Experiment: Waves and Acoustics

Contents

Mersenne: the laws of the vibrating string (1636)
Melde: standing waves on a driven string (1860)
Chladni: the nodal figures of a vibrating plate (1787)
Newton, Laplace and the speed of sound (1687–1822)
Kundt's tube: sound speeds from dust figures (1866)
Buys Ballot: the Doppler effect on a moving train (1845)
Summary of the evidence

Oscillations and Mechanical Waves reduces a large part of classical wave physics to four sharp claims. A string clamped at both ends sustains a discrete harmonic series $f_{n}=n f_{1}$ with $f_{1}=(1/2L)\sqrt{T/\mu}$. A bounded elastic sheet has its own discrete spectrum, whose modes are surfaces rather than curves and whose overtones are not integer multiples of a fundamental. A compressible fluid carries non-dispersive waves at the speed $\sqrt{\dd p/\dd\rho}$, whose value depends on which thermodynamic path the compressions follow. And a source in motion through the medium is heard at a shifted frequency. This chapter reports the measurements that established all four.

They are old — 1636 to 1866 — and they are, for that reason, unusually clean. Each is a counting experiment: a number of loops on a thread, a number of nodal lines on a plate, a spacing of dust ridges, a musical interval. Ratios of small integers and ratios of lengths are what the apparatus delivers, and ratios are exactly what the theory predicts without adjustable parameters. Two of the experiments do more than confirm: the sound-speed measurement of Section 35.4 exposed a $15\,\mathrm{\%}$ discrepancy that stood for one hundred and twenty-nine years and was resolved only by recognizing the compressions as adiabatic, so that an acoustic measurement became a thermodynamic one; and the Doppler test of Section 35.6 is the case in which the medium is real and at rest in the laboratory, which makes the source–observer asymmetry a physical fact here and its absence in optics (Experiments: Light, the Aether, and Time) a genuine surprise.

Mersenne: the laws of the vibrating string (1636)

Tests Equation (35.1). Assuming Equation (28.46).

The pitch of a stretched string was the first quantitative law of acoustics, and it was found a century before there was a wave equation to explain it. Its interest here is not only historical: the three proportionalities are precisely the content of the formula $f_{1}=(1/2L)\sqrt{T/\mu}$ that Oscillations and Mechanical Waves derives, and they were measured against an instrument of remarkable precision — the trained ear, which judges frequency ratios and not frequencies.

Apparatus

A monochord: one string of gut or of drawn brass stretched over a graduated soundboard between a fixed bridge and a movable one, tensioned by a weight hanging over a pulley at one end, so that the tension is known as a weight and not as a turn of a peg. A set of strings of several materials and several diameters, otherwise identical, allows the mass per unit length to be varied at fixed length and tension. For the absolute measurement a second apparatus is needed: a long, heavy cord, metres in length and slack enough that its fundamental period is a substantial fraction of a second, stretched horizontally between fixed supports.

Procedure

Three comparisons, each holding two of the three variables fixed. Displace the movable bridge and compare the pitch of the shortened string with that of the whole; the comparison is made by ear against the consonances, whose ratios are known in advance and are exact by definition — the octave $2:1$, the fifth $3:2$, the fourth $4:3$. Hang different weights on the same length and find the loads at which the pitch rises by a known interval. Substitute strings of different diameter at the same length and tension and do the same.

The absolute measurement is a separate operation and is the more ingenious of the two. The long cord vibrates slowly enough that its swings can be counted directly against a pendulum or a pulse, giving one frequency in absolute terms. The three laws are then used in the reverse direction, as a transfer standard, to carry that number up to the audible range: shorten the length and reduce the diameter by measured factors, and the frequency scales by factors that a ruler determines.

Observations and data

The three ratio laws, stated without any absolute frequency. Halving the length raises the pitch by an octave, and thirds and quarters of the length give the twelfth and the double octave, so that $f\propto1/L$ exactly over at least two octaves. Quadrupling the hanging weight raises the pitch by an octave, so that $f\propto\sqrt{T}$. Doubling the diameter of a string of the same material at the same length and tension lowers the pitch by an octave, so that $f\propto1/d$, and since $\mu\propto d^{2}$ for a cylinder of fixed density this is $f\propto1/\sqrt{\mu}$ [Mersenne:1636] [Galilei:1638].

Phenomenon 35.1 (Mersenne's laws of the stretched string).

The fundamental frequency of a uniform string clamped at both ends varies inversely as its sounding length, as the square root of its tension, and inversely as the square root of its mass per unit length; all three hold as exact proportionalities, over the full range of a musical instrument, before any theory of the vibration exists [Mersenne:1636] [Galilei:1638]. The higher partials that the same string sounds together with its fundamental stand to it in the ratios of the whole numbers.

Derivation. Derives Phenomenon 35.1. Take from Oscillations and Mechanical Waves the clamped-string spectrum

\begin{equation}\tag{35.1} f_{n}=\frac{n}{2L}\sqrt{\frac{T}{\mu}}\ec\qquad n\in\N\ep \end{equation}

Holding any two of $L$, $T$, $\mu$ fixed and varying the third reads off the three laws directly, and setting $n=2,3,4$ at fixed $L$, $T$, $\mu$ gives the whole-number partials. What must be derived here is the transfer, because the absolute measurement is made on one string and reported for another. Let a cord of length $L$ and diameter $d$ be compared with a string of the same material and the same tension, of length $L'$ and diameter $d'$. Since $\mu=\tfrac{1}{4}\pi\rho d^{2}$ for a cylinder of density $\rho$, Equation (35.1) gives

\begin{equation}\tag{35.2} \frac{f'_{1}}{f_{1}}=\frac{L}{L'}\,\frac{d}{d'}\ec \end{equation}

in which the tension, the density and the numerical factors have all cancelled. Every quantity on the right is a length measured with a ruler. A frequency of a few hertz, counted by eye on a cord metres long and a millimetre thick, therefore fixes the frequency of a string centimetres long and a tenth of a millimetre thick — three orders of magnitude away — with no clock faster than a pulse and no calibration of the material at all.

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Interpretation

Two things are established, and they are of different kinds. The first is the empirical law itself, which any theory of the vibrating string was thereafter obliged to reproduce; d'Alembert's wave equation [dAlembert:1747] and Bernoulli's modal superposition [Bernoulli:1753] were tested against it a century later, and it is the reason the eigenvalue problem of Ordinary Differential Equations and Sturm–Liouville Theory was posed at all. The second is methodological, and is the reason this experiment opens the chapter: Equation (35.2) shows that the whole of acoustics can be done in ratios. The ear is a poor absolute instrument and an excellent comparator, resolving a mistuned unison of a fraction of a per cent; the theory predicts ratios; and the apparatus is built so that what is read off is a ratio of lengths. The same discipline governs every experiment below.

