Doppler Effect · study.

Doppler measurements and wave-front geometry

The Doppler effect compares two rates. A source emits successive wave crests at its source frequency $f_{s}$ . A receiver records the arrival rate of those crests, called the received frequency $f_{r}$ . Relative motion can change the arrival rate even when the source oscillator maintains the same period. An approaching source–receiver pair gives $f_{r} > f_{s}$ ; a separating pair gives $f_{r} < f_{s}$ .

For sound and other mechanical waves, every velocity in the classical formula must be measured relative to the propagating medium. Air, water, a string, or another material carries the disturbance. The medium defines a physically distinguished reference frame. A car horn and a listener moving together through still air are both moving relative to the medium; source and receiver terms require their separate medium-frame velocities.

The word “frequency” needs a precise location. Source frequency counts oscillation cycles at the source. Received frequency counts crest arrivals at the receiver. A frequency counter connected to a microphone measures the latter. It does not directly measure the oscillator period at a source that is moving away from the microphone. An axial geometry gives the cleanest derivation. Let a sound wave travel from a source toward a receiver along the positive $x$ direction. Let $v$ be the sound speed relative to the medium. Let $u_{s}$ and $u_{r}$ be the source and receiver velocity components along positive $x$ , also relative to the medium. The signed convention is

u_{s} > 0 source moves toward a receiver ahead of it, u_{r} > 0 receiver moves with the wave .

With this convention, a receiver moving away from the source has positive $u_{r}$ and meets fewer crests per second. A source moving toward the receiver has positive $u_{s}$ and leaves successive crests closer together on the forward side. The signs enter different parts of the formula because source motion changes wavefront spacing, whereas receiver motion changes the rate at which a fixed spacing is sampled.

Signed axial convention for a wave travelling toward positive

x

. A source velocity

u_{s} > 0

points along the wave and compresses the forward crests; a receiver velocity

u_{r} > 0

carries the receiver with the wave and lowers its crest-arrival rate.

The basic classical result for an axial ray is

f_{r} = f_{s} \frac{v - u _{r}}{v - u _{s}} .

It is valid when all three velocities are relative to the same uniform medium, the source speed remains below $v$ , and the line from source to receiver has a stable direction during the interval being analyzed. The expression is easier to use after the physical geometry has been drawn. Substitute signs only after assigning the positive wave direction.

The axial formula applies locally to a three-dimensional measurement. Let $\overset{n}{^}$ point from source to receiver along the outgoing ray. Replace $u_{s}$ and $u_{r}$ by the corresponding medium-frame projections

u_{s} = v_{s} \cdot \overset{n}{^}, u_{r} = v_{r} \cdot \overset{n}{^} .

The ray direction can rotate while an object passes a sensor. A long data record then contains changing radial velocity even if the object travels at constant speed along a straight path. A constant frequency shift therefore supports a constant radial component, not automatically a constant speed magnitude.

Only the velocity component along the source-to-receiver ray produces an instantaneous Doppler shift. The transverse component changes the object's bearing; the radial projection sets the measured frequency ratio.

Convert to the medium frame before evaluating a uniform-wind case. Consider a listener and a horn both fixed to the ground during a uniform wind. Each has a nonzero velocity relative to the air, and those two medium-frame velocities are equal. Their numerator and denominator factors are the same, so received frequency equals source frequency. The wind changes ground-frame propagation speed and arrival time without changing the crest-count ratio between the two ground-fixed endpoints.

The word “approach” also requires care. Source and receiver can become closer in ground coordinates while a moving medium modifies their individual medium-frame velocities. The signed formula is the final test. Draw the wave direction, convert every object velocity into the medium frame, project along the ray, and then evaluate the numerator and denominator.

Moving source: wave-front geometry

Suppose the receiver is at rest in the medium, so $u_{r} = 0$ . The source emits one crest every source period $T_{s} = 1/ f_{s}$ . During one period, the earlier crest advances a distance $v T_{s}$ through the medium. During the same period, the moving source advances $u_{s} T_{s}$ . On the forward side, the gap between that earlier crest and the newly emitted crest is

λ_{front} = (v - u_{s}) T_{s} = \frac{v - u _{s}}{f _{s}} .

