Energy and momentum move from place to place; matter oscillates and stays. Everything in this question follows from that one sentence.
TRANSVERSE AND LONGITUDINAL. In a TRANSVERSE wave the particles of the medium oscillate at right angles to the direction of propagation — a wave on a string, a wave on a stretched membrane, and light and all electromagnetic waves, which need no medium at all. In a LONGITUDINAL wave the particles oscillate along the same line as the propagation, producing compressions and rarefactions; sound in air is the standard example. Transverse mechanical waves can only travel through a medium that resists shear, which is why sound waves in a gas or a liquid are longitudinal, and why the S-waves of an earthquake stop at the liquid outer core while the P-waves pass through — the observation that revealed the core's state.
SIMPLE HARMONIC MOTION is oscillation in which the restoring force is proportional to the displacement and directed towards the mean position: a = -(omega^2) y. Its solution is sinusoidal in time, its period is T = 2 pi / omega and its frequency f = 1 / T = omega / 2 pi. The energy of the oscillation is proportional to the square of the amplitude and to the square of the frequency. Every element of the string in this question is executing exactly this motion.
THE QUANTITIES OF A TRAVELLING WAVE, and which of them belongs to what:
FREQUENCY f and angular frequency omega = 2 pi f — set by the SOURCE, and unchanged when the wave enters a new medium. SPEED v — set by the MEDIUM. On a stretched string v = square root of (T / mu), where T is the tension and mu the mass per unit length. WAVELENGTH lambda — the consequence of the other two, since v = f lambda. WAVE NUMBER k = 2 pi / lambda, and v = omega / k. AMPLITUDE A — set by how hard the source is driven, and it governs the energy carried. PHASE — how far through its cycle a given element is; two elements a distance x apart differ in phase by k x.
PARTICLE VELOCITY AGAINST WAVE VELOCITY is the distinction most often tested. The particle velocity is A omega cos(omega t - k x): it changes every instant, peaks at A omega at the mean position and is zero at the extremes. The wave velocity is constant and depends only on the medium. The two are unrelated in size and are frequently confused.
STANDING WAVES arise when a travelling wave meets its own reflection, as on a string fixed at both ends. The result does not travel: it has NODES that never move and ANTINODES that oscillate with maximum amplitude, and only certain frequencies fit the length of the string — the fundamental and its harmonics. That is how a string instrument sounds a definite note, and it is the usual follow-up topic to this one.
General science on this APFC paper is a strand of about five questions, asked at school level and conceptually. This item prints no numbers and no figure at all: it is testing whether a candidate can distinguish the motion of the MEDIUM from the motion of the WAVE, which is the single most productive idea in the whole chapter.
The option set is built with care, and the way it is built is instructive. Two choices offer simple harmonic motion and two offer uniform circular motion; within each pair, one says the rate matches the wave and one says it does not. That is a two-by-two grid, and the efficient method is to decide the two questions separately — first the shape of the path, then the rate — rather than trying to judge four composite statements. Deciding the path removes half the page in one step.
The habit worth building for EPFO physics generally is to translate the words into the standard equation and read the answer off it. Writing y = A sin(omega t - k x) and then holding x fixed converts a question that sounds like it needs intuition into one line of algebra. The same move answers questions about particle speed, about phase difference and about where the energy of a wave is largest.
The paper's own wording is slightly uneven and should not be over-read: options (a) and (b) speak of 'frequency' while (c) and (d) speak of 'angular speed', and (a) says 'angular frequency' where (b) says only 'frequency'. Nothing turns on it — the item is asking about the rate of oscillation either way.
Key facts
- In a wave, energy and phase travel through the medium; the particles of the medium only oscillate about fixed mean positions and do not travel with the wave.
- For y = A sin(omega t - k x), holding x fixed gives y = A sin(omega t - constant), which is simple harmonic motion at the wave's own angular frequency omega.
- Every element of the string oscillates at the SAME frequency; elements differ only in PHASE, by k x for a separation x.
- Frequency is set by the source and does not change when a wave passes into another medium; speed and wavelength both change.
- Wave speed on a stretched string is v = square root of (T / mu), tension divided by mass per unit length — a property of the medium, not of the source.