One caution belongs with the result. The absolute frequency obtained by the transfer is only as good as the assumption that Equation (35.1) extrapolates unchanged across those three orders of magnitude — that is, that the string is non-dispersive and the vibration linear. That assumption is itself a prediction of the theory, and it is tested independently in Section 35.2, where the same law is checked at fixed frequency across a whole family of modes.

Primary references

[Mersenne:1636] [Galilei:1638]. Galileo's Discorsi states the same three rules, arrived at independently and published two years later; the two accounts are usually cited together and are so cited here.

Melde: standing waves on a driven string (1860)

Tests Equation (35.3). Assuming Equation (35.1).

Mersenne's laws are read from the sounding pitch of a freely vibrating string. Melde inverted the arrangement: drive the string at a fixed, known frequency and vary the tension instead, and the standing-wave pattern becomes visible, countable, and present only at discrete settings of the load. The experiment thereby shows the eigenvalue structure of the boundary-value problem directly, in a form that needs no ear at all.

Apparatus

A tuning fork of known frequency, maintained electromagnetically so that it runs continuously at constant amplitude. A light silk or cotton thread tied to one prong, led horizontally over a smooth pulley at a measured distance, and terminated in a scale pan carrying weights, so that the tension is a known weight. The fork is mounted on a rotatable block, so that its prong can be set to vibrate either perpendicular to the thread or along it.

Procedure

With the prong vibrating transversely, the thread is driven at the fork's frequency. Weights are added to the pan and the thread watched; at most loads it thrashes without pattern, but at particular loads it settles into a stationary figure of $n$ clean loops separated by points that do not move. The load is recorded for each $n$. Because the prong's excursion is a small fraction of the loops' amplitude, the driven end is a node to good approximation, and the thread behaves as if clamped at both ends.

The fork is then rotated through a right angle, so that it pulls the thread along its own length, periodically modulating the tension rather than displacing the end. Standing waves appear again, at different loads, and a direct comparison shows the thread completing one transverse cycle for every two cycles of the fork.

Observations and data

Discrete loads, and a pure-number law relating them. The pattern of $n$ loops appears at a tension $T_{n}$, and the tensions found for successive $n$ fall as the inverse square of the number of loops, so that four loops require one quarter the load of two and nine loops one ninth the load of three [Melde:1860]. In the second, longitudinal configuration the response is at half the driving frequency.

Phenomenon 35.2 (Standing waves appear only at discrete tensions).

A string of fixed length and mass per unit length, driven transversely at a fixed frequency $f$, shows a stationary pattern of $n$ loops only at the discrete tensions

\begin{equation}\tag{35.3} T_{n}=\frac{4L^{2}f^{2}\mu}{n^{2}}\ec\qquad n\in\N\ec \end{equation}

and shows no stationary pattern between them; the loads at which successive patterns appear stand in the ratios $1:\tfrac{1}{4}:\tfrac{1}{9}:\dots$ [Melde:1860]. Rests on Equation (35.1).

Derivation. Derives Phenomenon 35.2. A stationary pattern with $n$ loops on a string of length $L$ whose ends are both nodes requires $L=n\lambda/2$, so $\lambda=2L/n$. The wave travels at $v=\sqrt{T/\mu}$ and the pattern oscillates at the driving frequency, so $v=\lambda f$. Eliminating $\lambda$ and $v$,

\begin{equation}\tag{35.4} \sqrt{\frac{T_{n}}{\mu}}=\frac{2Lf}{n}\ec \end{equation}

which on squaring is Equation (35.3). Two features make this a stringent test rather than a demonstration. First, the ratios $T_{n}/T_{1}=1/n^{2}$ contain none of $L$, $f$ or $\mu$: the experiment can be run without knowing the fork's frequency, the thread's mass or even the distance to the pulley, and it still tests the theory, because a scale pan measures a ratio of loads directly. Second, the prediction is as much about the loads at which nothing happens as about those at which something does. A continuum of admissible frequencies would give a pattern at every load; the discreteness is the observable, and it comes from the boundary conditions, not from the wave equation.

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Phenomenon 35.3 (Half-frequency response to a modulated tension).

When the same string is driven not by displacing its end but by periodically varying its tension at frequency $f$, standing waves are again established, but the string oscillates at $f/2$ — one transverse cycle for every two cycles of the drive [Melde:1860]. The response is absent for weak modulation and sets in above a threshold that increases with the damping. Rests on Equations (35.1) and (35.3).

Derivation. Derives Phenomenon 35.3. Rotating the fork removes the transverse drive altogether: the prong now moves along the thread, stretching and relaxing it, so the end is not displaced sideways at all and what the drive modulates is the tension,

\begin{equation}\tag{35.5} T(t)=T_{0}\left(1+\epsilon\cos\omega t\right)\ec\qquad \omega=2\pi f\ec\quad 0<\epsilon\ll1\ec \end{equation}

with $\epsilon$ the fractional swing of the load and $f$ the fork's frequency. The length, the density and the end conditions are untouched, so the transverse displacement may still be expanded in the clamped-string modes, $y(x,t)=\sum_{n}q_{n}(t)\sin\left(n\pi x/L\right)$. Substituting that expansion into $\mu\,\pp_{t}^{2}y=T(t)\,\pp_{x}^{2}y$ and using the orthogonality of the sines, each mode separates:

\begin{equation}\tag{35.6} \ddot{q}_{n}+\omega_{n}^{2} \left(1+\epsilon\cos\omega t\right)q_{n}=0\ec\qquad \omega_{n}=\frac{n\pi}{L}\sqrt{\frac{T_{0}}{\mu}}=2\pi f_{n}\ec \end{equation}

From Equations (35.1) and (35.5) (inserting the modal expansion into the string equation and projecting on $\sin(n\pi x/L)$). with $f_{n}$ the free frequency of Equation (35.1) at the mean tension $T_{0}$. Each mode of the thread is therefore an oscillator whose stiffness — not whose applied force — is modulated, and Equation (35.6) is Mathieu's equation [Mathieu:1868]. That is exactly the system solved in Oscillations and Mechanical Waves, and its two conclusions may be taken over unchanged: the amplitude grows exponentially in a band of drive frequencies centred on $\omega=2\omega_{n}$, so that the response is at half the drive; and, once the mode's damping is restored, the growth occurs only above the threshold $\epsilon>2/Q_{n}$, with $Q_{n}$ the quality factor of that mode. Both features asserted in the phenomenon follow — the half-frequency response by construction, since a term $\cos\omega t$ multiplying $q_{n}$ feeds back on itself only through its half-frequency component, and the threshold because the modulation multiplies the amplitude instead of adding to it.