On the rear side, the source moves away from the earlier crest, so the spacing is

λ_{rear} = (v + u_{s}) T_{s} = \frac{v + u _{s}}{f _{s}} .

The source oscillator has emitted one cycle in both directions. Motion changes the distribution of those crests through space. Forward spacing is shorter and rear spacing is longer for a source moving toward positive $x$ .

Successive wavefronts are centered at successive emission points along the path. The source advances while each older crest keeps spreading, so the forward crests crowd together and the rear crests spread apart.

The stationary receiver samples those forward or rear spacings at speed $v$ relative to the medium. In front of the source,

f_{r, front} = \frac{v}{λ _{front}} = \frac{v}{v - u _{s}} f_{s} .

Behind the source,

f_{r, rear} = \frac{v}{v + u _{s}} f_{s} .

These two values refer to receivers on opposite sides of the source at the same source speed. A single source may therefore produce an upshift ahead and a downshift behind simultaneously. The observed pitch changes abruptly in an ideal pass-by model only because the receiver switches from sampling the forward wavefronts to sampling the rear wavefronts. The forward formula has a singular denominator as $u_{s}$ approaches $v$ from below. It signals the failure of a separated-wavefront description directly ahead of the source. The source catches up to previously emitted crests, and the linear small disturbance model no longer gives a regular train of forward waves. A supersonic source produces an envelope of wavefronts and a shock geometry.

Moving receiver: crest-arrival geometry

Now hold the source at rest in the medium. The source creates an undisturbed wavelength

λ = \frac{v}{f _{s}} .

The crest spacing is already established in the medium before the receiver samples it. A receiver moving along the positive wave direction has speed $u_{r} > 0$ and moves with each arriving crest. The closing speed between a crest and the receiver is $v - u_{r}$ . The time from one arrival to the next is therefore

T_{r} = \frac{λ}{v - u _{r}}, f_{r} = \frac{1}{T _{r}} = \frac{v - u _{r}}{λ} = (1 - \frac{u _{r}}{v}) f_{s} .

A receiver moving toward the source has $u_{r} < 0$ . Its closing speed exceeds $v$ , the arrival interval shortens, and received frequency increases. The wavelength does not change in this moving-receiver case because the source has not moved through the medium between emissions.

The source is at rest, so the crest spacing is fixed in the medium. A receiver moving toward the source raises the closing speed and meets each crest sooner, so its arrival rate rises.

Source and receiver motion combine by applying the two geometric effects in sequence. First find the wavelength produced by the moving source. Then find the rate at which the moving receiver meets that wavelength. The result is

f_{r} = f_{s} \frac{v - u _{r}}{v - u _{s}} .

The numerator belongs to the receiver because it is a crest-arrival speed. The denominator belongs to the source because it determines crest spacing. Interchanging the velocities gives a different physical problem and generally a different exact frequency ratio. Apply these sign checks when reading the ratio:

Source moves toward the receiver. Its forward crest spacing decreases. Use $u_{s} > 0$ , which makes $v - u_{s}$ smaller and raises $f_{r}$ .
Source moves away from the receiver. Its relevant component has $u_{s} < 0$ . The denominator grows and $f_{r}$ falls.
Receiver moves toward the source. Its velocity is opposite the positive propagation direction, so $u_{r} < 0$ . The numerator grows and $f_{r}$ rises.
Receiver moves with the wave. Its velocity has $u_{r} > 0$ . The numerator shrinks and $f_{r}$ falls.

These cases assume a receiver lies ahead of a source and the outward ray is the positive direction. A receiver behind the source requires a ray drawn in the direction of the observed wave. A verbal word such as “ahead” or “approaching” cannot replace the signed ray diagram when source and receiver have arbitrary three- dimensional paths.

Four axial cases share one positive wave direction. Source approach and receiver approach both raise the received frequency; source recession and receiver recession both lower it, through different terms of the ratio

f_{r} = f_{s} (v - u_{r}) / (v - u_{s})

The exact source and receiver effects differ even when their speeds have the same magnitude.

Equal source and receiver speed fractions give different Doppler ratios. The source-motion ratio

v / (v - u)

steepens toward the sound speed, while the moving-receiver ratio

(v + u) / v

stays linear across the subsonic range.

