Calculate Doppler shifts relative to the medium

Work in stationary uniform air with sound speed c. A source emits wavefronts with its own prescribed period Tₛ, treated as the same emission period in the classical medium frame. While one wavefront moves a distance cTₛ, a source moving toward the receiver advances a distance vₛTₛ. The forward spacing is their difference. Thus the wavelength in front of a subsonic source is smaller than the stationary-source wavelength. Behind the source, the spacing is larger.

For a receiver at rest in the medium, divide propagation speed by that local wavelength. The denominator contains c−vₛ for an approaching source. Here vₛ is a signed source speed toward the receiver along the propagation line, not an arbitrary speed measured relative to the receiver. The elementary formula assumes constant collinear velocities over many periods and no wind or inhomogeneous medium. It is not valid as an ordinary ahead-of-source periodic-front construction when the source is supersonic.

Now let the receiver move toward the incoming fronts at signed speed vₒ relative to the medium. It encounters fronts at relative speed c+vₒ while their spacing still comes from the source calculation. Combining the two effects gives a numerator c+vₒ and denominator c−vₛ. A receding receiver has negative vₒ in this convention; a receding source has negative vₛ. Ordinary positive-frequency reception along this direction also requires c+vₒ>0; a subsonic receiver satisfies this. Explicit sign definitions then handle approach or recession in this collinear setting.

Compare two experiments with equal speed magnitudes. A moving source approaching a resting receiver increases pitch by the factor 1/(1−v/c). A moving receiver approaching a resting source increases it by 1+v/c. These agree at first order when speed is much smaller than c, but differ at higher order. The distinction reflects sound propagation through a medium. It should not be erased by substituting a relativistic optical Doppler formula, which describes a different physical propagation problem.

Away from a straight approach, only the appropriate line-of-sight component enters each instantaneous kinematic factor, with source geometry evaluated at emission and receiver geometry at reception. As a source passes, this direction changes, so the observed frequency can vary even at constant speed magnitude. A moving source’s wavefronts are centered at earlier source positions. Using the current source-to-receiver direction without checking retarded geometry can therefore misdescribe a close or fast pass. The collinear derivation avoids that complication deliberately.

Use the vectors lab to isolate a projection before attempting such geometry. Regard its single vector as a normalized velocity, keep magnitude fixed, and predict its x component at angles zero, 60°, and 90°. The controls actually vary angle and magnitude and show Cartesian projections. They do not emit sound, compute retarded times, or distinguish source from receiver motion. The useful lesson is that a transverse component does not automatically enter a line-of-sight Doppler calculation; the complete acoustic formula still needs its own frame and sign choices.