The reduction also yields a prediction that a scale pan tests directly. Under the transverse drive the $n$-loop pattern needs $f_{n}=f$, which is Equation (35.3); under the longitudinal drive it needs $f_{n}=f/2$, so that $\left(n/2L\right)\sqrt{T'_{n}/\mu}=f/2$ and

\begin{equation}\tag{35.7} T'_{n}=\frac{L^{2}f^{2}\mu}{n^{2}}=\frac{T_{n}}{4}\ep \end{equation}

From Equations (35.3) and (35.6) (imposing $f_{n}=f/2$ in place of $f_{n}=f$ and dividing). Turn the fork through a right angle and the same figure of $n$ loops reappears at one quarter of the load that held it before. The ratio contains neither the fork's frequency nor the thread's mass, so it is another pure number read off a set of weights — and it is a prediction, not a datum: the loads for the longitudinal configuration are not quoted in the source, which reports the halving of the frequency.

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Remark 35.4 (What the reduction does not settle).

Equation (35.6) is linear, so it predicts unbounded exponential growth above threshold and nothing about the steady amplitude actually seen. What arrests the growth is the stretching of the thread by its own transverse excursion, which raises $T_{0}$ with amplitude and detunes the mode out of the unstable band; that saturation, and the further instabilities beyond it, belong to Nonlinear Dynamics and Chaos. The linear analysis also locates only the principal band; the full tongue structure in the $(\omega,\epsilon)$ plane is Floquet's theorem, which Oscillations and Mechanical Waves records as a result Part II does not yet carry. Nothing above uses it. Rests on Equation (35.6).

Interpretation

The transverse configuration makes the discreteness of the spectrum visible to the eye and countable, and it does so at fixed frequency, which is the complement of Mersenne's fixed-mode, variable-frequency measurement: between them the law $f_{n}=(n/2L)\sqrt{T/\mu}$ is checked along both of its independent directions. The nodes, in particular, are not an artefact of the ear or of any interpretation; they are points of a taut thread that visibly do not move while their neighbours a centimetre away sweep through centimetres, and their number is an integer that anyone in the room can count.

The longitudinal configuration is a different phenomenon, and it is the reason this experiment retains a place in modern physics. Driving a system by modulating one of its parameters rather than by applying an external force produces a response at half the drive frequency, above a threshold — the mechanical prototype of parametric amplification, and the classical ancestor of the parametric oscillators of Quantum Optics and the Photon. Melde was not the first to see it: Faraday had already reported a half-frequency response in the “crispations” of liquid on a vibrating plate [Faraday:1831], thirty years earlier and in a two-dimensional system. Melde's string is the case in which the mechanism is transparent, because the modulated parameter is a single measurable number, the tension.

Primary references

[Melde:1860]; [Faraday:1831] for the earlier observation of a half-frequency response.

Chladni: the nodal figures of a vibrating plate (1787)

Tests Equation (35.12). Assuming Equation (35.9).

A string has nodes at isolated points; a plate has them along curves, and the curves can be made to draw themselves. Chladni's figures are the first direct images of the eigenfunctions of a two-dimensional boundary-value problem, and they are still the standard demonstration that a bounded elastic body has a discrete spectrum whose modes have shape.

Apparatus

A flat plate of brass or of glass, square or circular, of the order of a tenth of a millimetre to a millimetre thick and some tens of centimetres across, held rigidly at its centre by a clamp on a stand so that the whole boundary is free. Fine, dry, sifted sand. A violin bow, well rosined.

Procedure

Sand is sprinkled evenly over the upper surface. One point of the rim is damped with a fingertip, which forces a node there, and the bow is drawn downwards across the rim at another point, which forces an antinode there. The plate sounds, and within a second or two the sand migrates off the regions in motion and settles in sharp ridges along the curves that are at rest. Changing the damped and bowed points selects a different mode, and the figure and the pitch change together. Each figure is recorded by drawing, and its pitch is recorded by comparison with a monochord or an organ pipe.

Observations and data

A catalogue of figures, each with its own pitch. The nodal sets are curves — for a circular plate, a family of diameters and concentric circles; for a square plate, systems of lines and hyperbola-like arcs that are not the modes of a rectangular membrane and are strikingly harder to guess. The number of nodal diameters $m$ and nodal circles $n$ labels the mode, and the frequencies of the circular plate were found to rise roughly as the square of $m+2n$ [Chladni:1787]. Crucially, the pitches are not in whole-number ratios: a plate has a discrete spectrum, like a string, but an inharmonic one, which is why a struck plate has no definite musical pitch.

Phenomenon 35.5 (Two-dimensional normal modes have shape).

A bounded elastic plate vibrates freely only at a discrete set of frequencies; each is accompanied by a stationary pattern whose nodal set is a system of curves on the surface, determined by the shape of the plate and by the boundary conditions and not by how the plate is excited. The frequencies are not integer multiples of the lowest, and for a circular plate labelled by $m$ nodal diameters and $n$ nodal circles they increase approximately as $\left(m+2n\right)^{2}$ [Chladni:1787].

Derivation. Derives Phenomenon 35.5. Continuum Mechanics and Elasticity establishes the equation governing the transverse deflection $w$ of a thin plate of density $\rho$, thickness $h$ and flexural rigidity $D$, together with the two conditions a free edge imposes; free harmonic motion $w=W\ee^{-\ii\omega t}$ reduces it to

\begin{equation}\tag{35.8} \nabla^{4}W=k^{4}W\ec\qquad k^{4}=\frac{\rho h\omega^{2}}{D}\ep \end{equation}

Everything asserted follows from that eigenvalue problem.

Discreteness and shape come first, and they are the cheapest. The operator in Equation (35.8) is self-adjoint on the plate's own domain under the plate's own edge conditions, so its spectrum is a discrete increasing sequence and each eigenvalue carries an eigenfunction determined by the domain and the boundary alone. The bow selects which eigenfunction is excited and how strongly; it has no other freedom. That is exactly the observed independence of the figure from where and how hard the plate is bowed.