For speed magnitudes much smaller than $v$ , expand the exact ratio to first order:

\frac{f _{r} - f _{s}}{f _{s}} ≃ \frac{u _{s} - u _{r}}{v} .

The numerator is the relative radial speed in the signed convention. This compact form often supports a quick estimate. Use the exact expression when source or receiver speed is an appreciable fraction of wave speed, when a reflected path has two shifts, or when the required accuracy is comparable with the omitted higher-order terms.

Passing paths and moving media

An object passing a stationary microphone at constant path speed usually has a changing Doppler frequency. Far before closest approach, the velocity component along the ray is mostly toward the microphone and the received frequency is high. At closest approach, the velocity is perpendicular to the ray and the instantaneous first-order shift is zero. After passing, the radial component reverses sign and the received frequency lies below the source frequency.

A straight constant-speed path past a stationary receiver gives a changing radial component: positive on approach, zero at closest approach, negative on recession, even though the path speed never changes.

The radial component of an object at coordinates $(x (t), b)$ relative to a receiver at the origin and moving with constant path speed $U$ along positive $x$ is

u_{rad} (t) = - \frac{U x ( t )}{x ^{2} ( t ) + b ^{2}} .

This sign follows the source-to-receiver ray convention. Before closest approach, $x < 0$ and the radial component is positive; after closest approach, $x > 0$ and it is negative. Its magnitude tends toward $U$ at large range and becomes zero at $x = 0$ . A measured frequency trace can therefore identify closest approach and estimate path speed or offset only after a geometric model has been specified.

The pass-by tone falls monotonically: highest on approach, crossing the source frequency near closest approach, and settling to a lower recession value even at constant path speed.

Moving media and the reference-frame conversion

The sound speed $v$ in the classical Doppler formula is measured relative to the medium. A uniform air flow or water current changes the ground-frame speed of a crest. The material-frame wave speed remains set by the medium’s elastic and inertial properties. Keep these two speeds separate:

V_{crest} = U_{m} + v,

A crest travelling in the positive direction has $U_{m}$ as its medium velocity in the ground frame. The crest speed over the ground is therefore $U_{m} + v$ . Use $v$ in the Doppler ratio after source and receiver velocities have been measured relative to the medium.

Let $U_{s}$ , $U_{r}$ , and $U_{m}$ be the source, receiver, and medium velocity components along the outgoing ray in one ground frame. Their medium-frame components are

u_{s} = U_{s} - U_{m}, u_{r} = U_{r} - U_{m} .

Substitution into the axial Doppler expression gives

f_{r} = f_{s} \frac{v - ( U _{r} - U _{m} )}{v - ( U _{s} - U _{m} )} .

Every velocity in this expression is a signed projection along the ray. A full three-dimensional wind vector matters only through its component along propagation for this axial model. A crosswind can bend a ray or alter the path through a nonuniform atmosphere, but its direct contribution to the one-dimensional frequency ratio is absent when its axial projection is zero. A source and receiver bolted to the ground have $U_{s} = U_{r} = 0$ . In a uniform flow, both have medium-frame velocity $- U_{m}$ . Their frequency ratio is

\frac{f _{r}}{f _{s}} = \frac{v + U _{m}}{v + U _{m}} = 1.

The arrival time changes. A downstream distance $L$ takes $L / (v + U_{m})$ , whereas an upstream distance takes $L / (v - U_{m})$ for a flow whose positive component is $U_{m}$ . Frequency remains unchanged because the source produces successive crests at the same material-frame spacing pattern and the ground-fixed receiver samples that pattern with the same medium-frame drift as the source.

In uniform flow both ground-fixed endpoints share the medium velocity, so the drift cancels from the frequency ratio. The crests travel faster over the ground, changing arrival time but not the received frequency.

For moving endpoints, the conversion must be done before judging “approach.” Suppose an aircraft and a ground microphone have a closing ground-frame range. A tailwind can reduce or increase the aircraft’s medium-frame speed, while the microphone also has a medium-frame velocity opposite the wind. The two converted radial components determine the sign of the Doppler shift. Uniform flow is an idealization. A temperature gradient, wind shear, turbulence, or water-current gradient makes local propagation speed and ray direction vary along the path. A receiver can then see frequency variation caused by changing path geometry or by a varying medium, even when the source oscillator and source trajectory are stable. State whether a measurement assumes uniform flow, measures flow independently, or treats unmeasured flow as an uncertainty contribution.

