Waves

A wave is a disturbance that travels through a medium, carrying energy and momentum from one point to another without the medium itself moving along with it. Drop a stone in a pond: the ripples spread outward, but a leaf floating on the surface only bobs up and down about its own position — it does not travel with the ripple. This is the central idea of wave motion: particles of the medium oscillate about their mean positions, and it is the pattern of oscillation, not the matter, that propagates.

Waves are broadly of two kinds. Mechanical waves — water waves, sound waves, waves on a string — need a material medium to travel through, because they work by one particle of the medium disturbing its neighbour through the medium's elasticity and inertia. Electromagnetic waves (light, radio waves) need no medium and can travel through vacuum; they are not part of this chapter but are worth knowing as the contrast. This chapter deals only with mechanical waves.

Two properties of the medium decide how fast a mechanical wave travels: its elasticity (how strongly it resists being deformed and springs back) and its inertia (its mass density, which decides how sluggishly it responds). A stiffer, lighter medium carries waves faster; a floppier, denser one carries them slower.

In a transverse wave, the particles of the medium oscillate perpendicular to the direction in which the wave travels. A wave sent along a stretched string by moving one end up and down is transverse — the string moves up-down while the wave itself moves along the string's length. Transverse mechanical waves can only exist in a medium that can sustain shearing stress, which is why they travel through solids and along string/wire surfaces but not through the bulk of a liquid or a gas (liquids and gases cannot support a shear).

In a longitudinal wave, the particles oscillate along the same direction as the wave travels. Sound in air is the standard example: air molecules move back and forth along the direction the sound travels, creating alternate regions of compression (high pressure/density) and rarefaction (low pressure/density) that move forward. Longitudinal waves need only volume elasticity, so they travel through solids, liquids and gases alike.

A useful exam distinction: only transverse waves can be polarised — restricted to oscillate in a single plane — because their oscillation direction is perpendicular to propagation and can point anywhere in that perpendicular plane. Longitudinal waves oscillate only along the line of travel, so there is nothing left to restrict; they cannot be polarised. This is one way experiments proved that light is a transverse wave.

A sinusoidal wave travelling in the +x direction is written as y(x, t) = a sin(kx − ωt + φ), where y is the displacement of the particle at position x and time t. Here a is the amplitude (maximum displacement), k is the angular wave number (k = 2π/λ, unit rad m⁻¹), ω is the angular frequency (ω = 2π/T = 2πν, unit rad s⁻¹), and φ is the initial phase. The quantity (kx − ωt + φ) is the phase of the wave at that point and instant.

Wavelength (λ) is the distance between two consecutive points of the medium that are in the same phase, e.g. two consecutive crests. Time period (T) is the time for one particle to complete one full oscillation, and frequency (ν) = 1/T is the number of oscillations per second. The wave repeats itself over one wavelength in space and one time period in time.

The wave speed follows from the fact that in one time period T, the wave pattern advances exactly one wavelength λ: v = λ/T = νλ. In terms of the wave equation this is the same as v = ω/k, because both are the rate at which a point of constant phase moves. A wave written as a function of (kx + ωt) instead travels in the −x direction — the sign in front of ωt tells you the direction of travel.

Do not confuse wave speed with particle speed. Wave speed v = νλ is how fast the pattern (crest, trough) moves along the medium — it is fixed by the medium. Particle speed is ∂y/∂t at a fixed x, which oscillates between 0 and aω as the particle goes through its cycle — it depends on where in the cycle the particle is, not on the medium alone.

Speed of a transverse wave on a stretched string. The speed depends only on the tension T in the string (the restoring force) and μ, the mass per unit length (the inertia). By dimensional analysis: let v = C T^a μ^b. Since [T] = MLT⁻² and [μ] = ML⁻¹, matching dimensions on both sides of [LT⁻¹] = [MLT⁻²]^a[ML⁻¹]^b gives a + b = 0, a − b = 1, and −2a = −1, so a = ½, b = −½. The constant C works out to 1 from the full mechanical treatment, giving v = √(T/μ). A heavier string (large μ) carries waves more slowly; a more tightly stretched string (large T) carries them faster.