The inharmonicity is read off the dispersion relation. Substituting a plane wave $W\propto\ee^{\ii\vect{k}\cdot\vect{x}}$ in Equation (35.8),

\begin{equation}\tag{35.9} \omega=k^{2}\sqrt{\frac{D}{\rho h}}\ec \end{equation}

From Equation (35.8) (taking $\nabla^{4}$ of a plane wave, which multiplies it by $k^{4}$, and solving for $\omega$). quadratic in the wavenumber where the string of Equation (35.1) is linear. That single difference is the whole of the inharmonicity: a clamped string has $k_{n}=n\pi/L$ in exact arithmetic progression, and a dispersion linear in $k$ carries an arithmetic progression of wavenumbers into an arithmetic progression of frequencies — the harmonic series. Squaring destroys it. Whatever sequence $k_{1}<k_{2}<\dots$ the boundary conditions select, the frequencies go as $k^{2}$, so equally spaced wavenumbers give ever-widening frequency intervals and no ratio of small integers survives.

The square law in the labels needs the geometry of the disc. Since $\nabla^{4}-k^{4}=\left(\nabla^{2}+k^{2}\right) \left(\nabla^{2}-k^{2}\right)$, every solution of Equation (35.8) is the sum of a propagating part solving the Helmholtz equation and an evanescent part solving the modified one. Separating both in polar coordinates on a disc of radius $a$ and discarding what is singular at the centre,

\begin{equation}\tag{35.10} W=\left[A\,J_{m}(kr)+B\,I_{m}(kr)\right] \cos m\left(\theta-\theta_{0}\right)\ec\qquad m\in\N\cup\{0\}\ec \end{equation}

with $J_{m}$ the Bessel function of Definition 9.93 and $I_{m}$ its modified counterpart; $m$ is the number of nodal diameters, and the two free-edge conditions fix $B/A$ and quantize $k$. Because $I_{m}(kr)$ grows monotonically, roughly as $\ee^{kr}/\sqrt{2\pi kr}$, the evanescent term is confined to within a few $1/k$ of the rim once $ka\gg1$, and the nodal circles in the interior are those of $J_{m}(kr)$, at $r=j_{m,s}/k$ with $j_{m,s}$ the $s$-th positive zero. Exactly $n$ of them lie inside the plate when $j_{m,n}\lesssim ka<j_{m,n+1}$, so the mode carrying $n$ interior circles has $ka\approx j_{m,n}$, up to a shift of order unity fixed by the edge condition. The large zeros of $J_{m}$ are asymptotically spaced by $\pi$,

\begin{equation}\tag{35.11} j_{m,n}=\left(n+\frac{m}{2}-\frac{1}{4}\right)\pi +O\!\left(\frac{1}{n}\right)\ec \end{equation}

whence $ka\approx\left(\pi/2\right)\left(m+2n\right)$ and, by Equation (35.9),

\begin{equation}\tag{35.12} \omega_{mn}\approx\frac{\pi^{2}}{4a^{2}} \sqrt{\frac{D}{\rho h}}\left(m+2n\right)^{2}\ec \end{equation}

From Equations (35.9) and (35.11) (substituting $k\approx\pi(m+2n)/2a$ into the dispersion relation). which is the measured law. The combination $m+2n$ is not a coincidence of bookkeeping: Equation (35.11) says a nodal circle advances $ka$ by a full $\pi$ while a nodal diameter advances it by $\pi/2$, because the angular index enters as the order of the Bessel function and each unit of order displaces the zeros by half a period. Note also that the prefactor of Equation (35.12) carries the thickness and the inverse square of the diameter, which is the scaling Continuum Mechanics and Elasticity obtains for geometrically similar plates without solving for any mode at all.

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Remark 35.6 (Part II owes the asymptotics of the Bessel zeros).

Equation (35.11) is the one step above that is neither physics nor a computation special to this plate: it is a statement about $J_{m}$, and it belongs beside the Bessel material of Ordinary Differential Equations and Sturm–Liouville Theory, which at present carries the series, the recurrences, the generating function and the orthogonality relation but says nothing about where the zeros lie. The missing statement is that the substitution $u=\sqrt{r}\,R$ puts the Bessel equation in the form $u''+\left[k^{2}-\left(m^{2}-\tfrac{1}{4}\right)/r^{2}\right]u=0$, whose potential term is negligible for $kr\gg m$, so that the zeros are eventually spaced by $\pi/k$ with the offset quoted. It is used here only for the asymptotic form of Equation (35.12), which is itself asserted as an approximation and is compared with an experiment that reports pitches to a fraction of a tone; nothing in this chapter would change if the offset $-1/4$ were carried differently. Rests on Equation (35.11) and Definition 9.93.

Remark 35.7 (What is not computed here).

The exact eigenvalues are not. Imposing the two free-edge conditions on Equation (35.10) gives a transcendental determinant in $ka$ whose roots depend on Poisson's ratio, and neither those roots nor the nodal figures of the square plate — which are not separable and which are the harder and more famous half of Chladni's catalogue — follow from the argument above. What is established is the structure the experiment actually reports: a discrete spectrum, figures owned by the plate rather than by the bow, an inharmonic sequence, and the approximate square law in $m+2n$. Rests on Equation (35.10).

Phenomenon 35.8 (The figure depends on the tracer).

Fine, light powder such as lycopodium does not settle where sand settles. Sprinkled on the same plate sounding the same mode, it collects at the antinodes — the regions of greatest motion — producing the photographic negative of the sand figure [Faraday:1831]. Rests on Phenomenon 35.5.

Half of this is already accounted for. Continuum Mechanics and Elasticity derives the sand figure from a threshold: a grain leaves the surface wherever the peak acceleration $\omega^{2}A$ of the plate exceeds $g$, is thrown, lands, and is thrown again until it reaches a neighbourhood of a node where the local amplitude is too small to eject it. That argument is about a grain coupled to the plate, and it predicts the sand figure and nothing else. The lycopodium figure is the complementary case, in which the grain is coupled to the air; what carries it is the steady second-order flow that an oscillating boundary drives in the fluid above it, and that flow is not obtained from any first-order acoustic quantity.

Full derivation in Appendix A.

Derives Phenomenon 35.8.

The argument there is a second-order one throughout, and that is the whole reason it does not fit here. Every first-order acoustic quantity oscillates with zero mean, so the steady transport of a tracer cannot appear until the viscous equations of Fluid Dynamics are expanded to second order in the amplitude. What emerges is Rayleigh's result: the oscillating viscous layer against the plate rectifies the flow above it into a steady slip velocity proportional to the gradient along the surface of the squared first-order amplitude, directed from the antinodes towards the nodes inside the layer and back the other way in the bulk above it. A grain that the plate never throws is carried by the outer branch of that circulation, and collects where the sand does not.

Remark 35.9 (The discriminator between the two powders).

