Why Newton's Own Formula for the Speed of Sound Was Wrong by 15%

Drop a pebble into a still pond and the disturbance it creates does not stay put -- it spreads outward as an expanding circle, and if you set cork pieces floating on the surface, you can watch them bob up and down in place without ever drifting outward with the circle itself; the water is not flowing away from the point of disturbance, only a moving pattern is. This same idea, a disturbance that propagates without the actual physical transfer of matter as a whole, is what defines a wave, and it appears throughout physics in genuinely different guises. Mechanical waves -- waves on a string, water waves, sound waves, seismic waves -- require a material medium and cannot exist in a vacuum; they arise because the constituent particles of that medium are elastically coupled to their neighbours, so displacing one particle drags on the next, exactly like a chain of springs connecting a series of masses, or the coupled bogies of a train where an engine's single push at one end propagates backward bogie by bogie without the whole train instantly lurching forward together. Electromagnetic waves -- light, radio waves, X-rays -- are entirely different in this one crucial respect: they do not require a medium at all, and travel through the vacuum of space at the same fixed speed, c = 299,792,458 m/s, which is exactly how sunlight and starlight cross the practical vacuum of interstellar space to reach us. A third, still more abstract category, matter waves, is associated with the quantum mechanical behaviour of particles like electrons and is explored in later study; this chapter, however, is concerned entirely with mechanical waves, the kind that must have some material medium to travel through at all.

A single sharp jerk on a stretched string sends a lone pulse racing down its length -- and watching any one point on the string reveals it moving up and down, perpendicular to the direction the pulse itself is travelling along the string. Waves where the medium's particles oscillate perpendicular to the wave's own direction of travel are called transverse waves. A piston oscillating back and forth inside a long air-filled tube, by contrast, produces alternating regions of compression and rarefaction that travel down the tube's length -- but here, the air itself oscillates back and forth parallel to the very direction the disturbance is travelling. Waves with this parallel relationship are called longitudinal waves, and sound is the most familiar example. This distinction is not just descriptive -- it determines which media can even support each type. Transverse waves require the medium to sustain a genuine shearing strain (one layer sliding sideways relative to its neighbour), a property only solids possess in useful measure; fluids (liquids and gases) simply flow rather than resist shear, so they cannot carry transverse waves at all. Longitudinal waves, however, only require the medium to resist compression, a property every state of matter has to some degree, so longitudinal waves can propagate through solids, liquids and gases alike. Steel, accordingly, can carry both transverse and longitudinal waves, while ordinary air can carry only longitudinal ones. Water's own surface waves are a genuinely mixed case: capillary waves (short-wavelength ripples restored by surface tension) and gravity waves (longer waves restored by gravity's pull) both involve water particles moving in a complicated loop, partly up-and-down and partly back-and-forth, making ocean waves a real combination of transverse and longitudinal motion.

Describing a travelling wave properly needs a single function that does two jobs at once: frozen at any one instant, it must give the wave's whole shape in space; and watched at any one fixed location, it must give that location's own motion in time. For a transverse wave travelling along the positive x-direction, this function is y(x,t) = a sin(kx - ωt + φ). Each symbol in this compact equation carries real meaning. The amplitude, a, is the maximum displacement of the medium's constituents from their own equilibrium position -- always taken as a positive constant. The whole quantity inside the sine, (kx - ωt + φ), is called the phase of the wave; it alone determines the exact displacement at any given position and any given instant. φ itself is the phase specifically at x=0 and t=0, and is therefore called the initial phase angle -- by suitably choosing the origin of position and the start of time, φ can always be set to zero without any real loss of generality. Freezing time at some fixed instant t=t0 turns the phase into kx+constant, so the wave's shape in space, at that instant, is a plain sine curve in x; freezing position instead at some fixed x=x0 turns the phase into constant-ωt, so that single location's displacement varies sinusoidally in time, exactly like simple harmonic motion. As t increases, x must increase too, to keep the whole phase (kx-ωt+φ) constant -- confirming this particular equation describes a wave travelling in the positive x-direction; flipping the sign inside, a sin(kx+ωt+φ), describes an otherwise identical wave travelling in the negative x-direction instead.