The transformation also clarifies an apparent asymmetry. A source moving through still air and a receiver moving through still air at the same ground-frame speed are different physical cases because each one occupies a different role in the wavefront-generation and crest-sampling process. A common translation of source, receiver, and medium leaves all medium-frame velocities unchanged and leaves the predicted frequency unchanged. A translation of source and receiver without the medium changes their medium-frame velocities and can change the result. Wind data can be used as an independent check on a timing experiment. Send pulses between two fixed instruments in both directions along a surveyed path. The sum and difference of the measured travel times constrain the still-medium speed and the mean axial flow under the uniform-path model. That is a travel-time measurement. It should not be described as a Doppler frequency result unless an emitted-versus- received crest rate is also measured.

Inferring velocity and wave speed

A Doppler frequency ratio becomes a velocity estimate only after the unknown motion has been specified. With a receiver stationary in the medium, $u_{r} = 0$ , solve the moving-source relation for the source radial component:

u_{s} = v (1 - \frac{f _{s}}{f _{r}}) .

An approaching source has $f_{r} > f_{s}$ and produces positive $u_{s}$ in the selected outgoing-ray convention. A receding source has $f_{r} < f_{s}$ and produces negative $u_{s}$ . The inferred quantity is the source velocity component along the ray. A speed magnitude follows when source motion is known to be collinear with that ray.

With a source stationary in the medium and a moving receiver, solve instead for

u_{r} = v (1 - \frac{f _{r}}{f _{s}}) .

The sign reverses relative to the source case because positive receiver velocity means motion with the outgoing wave. A receding receiver has $u_{r} > 0$ and lower received frequency. Writing the inversion before entering data helps prevent the common error of assigning the source-motion sign to a moving receiver. One frequency ratio cannot determine two unknown endpoint velocities. For example, the same value of $(v - u_{r}) / (v - u_{s})$ can arise from many pairs of $u_{s}$ and $u_{r}$ . An experiment needs an additional constraint: one endpoint may be stationary in the medium, one speed may be read from a track encoder, or two independent acoustic paths may yield separate equations. A reported “Doppler speed” should name the moving object, the assumed stationary reference, and the ray direction.

Frequency data must also be connected to a source-frequency reference. A stable electronic source can provide $f_{s}$ from a direct electrical monitor. A mechanical source may need a nearby reference microphone in the source rest geometry. Treat a nominal dial setting only as a prior value when the required velocity uncertainty is small. Source drift can imitate target motion in a one-channel recording. Estimate the frequency of a recorded sinusoid over an interval where range, source output, and medium condition are adequately stable. A duration $T$ provides an elementary Fourier-bin spacing of approximately $1/ T$ . Longer records can resolve a smaller frequency difference, but they also average over more source motion and more line-of-sight rotation. Select a duration that is short enough for a nearly constant radial component and long enough for the required frequency precision.

Zero-crossing timing, phase-slope fitting, and spectral-peak fitting can each estimate frequency. Their agreement is a data-quality check. The method should state sample rate, analysis duration, taper or window choice, frequency estimator, and whether the reported value is an instantaneous estimate, a segment mean, or a trajectory-model fit. A reproducible velocity estimate includes those analysis settings with the displayed peak location.

Reflected sound and the two-stage shift

A stationary instrument can infer the radial speed of a moving scatterer from a returned sound signal. The outgoing signal first reaches the moving scatterer. The scatterer then reradiates the disturbance toward the instrument. The velocity affects both legs, so a reflected signal has two classical shifts.

Let the instrument transmit frequency $f_{s}$ toward a scatterer moving away with positive radial speed $u$ . On the outgoing leg, the scatterer receives the wave as a moving receiver:

f_{1} = f_{s} \frac{v - u}{v} .

On the return leg, the scatterer emits as a moving source opposite the return-ray direction. The received return frequency at the stationary instrument is

f_{ret} = f_{1} \frac{v}{v + u} = f_{s} \frac{v - u}{v + u} .