It is tempting to separate sand from lycopodium by the ratio of the grain's Stokes relaxation time to the acoustic period, and that is wrong: at audio frequencies the ratio exceeds unity for both powders, so neither follows the oscillating air. The discriminator is whether the surface ejects the grain at all — the threshold $\omega^{2}A>g$ of Continuum Mechanics and Elasticity — and, for a grain it does not eject, which of the steady streaming drag and the residual contact friction wins. That comparison involves the grain's adhesion to the plate, a quantity this treatise does not model, so what the appendix derives is the streaming flow and the direction of its outer branch, not a predicted threshold radius separating the two powders.

Interpretation

The figures are the observational content of the statement that a two-dimensional wave problem has eigenfunctions. A string can be described without that vocabulary; a plate cannot, because its modes have to be drawn. What the sand traces is a level set of the eigenfunction, namely its zero set, and the fact that this set is independent of how hard the plate is bowed, and of where, so long as the same mode is selected, is precisely the statement that the mode belongs to the plate and not to the excitation.

The inharmonicity matters as much as the discreteness, and it is the sharpest way to see that “discrete spectrum” and “harmonic series” are different claims. The clamped string's exact integer ratios are a special consequence of its linear dispersion relation $\omega=vk$; the plate's governing equation is fourth order in space, its dispersion relation is quadratic in $k$, and the overtones drift away from whole-number ratios. The same distinction returns in the lattice of Phonons and Lattice Dynamics, where the departure from linearity in $\omega(k)$ is the physical content of dispersion.

Chladni's plates also forced a theory into existence, which is the cleanest possible illustration of the relation this treatise asserts between experiment and derivation. The figures were exhibited across Europe and could not be explained; the Paris Academy set them as a prize problem; and the answer, Sophie Germain's equation for the elastic surface [Germain:1821], later put on a consistent footing by Kirchhoff [Kirchhoff:1850], is the plate theory used in Continuum Mechanics and Elasticity to this day.

Finally, Phenomenon 35.8 is retained here as a standing warning about visualization. The sand figure is not the mode; it is what one particular tracer does in the presence of the mode, and a lighter tracer, coupling to the air rather than to the plate, reports the complementary set. An experiment that images a field through a tracer measures the tracer as well as the field.

Primary references

[Chladni:1787] [Faraday:1831]; [Germain:1821] [Kirchhoff:1850] for the plate theory the figures provoked.

Newton, Laplace and the speed of sound (1687–1822)

Tests Equations (35.13) and (35.16). Assuming Equation (35.14).

This is the experiment in which a measurement of a mechanical speed turned out to be a measurement of a thermodynamic quantity. Newton derived the speed of sound in the Principia from the elasticity of the air and obtained a value well below what the field measurements gave; the discrepancy was not an experimental error, and it did not go away for one hundred and twenty-nine years.

Apparatus

Two stations on a surveyed baseline of the order of ten kilometres, each with a gun and an observer; chronometers, or in the earliest work a seconds pendulum; thermometer, barometer and hygrometer at each station. Newton's own attempt used no baseline at all: he timed the returning echoes from the far end of the cloister of Neville's Court at Trinity College against a pendulum, adjusting the pendulum until its swing kept step with the echoes.

Procedure

The observer at each station records the interval between seeing the flash of the distant gun and hearing its report. Light crosses the baseline in a time utterly negligible on this scale, so the interval is the transit time of the sound and the speed follows from the surveyed distance. The essential refinement is reciprocal firing: guns are discharged alternately from the two ends and the two transit times are averaged, which cancels the component of the wind along the line to first order — the dominant systematic error, and one that no amount of repetition from a single end will remove [Arago:1822]. Temperature, pressure and humidity are logged so that the result can be reduced to a standard state, and the whole procedure is repeated on several nights.

Observations and data

The speed of sound in dry air at $0\,\mathrm{^\circ\mathrm{C}}$ is close to $331.3\,\mathrm{m}/\mathrm{s}$, rising to about $343\,\mathrm{m}/\mathrm{s}$ at $20\,\mathrm{^\circ\mathrm{C}}$. It is independent of the amplitude of the sound at ordinary intensities, independent of its frequency across the audible band, and — at fixed temperature — independent of the pressure of the air. Newton's theoretical value, computed from the same measured pressure and density, is $279.9\,\mathrm{m}/\mathrm{s}$: low by more than $15\,\mathrm{\%}$, a discrepancy some fifty times any plausible error in a ten-kilometre baseline [Newton:1687].

Quantity	Value	Origin
Pressure $p_{0}$ of the standard atmosphere	$101325\,\mathrm{Pa}$	exact by definition
Density $\rho_{0}$ of dry air at $0\,\mathrm{^\circ\mathrm{C}}$ and $p_{0}$	$1.293\,\mathrm{kg}/\mathrm{m}^{3}$	reference value
Ratio of specific heats $\gamma$ of a diatomic gas	$1.400$	theory
Newton's isothermal speed $\sqrt{p_{0}/\rho_{0}}$	$279.9\,\mathrm{m}/\mathrm{s}$	computed
Laplace's adiabatic speed $\sqrt{\gamma p_{0}/\rho_{0}}$	$331.2\,\mathrm{m}/\mathrm{s}$	computed
Speed of sound, dry air at $0\,\mathrm{^\circ\mathrm{C}}$	$331.3\,\mathrm{m}/\mathrm{s}$	reference value
Speed of sound, dry air at $20\,\mathrm{^\circ\mathrm{C}}$	$343.2\,\mathrm{m}/\mathrm{s}$	reference value

The sound speed in dry air, and the two theoretical values it is to be compared with. No uncertainty is attached to any entry because none of them is a measurement reported by a source verified for this treatise: the first two rows are the standard state, the middle rows are computed in the derivation below from those two numbers alone, and the last two are the reference values for dry air, quoted to the precision shown. They are given so that the reader can repeat the arithmetic, not so that a fit can be claimed.

Phenomenon 35.10 (The speed of sound is set by the adiabatic compressibility).

Sound propagates through a gas at the speed

\begin{equation}\tag{35.13} c=\sqrt{\frac{\gamma p_{0}}{\rho_{0}}}\ec \end{equation}

with $\gamma=c_{p}/c_{V}$ the ratio of the specific heats and not, as Newton assumed, at $\sqrt{p_{0}/\rho_{0}}$ [Laplace:1816]. Because $p_{0}/\rho_{0}=RT/M$ for an ideal gas, the speed rises as the square root of the absolute temperature, does not depend on the pressure at fixed temperature, and carries no dispersion.

