Oscillations & Waves

Simple harmonic motion, energy, and how a wiggle travels · Class 11 Physics

1The big idea

An oscillation is any motion that repeats itself back and forth about a fixed central point — a swinging pendulum, a mass bobbing on a spring, a plucked guitar string, the air pressure in an organ pipe. A wave is what you get when one oscillator nudges its neighbour, which nudges the next, so the wiggle travels outward even though no single particle goes anywhere. These two topics are really one story: a wave is an oscillation on the move.

Underneath almost every oscillation you will meet is a single, beautifully simple rule: whenever something is pushed away from its resting place, a force pulls it straight back, and that force is proportional to how far it has strayed. That one sentence — a restoring force proportional to displacement — is the definition of simple harmonic motion, and it is the engine of the whole chapter.

2Simple harmonic motion

Take a block of mass m on a frictionless table, tied to a spring. Pull it a distance x from its rest position and let go. Hooke's law says the spring pulls back with a force proportional to the stretch and opposite to it:

The minus sign is the whole point: it is what makes the motion turn around and repeat instead of running off. Here k is the spring constant (stiffness, in N m⁻¹). Now bring in Newton's second law, F = m a:

The acceleration is proportional to the displacement and points back toward the centre. We give the constant k/m the name ω² (you will see in a moment why the square is convenient), so the defining equation of SHM is

Any system obeying a = −ω²x — spring, pendulum, floating cork, vibrating molecule — moves in exactly the same shape of motion. That shape is a sine wave in time:

Read the symbols carefully — every one earns its place:

• Amplitude A — the farthest the object strays from centre. It sets the size of the swing, nothing else.
• Angular frequency ω (rad s⁻¹) — how fast the sine cycles. It is fixed by the system (ω = √(k/m) for the spring), not by how hard you pull.
• Phase constant φ — where in the cycle the motion starts at t = 0.

Why does x = A sin(ω t + φ) solve a = −ω²x? Differentiate twice with respect to time. The velocity is v = A ω cos(ω t + φ), and the acceleration is

It fits exactly — and now the square in ω² makes sense, because each time-derivative of a sine pulls out one factor of ω. The period (time for one full swing) and frequency (swings per second) follow from ω:

Notice what is missing from the period: the amplitude. A wide swing and a narrow swing of the same spring take exactly the same time. This amplitude-independence — called isochronism — is why a pendulum can keep a clock honest.

3Energy in SHM

As the block oscillates, energy sloshes back and forth between two forms. The spring stores potential energy when stretched or compressed, and the moving block carries kinetic energy. Let us write both at any instant, using x = A sin ω t and v = A ω cos ω t (taking φ = 0):

Here is the trick that makes it collapse. Since ω² = k/m, we have m ω² = k, so the kinetic energy is really ½ k A² cos²ω t. Add the two:

The trigonometric identity sin² + cos² = 1 wipes out the time dependence entirely. The total energy never changes — it just trades between kinetic and potential as the block moves:

• At the centre (x = 0): the spring is relaxed, so PE = 0 and all the energy is kinetic. The block is moving fastest here, with v_max = A ω.
• At the extremes (x = ±A): the block is momentarily at rest, so KE = 0 and all the energy is potential. This is the turning point.

One more thing worth carrying away: E = ½ k A² depends on the square of the amplitude. Double the swing and you quadruple the energy. The interactive below lets you watch this exchange happen in real time.

4Time periods — spring & pendulum

The spring. We already have everything. For a mass on a spring, ω = √(k/m), so

Heavier mass → slower (longer T); stiffer spring → faster. Gravity does not appear, so the same spring gives the same period lying flat or hanging vertically.

The simple pendulum. A bob of mass m on a string of length L, pulled to a small angle θ. Gravity pulls straight down, but only the component along the arc acts to restore it: F = −m g sin θ. For small angles (in radians) sin θ ≈ θ, and the arc displacement is x = L θ, so θ = x/L. Substitute:

Compare this with F = −k x: the pendulum behaves like a spring whose effective stiffness is k = m g / L. Drop that into T = 2π√(m/k) and the mass cancels:

Two remarkable facts fall out. The mass has vanished — a heavy bob and a light bob swing in step. And the amplitude is absent too (as long as θ stays small, roughly under 15°), which is exactly the sin θ ≈ θ approximation talking. A longer pendulum swings more slowly; to double the period you must quadruple the length.

5Waves — the travelling disturbance

Now let each oscillator pull on its neighbour. The disturbance marches along while the particles themselves only jiggle in place. Waves come in two flavours, split by the direction of that jiggle relative to the travel:

• Transverse — particles vibrate across the direction of travel. A string, ripples on water, light. You see crests and troughs.
• Longitudinal — particles vibrate along the direction of travel. Sound in air is the classic case: the medium bunches into compressions and spreads into rarefactions.