The wavelength, λ, is the minimum distance between two points sharing exactly the same phase -- most simply, the distance between two consecutive crests or two consecutive troughs. Since the sine function repeats identically every 2π change in its own argument, the wave's own spatial pattern must repeat every time kx increases by exactly 2π, giving directly λ = 2π/k, where k, called the angular wave number, has SI units of radian per metre. Watching one single fixed location instead, the time period, T, is the time taken for that location to complete one full oscillation -- and since the sine function again repeats every 2π, this time in ωt, exactly the same reasoning gives T = 2π/ω, where ω, the angular frequency, has SI units of radian per second. The ordinary frequency, ν, simply counts oscillations per second rather than radians, so ν = 1/T = ω/2π, measured in hertz. For a longitudinal wave, exactly the same displacement equation applies, only the displacement itself, now parallel rather than perpendicular to the direction of travel, is conventionally written s(x,t) = a sin(kx-ωt+φ) instead of y(x,t), to make the distinction visually clear. A single fully-worked example ties all of these together: given the equation y(x,t) = 0.005 sin(80.0x - 3.0t) in SI units, comparing directly against the general form immediately identifies the amplitude as 0.005 m (5 mm), the angular wave number as k=80.0 m⁻¹ (giving λ=2π/80.0≈7.85 cm), and the angular frequency as ω=3.0 s⁻¹ (giving T=2π/3.0≈2.09 s and ν=1/T≈0.48 Hz) -- and substituting any specific x and t directly into the original equation gives the exact displacement at that particular position and instant.

Tracking any single point of constant phase on a travelling wave -- most conveniently, a crest -- and watching how far it moves in a given time reveals the wave's own speed. Requiring the phase (kx-ωt) to stay constant as both x and t change together gives k(dx) = ω(dt), or dx/dt = ω/k -- this ratio, ω/k, is the wave's speed, v. Relating ω to the time period and k to the wavelength turns this into a far more useful, universal form: v = ω/k = λ/T = λν, valid for every kind of progressive wave, whatever is actually causing it. This single relation carries a genuinely important implication: it is the medium, through its own elastic and inertial properties, that fixes the wave's speed -- not the wave's source. A source can only ever choose the frequency it generates; once that frequency enters a given medium at that medium's own fixed speed, the wavelength is entirely determined in turn, λ=v/ν. A higher-frequency source produces a correspondingly shorter wavelength in the same medium, and a lower-frequency source a correspondingly longer one, but the speed itself never changes just because the source's own frequency does.

A wave's speed through any medium is set by exactly two competing properties: how strongly the medium resists being disturbed (its elasticity, providing the restoring force) and how much inertia it has to overcome (its mass density, resisting acceleration) -- a stiffer medium supports a faster wave, and a denser one supports a slower one. For a wave on a stretched string, the restoring force comes from the string's own tension, T, and the relevant inertia is its linear mass density, μ (mass per unit length); dimensional analysis alone (T has dimensions of force, μ of mass per length) shows T/μ must have the dimensions of speed-squared, and the full derivation confirms the constant of proportionality is exactly 1, giving v=√(T/μ). A steel wire 0.72 m long with a mass of 5.0x10⁻³ kg, for instance, has μ=5.0x10⁻³/0.72≈6.9x10⁻³ kg/m, and under a tension of 60 N supports transverse waves at v=√(60/6.9x10⁻³)≈93 m/s. For longitudinal waves, including sound, the relevant restoring property instead is the bulk modulus, B, defined by B=-ΔP/(ΔV/V) (how strongly a material resists a given fractional volume change under a given pressure change), and the relevant inertia is the ordinary mass density, ρ, giving v=√(B/ρ) for fluids in general. In a long, thin solid bar, where lateral expansion is negligible, the more specific Young's modulus, Y, takes over from B, giving v=√(Y/ρ) instead. Since solids and liquids both have considerably higher density than gases, yet an even larger corresponding increase in stiffness (their B or Y values), the net effect, once the two are combined under the square root, is that sound genuinely travels faster in solids and liquids than in gases, exactly matching the everyday ordering already observed.

Applying v=√(B/ρ) to an ordinary gas needs one further step: what, exactly, is a gas's bulk modulus? Newton assumed the temperature stays constant as a sound wave's compressions and rarefactions pass through -- an isothermal process -- and for an ideal gas held at constant temperature, this assumption directly gives B=P, the gas's own pressure, so Newton's formula for the speed of sound becomes v=√(P/ρ). Estimating the speed of sound in air at standard temperature and pressure this way, using air's own density there (ρ≈1.29 kg/m³, from 29.0x10⁻³ kg per mole occupying 22.4 litres), gives v≈280 m/s -- a real, specific prediction, and a real, specific problem: the measured speed of sound in air is about 331 m/s, meaning Newton's own formula falls short by roughly 15%, a sizeable, genuinely embarrassing gap for a formula built on Newton's own laws of motion. The resolution came from Pierre-Simon Laplace, who identified precisely which assumption was wrong: sound's compressions and rarefactions happen far too quickly for heat to flow and equalize temperature across each tiny region, so the process is actually adiabatic, not isothermal. For an adiabatic process in an ideal gas, the correct bulk modulus works out to B=γP instead of simply P, where γ (the ratio of the gas's two specific heats, Cp/Cv) is about 7/5 for air -- giving the corrected formula v=√(γP/ρ), now known as the Laplace correction. Recomputing the speed of sound in air at STP using this corrected formula gives v≈331.3 m/s, matching the measured value almost exactly -- a clean, satisfying resolution that fixed one specific wrong assumption inside an otherwise perfectly sound formula, rather than requiring any new physics at all.