Solving for the scatterer radial speed gives

u = v \frac{f _{s} - f _{ret}}{f _{s} + f _{ret}} .

Positive $u$ gives a return frequency below the transmitted frequency. Negative $u$ gives an upshift. For $∣ u ∣ ≪ v$ ,

\frac{f _{ret} - f _{s}}{f _{s}} ≃ - \frac{2 u}{v} .

The factor of two arises from the outgoing and return legs. It is a path-count result for a reflected signal, not a generic multiplier for every Doppler measurement.

A moving scatterer imposes two classical shifts. It first receives the outgoing wave as a moving receiver, then reradiates as a moving source; the radial speed enters both legs, giving the factor of two.

The reflected model assumes a scatterer with a stable, identifiable return component. An extended rotating object can produce a range of radial speeds and a broadened return spectrum. Multiple surfaces can create several return paths with different delays and frequency shifts. Analyze a time interval and frequency region associated with one modeled path before applying the single-scatterer inversion. The transmitted and returned frequencies can be estimated with separate reference records or with a synchronized digital generator and receiver. Record the source frequency before and after the motion interval. A drift correction derived from those reference records is valid only if source behavior changes smoothly over the interval. A sudden source-frequency jump is indistinguishable from a target-speed change without an independent source monitor.

Laboratory frequency-data workflow

A laboratory Doppler experiment separates geometric variables from signal-analysis variables. The geometric record establishes source and receiver locations, source axis, track direction, medium state, and any flow along the acoustic path. The signal record establishes source frequency, received frequency, sample rate, time gate, and frequency-estimation method. Combining both records produces a radial-velocity estimate with assumptions that can be checked later.

A simple single-pass arrangement uses a tone source and a microphone on a rail. Hold the microphone fixed in the medium frame and move the source at a known track speed, or hold the source fixed and move the microphone. A rail aligned with the acoustic ray makes the radial component equal to the cart speed. An off-axis rail requires a position-dependent projection; it supports pass-by data analysis but should not be analyzed with an axial constant-shift formula. Before a motion run, verify the source frequency using an electrical monitor or a nearby stationary reference sensor. Verify the sign convention on a slow trial: move the source toward the receiver and confirm that the reported received frequency increases. Repeat with recession and confirm that it decreases. These two tests expose swapped cables, an inverted velocity sign, and a frequency-analysis label that has been assigned to the wrong channel.

The sound speed used in the reduction should match the actual medium condition. Air measurements should record temperature near the propagation path and any axial flow. A measured time-of-flight speed is a direct path-specific value. A nominal room-temperature number can be sufficient for a qualitative demonstration, but it contributes directly to every inferred velocity through the factor $v$ . In water or another fluid, record composition and temperature because both affect speed.

Time gates and frequency estimates

Each data gate must be short enough that the radial component is approximately constant across the gate. It must also be long enough to resolve the required frequency difference. A spectral estimate from record length $T$ has a natural bin spacing of $1/ T$ . If a target speed creates a predicted shift of only $0.5 Hz$ , a one-second record has insufficient raw bin spacing for a simple discrete-bin readout. Longer records, interpolation around a spectral peak, or phase-based fitting can improve precision, provided radial motion remains stable over the selected interval. Keep the source-reference and received-signal time bases synchronized when using phase-slope or sample-count methods. A drifting acquisition clock changes the frequency ratio even if the acoustic source and geometry remain stable. A shared clock, a recorded reference channel, or a documented synchronization procedure gives the conversion from sample count to physical frequency. The required clock accuracy should be compared with the fractional Doppler shift being measured.

Sampling rate must exceed twice the highest retained frequency component. A higher rate also permits a more detailed time trace and broader analysis margin, but it does not by itself improve frequency resolution. Record the anti-alias acquisition band, input range, and any resampling step. A frequency estimate produced after undocumented decimation cannot be independently reproduced from the stored data. A direct source-motion example illustrates the reduction.