A wave travelling in the +x direction is captured by a sine that depends on both position and time:

Freeze time and it is a sine in space with wavelength λ; stand at one spot and it is a sine in time with period T. The two new symbols are the wave number k = 2π/λ (careful — a different k from the spring constant) and, as before, ω = 2π/T. The combination (k x − ω t) is what makes the pattern move: to stay on a given crest you must increase x as t grows, and the speed at which you do so is the wave speed. Dividing ω by k:

So v = f λ — one crest passes every period, covering one wavelength, and that is all this famous formula says. The wave speed is set by the medium, not by the source. For a stretched string with tension T and mass-per-length μ (in kg m⁻¹):

Tighter string → faster wave → higher pitch (which is why tuning pegs work); heavier string → slower wave → lower pitch (which is why the bass strings are the thick ones). If you tighten the string, v rises; since the string length fixes λ, the frequency f = v/λ rises with it.

6Superposition & standing waves

When two waves overlap, the medium simply adds their displacements — this is the principle of superposition. Where two crests meet you get a bigger crest (constructive); where a crest meets a trough they cancel (destructive). Nothing is lost; after passing through, each wave carries on unchanged.

Send a wave down a fixed string and it reflects off the end and comes back. The outgoing and returning waves superpose, and for the right frequencies they lock into a pattern that looks stationary — a standing wave. Some points, the nodes, never move; halfway between them the antinodes swing with full amplitude. Only special wavelengths fit: the string must hold a whole number of half-loops, L = n λ/2. The allowed frequencies are the harmonics:

The lowest (n = 1) is the fundamental; the rest are its integer multiples, and it is their mix that gives an instrument its character. Air columns in pipes do the same, and here a genuinely useful distinction appears between the two kinds of pipe:

• Open pipe (both ends open): antinodes at both ends, fundamental f₁ = v/2L, and it plays every harmonic — f_n = n v/2L for n = 1, 2, 3, …
• Closed pipe (one end closed): a node sits at the closed end and an antinode at the open end, so only a quarter-wavelength fits at the bottom. Its fundamental is f₁ = v/4L — an octave lower than an open pipe of the same length — and it plays only the odd harmonics, f_n = (2n − 1) v/4L.

7Beats, Doppler & resonance

Beats. Sound two notes of nearly-equal frequency together and superposition makes the combined loudness swell and fade, swell and fade. Each throb is a beat, and they arrive at a rate equal to the difference of the two frequencies:

A piano tuner listens for the beats to slow to a standstill — when they vanish, the two frequencies match. If a 256 Hz fork and a string give 4 beats per second, the string is at 252 or 260 Hz.

The Doppler effect. When a source and listener move relative to one another, the pitch shifts — the rising-then-falling wail of a passing siren. Motion squeezes the wavefronts ahead and stretches them behind, changing the frequency you actually receive:

The upper signs are for motion that brings source and listener together (higher pitch); the lower signs are for moving apart (lower pitch). Read it as a rule rather than a memorised sign soup: approaching always raises the pitch, receding always lowers it. A moving observer v_o changes the top; a moving source v_s changes the bottom.

Resonance. Every object has natural frequencies at which it loves to oscillate — its harmonics. Push it at exactly one of those frequencies and the amplitude builds up dramatically, because each push arrives perfectly in step with the swing. A child on a swing goes higher when pumped at the swing's own rhythm; a wine glass shatters when a singer hits its natural note; a radio picks one station by tuning its circuit to resonate with that broadcast. Resonance is what turns a small, well-timed nudge into a large response.

8Common confusions to clear up

• Amplitude does not affect the period. A big swing and a small swing of the same pendulum (or spring) take the same time — that is the whole idea behind pendulum clocks.
• The two k's are different. In SHM, k is the spring constant (N m⁻¹). In waves, k = 2π/λ is the wave number. Same letter, unrelated meanings — read the context.
• At the extremes the block is at rest but not force-free. Speed is zero at the turning point, yet the restoring force (and acceleration) is maximum there — which is exactly why it turns around.
• Wave speed is not set by frequency. The medium fixes v (e.g. √(T/μ) on a string). Raise the frequency and the wavelength shrinks to keep v = f λ — the speed itself does not change.
• A closed pipe is not simply "louder" or "quieter." Its fundamental is an octave lower than an open pipe of equal length, and it is missing the even harmonics — that is why the two sound so different.
• In a longitudinal wave nothing travels bodily along. Air molecules only jostle back and forth about their spots; it is the compression pattern — the energy — that moves forward.

9Check yourself

Class 11 Physics · Oscillations & Waves · aligned to NCERT Chapters 14–15 · SmartStudy.School