Derivation. Derives Phenomenon 35.10. Oscillations and Mechanical Waves linearizes the equations of a compressible fluid about rest and obtains a wave equation with speed $c^{2}=\dd p/\dd\rho$, the derivative to be taken along the thermodynamic path the gas actually follows. Newton took it at constant temperature. For dry air at $0\,\mathrm{^\circ\mathrm{C}}$, with $p_{0}=101325\,\mathrm{Pa}$ and $\rho_{0}=1.293\,\mathrm{kg}/\mathrm{m}^{3}$, the isothermal path $p\propto\rho$ gives

\begin{equation}\tag{35.14} c_{T}=\sqrt{\frac{p_{0}}{\rho_{0}}} =\sqrt{\frac{101325}{1.293}}\,\mathrm{m}/\mathrm{s} =279.9\,\mathrm{m}/\mathrm{s}\ep \end{equation}

A sound wave, however, compresses the air far faster than heat can diffuse across a wavelength, so the correct path is adiabatic, $p\propto\rho^{\gamma}$, whence $\dd p/\dd\rho=\gamma p_{0}/\rho_{0}$ and Equation (35.13) follows. With the diatomic value $\gamma=1.40$,

\begin{equation}\tag{35.15} c=\sqrt{\gamma}\,c_{T}=1.1832\times279.9\,\mathrm{m}/\mathrm{s} =331.2\,\mathrm{m}/\mathrm{s}\ec \end{equation}

which agrees with the measured $331.3\,\mathrm{m}/\mathrm{s}$ to better than one part in a thousand. The temperature dependence follows from $p_{0}/\rho_{0}=RT/M$: at $20\,\mathrm{^\circ\mathrm{C}}$ the speed is $331.3\,\mathrm{m}/\mathrm{s}\times\sqrt{293.15/273.15}=343.2\,\mathrm{m}/\mathrm{s}$, as observed. The absence of dispersion follows because the coefficient in the wave equation contains no $k$, so all audible frequencies travel at one speed.

∎

Phenomenon 35.11 (The shortfall is the square root of the specific-heat ratio).

The ratio of the measured speed to the isothermal prediction is $\sqrt{\gamma}$, so that the acoustic measurement determines the ratio of the specific heats of the gas without any calorimetry [Newton:1687] [Laplace:1816]. Rests on Equations (35.13) and (35.14).

Derivation. Derives Phenomenon 35.11. Dividing Equation (35.13) by Equation (35.14) gives $c/c_{T}=\sqrt{\gamma}$, so

\begin{equation}\tag{35.16} \gamma=\left(\frac{c}{c_{T}}\right)^{2} =\frac{c^{2}\rho_{0}}{p_{0}}\ep \end{equation}

With the measured $c=331.3\,\mathrm{m}/\mathrm{s}$ and the same $p_{0}$ and $\rho_{0}$ as above, $c/c_{T}=1.1836$ and $\gamma=1.401$ — the value of $c_{p}/c_{V}$ for a diatomic gas, obtained from a stopwatch, a survey and a barometer. The shortfall is therefore not a defect in the measurement but a datum: it is $1-1/\sqrt{\gamma}=15.5\,\mathrm{\%}$, and it is the thermodynamics.

∎

Interpretation

Three separate results sit in this one measurement, and they are worth separating.

The first is the speed itself, which is the input to every acoustic calculation in this part and which fixes the scale against which the Doppler shifts of Section 35.6 are read.

The second is the null result on dispersion, which is among the sharpest in physics and is almost never stated as one. A chord fired from a distant gun arrives as a chord; thunder from a kilometre away is a rumble but not a glissando; an orchestra heard from the back of a hall is not smeared in time by pitch. Each is an observation that $\dd c/\dd f$ vanishes to the precision of the ear over three decades of frequency, and each is a direct check on the linearity of the wave equation of Oscillations and Mechanical Waves. The linearity does fail: at large amplitude the compressions steepen into shocks, and that regime belongs to Fluid Dynamics.

The third is the thermodynamic one, and it is the reason this experiment is included in a mechanics part at all. In 1687 there was no thermodynamics, no specific heat and no notion of an adiabatic process; Newton's calculation was correct given his assumption, and the assumption was the error. Equation (35.16) converts a mechanical measurement into a determination of $\gamma$, hence of the number of thermally active degrees of freedom per molecule, which is the subject of Kinetic Theory of Gases and Classical Thermodynamics. That a discrepancy of $15\,\mathrm{\%}$ was allowed to stand, unexplained and undisguised, for well over a century is a better model of scientific practice than the outcome; Newton himself attempted to close it by invoking the finite size of the air particles and the presence of vapour, and that adjustment was wrong.

Primary references

[Newton:1687] [Laplace:1816]. The field determinations quoted in outline above are those of the eighteenth-century academies and, most carefully, of the Bureau des Longitudes commission that fired reciprocal cannon over a surveyed baseline south of Paris in 1822 and published in the Annales de Chimie et de Physique; that memoir is not among the entries of this treatise's bibliography, and no measured figure is attributed to it here. The reference values for the speed and for the density and pressure of dry air are standard and are used above only as inputs to a derivation the reader can repeat.

Kundt's tube: sound speeds from dust figures (1866)

Tests Equations (35.17) and (35.19). Assuming Equation (35.13).

A ten-kilometre baseline and a pair of cannon are a heavy way to measure a speed, and they cannot be pointed at an arbitrary gas. Kundt reduced the whole measurement to a bench-top apparatus and, more remarkably, to a ruler: his method eliminates the clock entirely, by comparing two wavelengths at one frequency instead of timing one wave.

Apparatus

A horizontal glass tube, closed at one end by a movable piston and at the other by a light disc mounted on the end of a metal rod that is clamped rigidly at its midpoint and passes into the tube through a sliding seal. Fine dry cork dust, or lycopodium powder, spread in a thin line along the floor of the tube. A rosined cloth for exciting the rod, and a supply of whichever gas is to fill the tube.

Procedure

The rod is stroked lengthwise with the rosined cloth until it sounds its longitudinal fundamental: clamped at the middle and free at both ends, it has a displacement node at the clamp and antinodes at its ends. The disc at its inner end therefore drives the gas column at the rod's frequency. The piston is then slid along the tube until the gas column is resonant, at which point the dust leaves the smooth line and springs into a regular series of sharp transverse ridges. The spacing of the ridges is measured with a scale laid along the tube, and the free length of the rod is measured with the same scale. The tube is then flushed and refilled with a different gas and the measurement repeated, the rod, and hence the frequency, being unchanged.

Observations and data

Sharp, evenly spaced ridges of dust, whose spacing depends strongly on the gas filling the tube and not at all on how hard the rod is stroked [Kundt:1866]. The measured quantities are two lengths: the ridge spacing $d$ and the half-length $L_{r}$ of the rod from clamp to end.

Phenomenon 35.12 (Dust ridges mark the half-wavelength spacing of a standing wave in a gas).