Speed of a longitudinal wave. The same reasoning with elasticity in place of tension gives, for a wave in a fluid, v = √(B/ρ), where B is the bulk modulus of the medium and ρ is its density. For a longitudinal wave in a long solid rod, v = √(Y/ρ), using the Young's modulus Y in place of B. In general: the stiffer the medium (larger elastic modulus) and the lighter it is (smaller density), the faster sound travels through it — which is why sound is fastest in solids, slower in liquids, and slowest in gases.

Speed of sound in a gas. Newton first proposed that sound travels through a gas isothermally (temperature stays constant during compression and rarefaction), giving v = √(P/ρ), where P is the pressure. For air at STP this gives about 280 m/s — roughly 15% below the experimentally measured 332 m/s. Laplace corrected this: compressions and rarefactions happen too fast for heat to flow between them, so the process is adiabatic, not isothermal. Using PV^γ = constant instead of PV = constant leads to v = √(γP/ρ), where γ = C_p/C_v is the ratio of specific heats (γ = 1.4 for air). This matches the measured speed of sound closely and is the accepted formula.

When two or more waves travel through the same region of a medium at the same time, the resultant displacement at any point is simply the algebraic sum of the displacements each wave would have produced there on its own: y = y₁ + y₂ + y₃ + ... This is the principle of superposition, and it holds because the wave equation is linear in y for the small oscillations considered in this chapter. The waves pass through each other completely unaffected — after they cross, each continues exactly as it would have if the other had never been there.

Superposition is the reason two waves can combine to reinforce each other (constructive interference, when crests meet crests) or cancel each other (destructive interference, when a crest meets a trough). It is also the tool used to build up every other result in this chapter: standing waves (superposition of a wave with its own reflection) and beats (superposition of two waves of slightly different frequency) are both direct applications of this one simple rule.

When a wave reaches the end of the medium — a rigid wall, a fixed end of a string — it reflects. At a rigid boundary the reflected wave undergoes a phase reversal (a crest returns as a trough); at a free/open boundary it reflects without phase reversal. A standing wave (or stationary wave) is what results when two identical waves of the same amplitude and frequency travel through the same medium in opposite directions — exactly the situation of an incident wave and its own reflection.

Derivation. Let y₁ = a sin(kx − ωt) be the incident wave and y₂ = a sin(kx + ωt) the wave travelling in the opposite direction. By superposition, y = y₁ + y₂ = a[sin(kx − ωt) + sin(kx + ωt)]. Using the identity sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2), this simplifies to y(x, t) = 2a sin(kx) cos(ωt). Notice this is not a travelling wave at all — the space part sin(kx) and time part cos(ωt) are separate. Every particle at a given x oscillates with the same phase and amplitude 2a sin(kx), which depends on position.

Points where sin(kx) = 0, i.e. x = 0, λ/2, λ, 3λ/2, ... never move — these are nodes. Points where sin(kx) = ±1, i.e. x = λ/4, 3λ/4, ... oscillate with the maximum amplitude 2a — these are antinodes. Nodes and antinodes alternate, each λ/4 apart.

Normal modes of a string fixed at both ends (length L). Both ends must be nodes, so L must be an integer number of half-wavelengths: L = nλ/2, giving allowed frequencies ν_n = nv/2L for n = 1, 2, 3, ... The lowest, ν₁ = v/2L, is the fundamental or first harmonic; ν₂ = 2ν₁ is the second harmonic (first overtone), and so on — a stretched string (and hence a sonometer or guitar string) supports all harmonics.