The result applies to that gate’s radial component. A track aligned to the source- receiver ray gives a cart-speed estimate. A pass-by with a nonzero offset gives one point of a radial-velocity trace and needs the path geometry before it can be converted into constant cart speed. The laboratory record should distinguish raw observations from inferred quantities. Store the received time series or a reproducible frequency table. Store source reference data, cart position or velocity reference, temperature and flow record, coordinate convention, and the formula used for inversion. A final velocity column without the accompanying frequency ratio and geometry cannot be reanalyzed if a later calibration correction is needed.

Uncertainty, path geometry, and model boundaries

For the stationary-receiver source inversion

u_{s} = v (1 - \frac{f _{s}}{f _{r}}),

frequency and sound-speed uncertainties propagate through different terms. Small independent standard uncertainties can be approximated by

σ_{u}^{2} ≃ (\frac{f _{s} v}{f _{r}^{2}} σ_{f_{r}})^{2} + (\frac{v}{f _{r}} σ_{f_{s}})^{2} + [(1 - \frac{f _{s}}{f _{r}}) σ_{v}]^{2} .

The measured-frequency term becomes more important when the estimate requires a small difference between two nearby frequencies. The sound-speed term becomes more important as the inferred velocity fraction grows. These expressions omit geometry error, source acceleration within a gate, multipath, and flow variation; add those terms when the experiment includes them. An off-axis path gives a separate geometry uncertainty. If a moving object has speed $U$ and its velocity makes an angle $ψ$ with the source-to-receiver ray, then

u_{rad} = U cos ψ .

Near $ψ = 9 0^{\circ}$ , a small error in angle can dominate a small radial component. A single Doppler sensor cannot recover the transverse component without additional paths or an external trajectory model. Two or more source–receiver directions can recover more velocity information when their ray geometry is surveyed and their measurements are synchronized. The classical mechanical-wave model has explicit boundaries:

Subsonic source motion: the ordinary forward-wavefront formula requires $∣ u_{s} ∣ < v$ along the relevant ray.
Uniform medium during a gate: the standard ratio uses one wave speed and one medium frame. Flow gradients or temperature gradients require a propagation model or an uncertainty allowance.
One identified path: direct sound, a selected reflected return, and other multipath contributions must not be mixed into one unqualified frequency estimate.
Stable radial component within a gate: acceleration and line-of-sight rotation must be small enough for the assigned gate model.
Mechanical-wave domain: the material-medium formula does not apply to light in vacuum. Electromagnetic Doppler calculations require their own relativistic model. At a source speed above the wave speed, no regular forward crest train remains. Wavefronts emitted at earlier positions form an envelope. The envelope angle of a source with speed magnitude $U > v$ obeys

sin α = \frac{v}{U}, M = \frac{U}{v},

where $M$ is the Mach number. A forward microphone receives a shock arrival rather than the divergent value predicted by taking $u_{s}$ arbitrarily close to $v$ in the ordinary subsonic formula. The shock geometry is a boundary of the Doppler model, not a route to an unlimited audible frequency.

Supersonic motion (

M = U / v > 1

) collapses the wavefronts into a Mach cone whose half-angle satisfies

sin α = v / U

. No regular forward crest train reaches a receiver ahead of the source.

Required measurement record

Medium state: propagation material, temperature, relevant flow component, and sound-speed value or speed-measurement method.
Geometry: source and receiver coordinates, ray direction, axis convention, track or trajectory description, and gate positions.
Frequency data: source reference, received estimate, sample rate, gate duration, estimator, and source-monitor checks.
Inference model: single-pass or reflected path, moving endpoint identified, exact formula, radial-component definition, and sign convention.
Uncertainty and residuals: frequency scatter, sound-speed uncertainty, geometry contribution, calibration drift, background paths, and deviations from the selected model.

A result reported as a signed radial velocity, with this record attached, retains its physical meaning if the source frequency, air condition, or coordinate convention is revisited later. This convention supports comparison among an axial rail experiment, a moving-receiver observation, and a reflected-scatterer measurement without treating their different Doppler geometries as interchangeable.

Controlled wave-speed measurements

The Doppler relation can be inverted for wave speed when a radial endpoint velocity is independently controlled and measured. This reverses the usual velocity-sensing use of the effect. It requires a sufficiently accurate motion reference and a frequency shift large enough to resolve. The result is a medium-relative wave speed; ground-frame crest speed in a moving medium needs a separate flow conversion.