- Particle velocity, A omega cos(omega t - k x), varies with time and is greatest at the mean position; wave velocity, omega / k, is constant. The two are different quantities.
- In a transverse wave the particles move perpendicular to the direction of propagation; in a longitudinal wave they move along it.
- Transverse mechanical waves need a medium that resists shear, which is why sound in air and in liquids is longitudinal and why earthquake S-waves do not cross the liquid outer core.
- Simple harmonic motion is the projection of uniform circular motion on a diameter — related to it, but not the same motion.
- Standing waves on a string fixed at both ends have fixed nodes and antinodes and occur only at the fundamental frequency and its harmonics.
Study next
Common traps
- Believing the medium travels with the wave. Only the disturbance travels; each particle stays put and oscillates.
- Confusing particle velocity with wave velocity. The first changes every instant and depends on amplitude; the second is fixed by the medium.
- Thinking a point on a string moves in a circle. Circular particle paths belong to surface water waves, not to a wave on a string.
- Assuming frequency changes when a wave enters a new medium. Frequency is fixed by the source; speed and wavelength adjust.
- Treating amplitude and frequency as linked. They are independent, though the energy carried depends on both.
- Forgetting that different points differ in phase, which is exactly what makes the wave appear to move.
Waves and oscillations appear on EPFO papers as one-line conceptual items with four short options and no diagram: what kind of motion a particle of the medium performs, which wave property stays the same across a boundary, whether a named wave is transverse or longitudinal, or which quantity depends on the medium and which on the source. Numerical versions, when they come, are a single substitution into v = f lambda or into the expression for the speed on a string. The option sets are built as small grids in which one attribute is varied against another, so a candidate who reasons about the whole option as a unit is doing four judgements where two would do. Learn the standard wave equation, the meaning of each symbol in it, and the short list of which quantities belong to the source and which to the medium; that is the whole of what these papers ask.
Related PYQs
EPFO_APFC_2016_Q74In raising an object to a given height by means of an inclined plane, as compared with raising the object vertically, there is a reduction in
- (a) Force to be applied
- (b) Work required
- (c) Distance covered
- (d) Friction force
Answer(a) Force to be applied
The inclined-plane item on this same paper — the other school-mechanics question here, and the same structure of four options that differ only in which quantity moves and in which direction.
EPFO_EOAO_2020_Q47Which one of the following statements is correct ? Wavelength of microwaves ranges between
- (a) infrared waves and radio waves.
- (b) visible waves and infrared light.
- (c) γ-rays and X-rays.
- (d) X-rays and visible waves.
Answer(a) infrared waves and radio waves.
Where microwaves sit in the electromagnetic spectrum — waves asked as a placement question, and a reminder that electromagnetic waves are transverse and need no medium.
EPFO_APFC_2023_Q51Two lenses of powers +2·0 D and –2·5 D are combined to make an optical instrument. The combination will
- (a) act as a convex lens
- (b) act as a concave lens
- (c) act as a simple mirror
- (d) not form any image
Answer(b) act as a concave lens
Combining lens powers of +2·0 D and -2·5 D — school physics asked as one application of a standard relation, which is the numerical form this strand takes on EPFO papers.
Practice
- practice — not a real PYQ
A transverse wave on a string is described by y = 0·05 sin(20t - 4x), where all quantities are in SI units. The speed of the wave is
- (a)4 m/s
- (b)5 m/s
- (c)20 m/s
- (d)80 m/s
Answer(b) 5 m/s — comparing with y = A sin(omega t - k x) gives omega = 20 rad/s and k = 4 rad/m, so the wave speed is v = omega / k = 20 / 4 = 5 m/s. The maximum speed of a particle of the string is a different quantity, A omega = 0·05 x 20 = 1 m/s.
- practice — not a real PYQ
When a wave travels from one medium into another, which one of the following remains unchanged ?
- (a)Speed
- (b)Wavelength
- (c)Frequency
- (d)Amplitude
Answer(c) Frequency — it is imposed by the source and is carried across the boundary unchanged. The speed is a property of the new medium and changes, the wavelength changes with it through v = f lambda, and the amplitude changes as energy is partly reflected and partly transmitted.