A powder light enough to be moved by the gas but too heavy to remain suspended collects at the displacement nodes of a standing sound wave, forming ridges spaced by half a wavelength [Kundt:1866]. The wavelength in the gas and that in the driving rod are therefore both measurable with a scale at one and the same frequency, and their ratio gives the ratio of the sound speeds,

\begin{equation}\tag{35.17} \frac{c_{\text{gas}}}{c_{\text{rod}}}=\frac{d}{L_{r}}\ep \end{equation}

Rests on Equation (35.13).

Derivation. Derives Phenomenon 35.12. The rod is clamped at its midpoint and free at both ends, so its longitudinal fundamental has a displacement node at the clamp and antinodes at the ends: each half of the rod is a quarter wavelength, and the whole rod is a half wavelength, $\lambda_{\text{rod}}=2L_{r}$ with $L_{r}$ the clamp-to-end length. In the gas the ridges form at the displacement nodes, which in a standing wave are spaced by half a wavelength, so $\lambda_{\text{gas}}=2d$. Both waves are sustained by the same source and therefore share one frequency $f$, so

\begin{equation}\tag{35.18} \frac{c_{\text{gas}}}{c_{\text{rod}}} =\frac{f\lambda_{\text{gas}}}{f\lambda_{\text{rod}}} =\frac{2d}{2L_{r}} =\frac{d}{L_{r}}\ec \end{equation}

which is Equation (35.17): the frequency cancels, and with it every requirement for a clock, a calibrated fork or a known pitch. Two gases compared with the same rod give, by Equation (35.13),

\begin{equation}\tag{35.19} \frac{d_{1}}{d_{2}}=\frac{c_{1}}{c_{2}} =\sqrt{\frac{\gamma_{1}M_{2}}{\gamma_{2}M_{1}}} \end{equation}

at a common temperature, so that a ratio of two ruler readings determines a ratio of specific-heat ratios once the molar masses are known.

∎

Interpretation

Kundt's tube is Chladni's method transplanted into a gas: a powder is used to make a nodal set visible, and the figure is read as a geometric object rather than as a pitch. What the transplant buys is decisive. In the open-air determination of Section 35.4 the measured quantities are a distance of ten kilometres and a time of thirty seconds, and the systematic error is the wind. Here the measured quantities are two lengths of a few centimetres on the same bench, read with the same scale, and Equation (35.17) shows that the frequency — the one quantity that is hard to hold and hard to calibrate — has dropped out of the answer altogether. Reducing a measurement to a ratio of two lengths taken with one instrument is a recurring strategy in this treatise, and it is the same move that Equation (35.2) makes for the string.

The consequence that reached furthest was thermodynamic. Because Equation (35.19) yields $\gamma$ for any gas that can be put in a tube, the method turned the ratio of specific heats into a routine measurement, including for vapours that no calorimeter of the period could handle. Kundt and Warburg applied it a decade later to mercury vapour and found a ratio of about $1.67$ — the value $\tfrac{5}{3}$ expected for a gas whose molecules have translational degrees of freedom and nothing else [KundtWarburg:1876]. That was the first clear evidence that a molecule can be a structureless point as far as heat is concerned, and it belongs to the equipartition argument of Kinetic Theory of Gases.

Primary references

The primary source is Kundt's memoir on a new kind of acoustic dust figure and its use for determining the speed of sound in solids and in gases [Kundt:1866], together with the later application to vapours by Kundt and Warburg [KundtWarburg:1876], which returns the monatomic ratio for mercury vapour. Both are printed in Poggendorff's Annalen der Physik und Chemie — volumes 127 and 157 of that series, which the publisher's continuous numbering records as 203 and 233. The physics the method tests is that of [Laplace:1816], and the dust-figure technique is that of [Chladni:1787].

Buys Ballot: the Doppler effect on a moving train (1845)

Tests Equations (35.20) and (35.21). Assuming Equation (35.13).

Doppler predicted in 1842 that motion of a source relative to the medium shifts the frequency received, and applied the idea — wrongly — to the colours of double stars [Doppler:1842]. Three years later Buys Ballot tested the acoustic half of the prediction with the fastest vehicle then available and the most sensitive frequency comparators then available: a locomotive, and musicians.

Apparatus

An open flat wagon drawn by a locomotive on a straight, level stretch of the Dutch Rhenish Railway between Utrecht and Maarssen. A group of trumpeters on the wagon, sounding and holding a note agreed in advance. A second group of trumpeters stationed beside the track. Observers of trained absolute pitch, some standing at the trackside, some riding on the wagon, each writing down the note heard. No instrument other than the human ear is used, and none was needed: the quantity to be measured is a frequency ratio, and a musician reports a frequency ratio as an interval.

Procedure

The train is run past the trackside observers at the highest speed the locomotive will hold, with the wagon trumpeters sounding continuously. Each trackside observer records the pitch heard while the source approaches and again while it recedes, expressed as an interval from the note actually being played. The arrangement is then reversed: the trumpeters stand still beside the track and the observers ride past them. Runs are repeated at several speeds, and over several days, the speed of the train being known from the timetable of the run over a measured stretch.

Observations and data

The pitch heard is higher than the pitch sounded while the source approaches and lower while it recedes; the change occurs as the source passes, and the effect reverses with the direction of travel. The size of the shift grows with the speed of the train and is of the order of a semitone at the speeds attainable. Reversing the roles of source and observer gives the same effect to the accuracy of the method [BuysBallot:1845].

Phenomenon 35.13 (Doppler shift of sound).

The frequency received from a source moving through still air differs from the frequency emitted, being raised on approach and lowered on recession by

\begin{equation}\tag{35.20} f'=f\,\frac{c}{c\mp v_{s}}\ec \end{equation}

with the upper sign for approach; and an observer moving through still air toward or away from a stationary source receives

\begin{equation}\tag{35.21} f'=f\,\frac{c\pm v_{o}}{c}\ep \end{equation}

Both are observed, and the shift is of order a semitone for the speeds of a nineteenth-century locomotive [Doppler:1842] [BuysBallot:1845].

Derivation. Derives Phenomenon 35.13. Work in the frame in which the air is at rest, where the wave speed is $c$ and depends on nothing else. Let the source emit $f$ crests per second while moving toward the observer at $v_{s}$. In the time $1/f$ between crests the first crest travels $c/f$ while the source advances $v_{s}/f$, so successive crests are separated in space by $\lambda'=(c-v_{s})/f$. A stationary observer meets crests at $f'=c/\lambda'=fc/(c-v_{s})$, which is Equation (35.20); recession changes the sign of $v_{s}$. If instead the source is at rest and the observer advances at $v_{o}$, the spacing of the crests in the air is unchanged at $\lambda=c/f$, but the observer sweeps through them at closing speed $c+v_{o}$, so $f'=(c+v_{o})/\lambda=f(c+v_{o})/c$, which is Equation (35.21).