A source moving toward a stationary receiver in a still medium has

f_{r} = f_{s} \frac{v}{v - u _{s}} .

Solving for $v$ gives

v = \frac{u _{s} f _{r}}{f _{r} - f _{s}} .

The denominator is a measured frequency difference. When $u_{s} ≪ v$ , that difference is small compared with either frequency. Small errors in $f_{r} - f_{s}$ then produce a large fractional error in the inferred wave speed. Controlled motion can therefore demonstrate the Doppler relation well before it supports a high-precision sound-speed measurement.

The calculation uses the exact moving-source relation. A first-order expression would give $v ≃ u_{s} f_{s} / (f_{r} - f_{s})$ and differs slightly because it replaces $f_{r}$ by $f_{s}$ in the numerator. The exact form should be retained when the experimental frequency precision is good enough for that difference to matter.

The sensitivity of the wave-speed estimate follows directly from the two measured frequencies:

\frac{\partial v}{\partial f _{r}} = - \frac{u _{s} f _{s}}{( f _{r} - f _{s} ) ^{2}}, \frac{\partial v}{\partial f _{s}} = \frac{u _{s} f _{r}}{( f _{r} - f _{s} ) ^{2}} .

As the frequency difference becomes small, both derivatives grow in magnitude. Increasing source frequency at the same controlled speed increases the absolute Doppler difference because the fractional shift is approximately $u_{s} / v$ . Higher tone frequency can therefore make frequency estimation easier, provided the source and receiver have a stable response in the selected band and the motion remains accurately axial. An alternative wave-speed arrangement uses a controlled moving receiver. With a source stationary in the medium,

f_{r} = f_{s} (1 - \frac{u _{r}}{v}), v = \frac{u _{r} f _{s}}{f _{s} - f _{r}} .

The signed receiver velocity must be along the outgoing ray. A receiver travelling toward the source has $u_{r} < 0$ and $f_{r} > f_{s}$ ; the numerator and denominator both carry signs that still yield positive $v$ . State the signed convention in the calculation. Insert the signed axial component defined by the ray convention.

The moving-source and moving-receiver estimates test different pieces of the model: source motion changes emitted spacing, and receiver motion changes sampled arrival rate. Agreement between the two measurements, made under comparable medium conditions, supports the classical medium-frame description. A discrepancy can indicate motion-reference error, an unrecognized air flow, source-frequency drift, or a geometry mismatch. A moving medium complicates wave-speed inference. The controlled Doppler relation still uses endpoint speeds relative to the medium. If the rail and receiver are fixed to the ground while air has uniform axial flow, a source ground speed must be converted by subtracting the flow component. A separate bidirectional pulse timing measurement can estimate the flow and still-medium speed. Combining a ground-frame speed with an unmeasured medium flow produces an apparent wave speed that depends on travel direction. Wave-speed reporting should identify the frame. “Speed of sound” commonly means the speed relative to the medium. “Arrival speed over ground” includes the medium drift component along the ray. A pulse-timing experiment can measure the latter directly. A Doppler experiment with controlled motion can infer the former only when the medium-frame endpoint speed is known. These quantities agree in a still medium and separate in a flowing medium.

Cross-checks for a controlled-motion run

Static reference: with both endpoints stationary relative to the medium, the received frequency should agree with the direct source reference within the frequency-estimation uncertainty.
Reversed motion: reverse the controlled radial motion. The shift should reverse sign with comparable magnitude when source output, rail alignment, and medium state remain unchanged.
Speed series: repeat several known speeds. The low-speed frequency difference should scale linearly with radial speed; residual curvature or an intercept signals a model or reference problem.
Geometry return: repeat one surveyed position after the speed series. A changed received frequency at the same geometry points to source drift or medium change.
Independent timing: compare the Doppler-derived medium speed with a pulse time-of-flight result under the same temperature and flow condition. The controlled-motion reduction has a distinct failure signature. If the fit slope changes after a source-frequency reference is recalibrated, source drift has entered the data. If residuals depend on rail position, path alignment or a reflecting boundary is a likely cause. If the inferred speed differs for the two travel directions, the medium may have axial flow or a temperature gradient. Each pattern points to a specific part of the physical model that needs revision, which a single averaged speed hides.