Normal modes of air columns. An open pipe (open at both ends) has an antinode at each end, giving the same condition L = nλ/2, ν_n = nv/2L — all harmonics present. A closed pipe (closed at one end, open at the other) has a node at the closed end and an antinode at the open end, giving L = (2n−1)λ/4, so ν_n = (2n−1)v/4L — only odd harmonics are present, and the fundamental ν₁ = v/4L is half that of an equally long open pipe.

When two sound waves of slightly different frequencies (ν₁ and ν₂, close to each other) reach the same point together, the ear hears a sound whose loudness rises and falls periodically — this waxing and waning is called beats. Each rise-and-fall cycle is one 'beat'.

Derivation. Take two waves of equal amplitude a at a fixed point: y₁ = a cos(ω₁t) and y₂ = a cos(ω₂t). By superposition, y = y₁ + y₂ = 2a cos((ω₁−ω₂)t/2) · cos((ω₁+ω₂)t/2). This is a wave of frequency (ν₁+ν₂)/2 — essentially the average, which is what we hear as the pitch — whose amplitude itself is not constant but slowly varies as 2a cos((ω₁−ω₂)t/2). Loudness is maximum whenever this envelope amplitude is at its largest magnitude, which happens twice in every cycle of the slow cosine term; working through the timing gives a beat frequency Δν = |ν₁ − ν₂| — simply the difference of the two frequencies.

Beats are used practically to tune musical instruments: a musician plays a note next to a reference tone and adjusts the string tension until the beats slow down and disappear, meaning the two frequencies have become equal. A related trick used in numericals: loading a tuning fork with wax lowers its frequency, and clipping (filing) a fork raises it — watching whether the beat rate increases or decreases after loading tells you whether the unknown frequency was originally above or below the known one.

The Doppler effect is the apparent change in the frequency of a wave when there is relative motion between the source and the observer. You hear this every time a fast car or train passes by: the pitch of the horn is higher as it approaches and drops as it recedes, even though the source is emitting a constant frequency ν throughout. Let v be the speed of sound in air (relative to the medium, taken as still).

Source moving, observer at rest. If the source moves toward the observer with speed v_s, each successive wave crest is emitted from a position slightly closer to the observer than the last, compressing the wavelength to λ' = (v − v_s)/ν. The observed frequency is therefore ν' = v/λ' = νv/(v − v_s) — higher than ν. If the source recedes, the same reasoning with a sign flip gives ν' = νv/(v + v_s) — lower than ν.

Observer moving, source at rest. If the observer moves toward the source with speed v_o, sound crests reach the observer faster because the observer effectively meets them at a relative speed (v + v_o); the observed frequency is ν' = ν(v + v_o)/v. If the observer recedes, ν' = ν(v − v_o)/v.

General case (both moving). Combining both effects gives the master formula ν' = ν(v ± v_o)/(v ∓ v_s), where the upper sign is used when the motion is toward the other and the lower sign when it is away — take + for v_o and − for v_s when they move to close the gap, and flip the sign for whichever one is moving apart.

Note carefully: this classical Doppler formula for sound depends on the speed of source and observer relative to the medium (still air), which is why the source and observer formulas are not symmetric — unlike light, where only relative velocity matters. Applications include radar speed guns, weather radar (Doppler radar), and estimating the speed of stars and galaxies from the shift in the frequency of their light (this optical case is covered again in Class 12).

A travelling wave carries energy — this is exactly how energy from an earthquake, a loudspeaker or a struck string reaches a distant point without any net transport of matter. For a wave on a string of mass per unit length μ, amplitude a and angular frequency ω, each element of the string undergoes simple harmonic motion, and the energy per unit length of the string works out to ½μω²a². As the wave moves at speed v, the average power transmitted across any point is P_av = ½μω²a²v. This shows the important scaling result you should remember for concept-based MCQs: the energy (and power) carried by a wave is proportional to the square of the amplitude and to the square of the frequency — doubling the amplitude quadruples the energy transported.