When two or more waves overlap in the same region of a medium, the principle of superposition governs exactly what happens: the net displacement at any point and instant is simply the algebraic sum of the individual displacements each wave would have produced alone, y=y1+y2 -- each wave behaves as if the others were not even there. Two identical waves travelling in the same direction, differing only by a constant phase φ, combine into a single wave of the same frequency but with amplitude A(φ)=2a cos(φ/2): when φ=0 (perfectly in phase), A reaches its largest possible value, 2a, called constructive interference; when φ=π (perfectly out of phase), A collapses to exactly zero everywhere, called destructive interference. Reflection introduces a further twist. A travelling wave meeting a rigid, fixed boundary must have zero displacement at that boundary at all times, which is only possible if the reflected wave is exactly inverted relative to the incident one -- a phase change of π; at a free, open boundary, by contrast, the reflected wave keeps the same phase as the incident wave, with no reversal at all. When a wave reflects repeatedly between two boundaries, such as on a string fixed at both ends, or in an air column closed or open at its ends, the incident and reflected waves continuously overlap and combine into a standing (stationary) wave pattern: y(x,t)=2a sin(kx) cos(ωt), the superposition of two identical waves travelling in opposite directions. This pattern looks and behaves completely differently from an ordinary travelling wave -- it does not move left or right at all; instead, certain fixed points, called nodes (where sin(kx)=0), never move, while points exactly halfway between them, called antinodes, oscillate with the largest possible amplitude, 2a. For a string of length L fixed at both ends, the boundary condition that both ends must be nodes restricts the allowed wavelengths to λ=2L/n, and hence the allowed frequencies to a discrete series, ν=nv/2L for n=1,2,3,..., called the normal modes or harmonics of the string -- the lowest of these, n=1, is the fundamental or first harmonic. A pipe closed at one end and open at the other instead only allows odd harmonics, ν=(n+½)v/2L for n=0,1,2,..., since the closed end must be a node while the open end must be an antinode; a pipe open at both ends, by contrast, allows every harmonic, exactly like the fixed string. This same underlying principle, constrained boundaries permitting only certain discrete frequencies, explains why a plucked or bowed sitar string, or a struck tabla membrane, or the deliberately-carved musical pillars of the Nellaiappar temple in Tamil Nadu, each produce their own clean, specific musical notes rather than an arbitrary jumble of frequencies.

Two harmonic sound waves of nearly, but not quite, equal frequency, ν1 and ν2, heard together produce something genuinely distinctive: not a steady tone at either frequency, but a single tone at their average frequency, whose loudness audibly waxes and wanes, rising and falling repeatedly -- this rhythmic pulsing is called beats. Superposing s1=a cos(ω1t) and s2=a cos(ω2t) mathematically, using a standard trigonometric identity, gives the combined displacement as s=[2a cos(ωbt)]cos(ωat), where ωa=(ω1+ω2)/2 is close to both original frequencies, and ωb=(ω1-ω2)/2 is small, since the two original frequencies are close together. The resulting wave still oscillates at essentially the average angular frequency ωa, but its own amplitude, the bracketed term 2a cos(ωbt), is itself slowly varying rather than fixed -- reaching its maximum magnitude, 2a, whenever cos(ωbt) hits +1 or -1. Since intensity depends on the square of amplitude, this slow amplitude variation is heard as a periodic loudness pulsing, and working through the frequency at which this loudness cycle repeats (twice per period of ωb, since both +1 and -1 produce a loudness peak) gives a wonderfully simple final result: the beat frequency is exactly the difference between the two original frequencies, νbeat=ν1-ν2. Musicians have exploited this effect for centuries without necessarily knowing its mathematics: sitar players tuning two strings to the identical note listen specifically for beats, and adjust the tension until the beating slows and finally vanishes entirely, confirming the two strings finally share the exact same frequency. This same logic solves a genuinely subtle reasoning problem directly: if two sitar strings A and B, playing the same note, produce beats at 5 Hz, and then slightly increasing the tension in string B (which necessarily raises its own frequency) causes the beat frequency to fall to 3 Hz rather than rise, this can only mean B's original frequency was lower than A's to begin with -- since raising an already-higher frequency further would only have widened the gap between them, not narrowed it.