Store the source reference, received-frequency values, known motion values, medium state, and geometry for every speed-series point. Refit the series after excluding only measurements with a documented instrument or geometry fault. An unexplained outlier belongs in the data record; hiding it by adjusting a source frequency or changing a sign convention after inspection destroys the diagnostic value of the residual pattern.

Emission time, reception time, and accelerating paths

A frequency measured at reception time describes a crest emitted earlier. A source at range $R (t_{e})$ in a still medium has emission time $t_{e}$ and reception time $t_{r}$ related by

t_{r} = t_{e} + \frac{R ( t _{e} )}{v} .

The source radial velocity in a moving-source Doppler formula belongs to the emission state. Over a short range, a slowly moving source can use its reception-time position as an acceptable approximation. A rapidly moving or accelerating source can have a propagation delay long enough that the reception-time position and the emission-time position differ appreciably.

A received crest carries the source state at emission, not at reception. Propagation delay maps a reception-time frequency estimate back to an earlier position and radial velocity on the trajectory.

Construct the radial-velocity model for a pass-by data set at the retarded emission times. A practical procedure starts with surveyed source positions or a track encoder time series. For each receiver time gate, solve the propagation-delay relation using the source range in the appropriate medium model. Evaluate the source position and velocity at the resulting emission time, project that velocity onto the emission-to- receiver ray, and compare the predicted frequency with the received estimate.

The retarded-time correction becomes important when the object traverses a meaningful fraction of the source–receiver range during sound propagation. A source moving at $20 m s^{- 1}$ with a receiver $100 m$ away in air has nominal travel time of about $0.29 s$ . The source can move about $5.8 m$ during that interval. Using the reception-time location would misplace the ray and can bias the radial projection.

An accelerating source also broadens a frequency estimate made over a long gate. A single spectral peak represents an average over a changing arrival rate. Shorten the gate, fit a time-dependent phase model, or report the estimate as a gate average with the corresponding mean geometry. Select the duration from source acceleration, range change, and required velocity resolution.

Data reconciliation needs a common clock. A cart encoder, source reference channel, and microphone recorder must share timestamps or a measured synchronization offset. If the source clock begins one gate later than the receiver clock, a velocity curve can acquire an artificial slope that resembles acceleration. Preserve timestamps, clock-rate calibration, and any offset correction with the frequency table.

A uniform moving medium requires the delay relation to use the appropriate ground-frame crest speed along the ray. A spatially varying flow or bent ray requires integration along the actual path. In that situation, a simple retarded-time correction based on $R / v$ is an approximation and should be identified as such. The same limitation applies to every inferred source state, including the radial component used in the Doppler formula.

Repeated gates should retain paired source-reference and received-frequency estimates. For each gate, first form the frequency ratio with the source reference measured for that same interval, then apply the appropriate Doppler inversion. Average the resulting radial components only across gates with comparable geometry and medium state. Averaging displayed frequencies from different ranges, then inserting one average into a nonlinear inversion, can combine source drift, path rotation, and velocity variation into a value that corresponds to no physical state.

Fit the raw frequency ratio or frequency difference in a controlled speed series against the independently measured radial speed. Preserve the covariance of source and received frequency estimates when both share a clock or a common calibration source. A fit residual sequence resolves more structure than a single scatter number: smooth curvature indicates ray-angle error, a step indicates an instrument range change, and a monotonic trend indicates source drift or medium change. Report the fitting variable, weighting rule, retained gates, and rejection criterion with the inferred speed or wave-speed result.

The distinction between emitted and received frequency also affects data labels. A table indexed by receiver timestamp should store the corresponding estimated emission timestamp whenever source position or velocity is used in the same row. State whether range, radial speed, and source frequency refer to emission time, reception time, or a gate average. This prevents an apparently small timing convention from becoming an untraceable disagreement between a trajectory record and a frequency record.

With a stationary source and moving receiver, the receiver state belongs naturally to the reception time because the crest-arrival rate is sampled at the receiver. A reflected measurement contains both an outgoing reception event at the scatterer and a later return reception event at the instrument. A high-accuracy reflected-path model associates each stage with its own event time. That bookkeeping becomes essential when the scatterer accelerates or the path length changes rapidly.