The size is what makes the experiment feasible. An equal-tempered semitone is the ratio $2^{1/12}=1.0595$; from Equation (35.20) an approaching source produces it when $c/(c-v_{s})=1.0595$, that is when

\begin{equation}\tag{35.22} v_{s}=c\left(1-\frac{1}{1.0595}\right) =0.0562\,c\approx19.3\,\mathrm{m}/\mathrm{s}\ec \end{equation}

taking for $c$ the summer-afternoon value $343\,\mathrm{m}/\mathrm{s}$ of Section 35.4: some $69\,\mathrm{km}/\mathrm{h}$, at the upper end of what a locomotive of 1845 could sustain, and a shift that a musician identifies at once because a semitone is the smallest interval the Western scale names. Note also that the shift is a ratio: the observer needs to know neither $c$ nor the absolute pitch of the trumpet, only the interval.

∎

Interpretation

The experiment confirms Phenomenon 35.13, and it does so in the one arrangement where the confirmation is unambiguous: the source motion is real, mechanical and reversible, the medium is demonstrably at rest, and the effect changes sign as the train goes by. It is the earliest quantitative test of a prediction that is now the basis of radar, of medical flow imaging, of stellar radial velocities and of the expansion of the universe.

Its deeper content is the asymmetry between Equation (35.20) and Equation (35.21). The two describe physically different situations — source moving through the air, observer moving through the air — and they are not the same function. Expanding both for small speeds,

\begin{equation}\tag{35.23} \frac{f'_{s}}{f}=1+\frac{v}{c}+\frac{v^{2}}{c^{2}}+\dots\ec \qquad \frac{f'_{o}}{f}=1+\frac{v}{c}\ec \end{equation}

so that they agree at first order and differ at second, by $v^{2}/c^{2}$. For $v/c=0.056$ that is about three parts in a thousand, roughly five cents of pitch, far below what an ear can judge against a moving background — which is why Buys Ballot found the two arrangements equivalent and why the asymmetry, though real, is not among his results. Stated positively: sound has a preferred frame, the rest frame of the air, and only relative motion with respect to it matters; a sufficiently precise acoustic Doppler measurement can therefore detect motion through the medium. That the corresponding optical experiment finds no such frame — and no such second-order asymmetry, only the symmetric relativistic formula — is the content of Experiments: Light, the Aether, and Time and the point of departure for Lorentz Transformations. The contrast is sharpest when the two are set side by side, which is why the acoustic case is worked out here in full.

A word on Doppler's own paper, in the spirit of the honesty rule that governs this treatise. Doppler applied his result to the colours of double stars and concluded that the colours are caused by orbital motion [Doppler:1842]. That conclusion is wrong: the colours of stars are set by their surface temperatures, and the velocities involved are far too small to shift a star out of the visible band. Buys Ballot said so, in the very memoir that confirmed the acoustic effect [BuysBallot:1845]. The prediction was right, the application was wrong, and the experiment separated them — the optical shift is real but had to wait for spectroscopy, where a shifted absorption line can be measured against a laboratory standard, rather than for the eye.

Primary references

[Doppler:1842] [BuysBallot:1845].

Summary of the evidence

Experiment	Prediction tested	Result
Mersenne 1636	$f_{1}\propto L^{-1}T^{1/2}\mu^{-1/2}$	all three proportionalities confirmed by musical interval; absolute frequency transferred from a slow cord by ratios of lengths
Chladni 1787	discrete inharmonic plate modes	nodal curves drawn by sand; frequencies of a circular plate rise as $(m+2n)^{2}$, which follows from the plate's quadratic dispersion and the spacing of the Bessel zeros; the spectrum is not harmonic
Melde 1860	$T_{n}\propto n^{-2}$ at fixed drive	loops appear only at discrete loads in the ratios $1:1/4:1/9$; a modulated tension gives a half-frequency response, predicted to hold the same figure at one quarter of the load
Newton 1687, Laplace 1816	$c=\sqrt{\gamma p_{0}/\rho_{0}}$	measured $331.3\,\mathrm{m}/\mathrm{s}$ in dry air at $0\,\mathrm{^\circ\mathrm{C}}$ against $279.9\,\mathrm{m}/\mathrm{s}$ isothermal; implies $\gamma=1.401$
Kundt 1866	$c_{\text{gas}}/c_{\text{rod}}=d/L_{r}$	sound speeds and $\gamma$ from two ruler readings, with no clock and no known frequency
Buys Ballot 1845	$f'=fc/(c-v_{s})$	pitch raised on approach and lowered on recession, of order a semitone, reversing as the train passes

Tests of the wave predictions of Oscillations and Mechanical Waves reported in this chapter. Each result is a ratio or a pure number, which is why apparatus of the seventeenth to the nineteenth century suffices to test them.

Every entry in Table 35.2 is a consequence of one linear wave equation together with a boundary condition, and none of them carries a free parameter. Two of them reach outside mechanics: the sound speed is a measurement of $c_{p}/c_{V}$ and therefore of the internal structure of molecules, and the Doppler shift is the acoustic prototype of the kinematic effect whose optical counterpart forces the transformations of Lorentz Transformations. One derivation is still pending — the acoustic streaming behind Phenomenon 35.8 — and it is of a kind that recurs throughout the chapter: it concerns how the medium moves the tracer, never whether the wave picture holds. Every claim about the waves themselves is derived here or in the theory chapters this one answers to.

Quantity	Value	Origin
Pressure $p_{0}$ of the standard atmosphere	\(101325\,\mathrm{Pa}\)	exact by definition
Density $\rho_{0}$ of dry air at \(0\,\mathrm{^\circ\mathrm{C}}\) and $p_{0}$	\(1.293\,\mathrm{kg}/\mathrm{m}^{3}\)	reference value
Ratio of specific heats $\gamma$ of a diatomic gas	\(1.400\)	theory
Newton's isothermal speed $\sqrt{p_{0}/\rho_{0}}$	\(279.9\,\mathrm{m}/\mathrm{s}\)	computed
Laplace's adiabatic speed $\sqrt{\gamma p_{0}/\rho_{0}}$	\(331.2\,\mathrm{m}/\mathrm{s}\)	computed
Speed of sound, dry air at \(0\,\mathrm{^\circ\mathrm{C}}\)	\(331.3\,\mathrm{m}/\mathrm{s}\)	reference value
Speed of sound, dry air at \(20\,\mathrm{^\circ\mathrm{C}}\)	\(343.2\,\mathrm{m}/\mathrm{s}\)	reference value