Doppler Effect Calculator - Sound, Radar and Redshift

The Doppler effect, from a passing siren to a receding galaxy

Why does a 700 Hz siren arrive at 759.3 Hz and leave at 649.3 Hz when the ambulance does 60 mph? Because motion squeezes the waves ahead of a source and stretches them behind it. This calculator works that out for sound, with the source and the listener moving separately and the speed of sound set by the air temperature, for a radar gun, where the wave bounces off the car and is shifted twice, and for light, with the relativistic formula and the redshift z. It also runs backward: from a pitch you heard, a shift a radar measured or a spectral line you observed, it gives the speed.

2.71

semitones a siren drops as it passes at 60 mph

72.02 Hz

shift per mph on a K-band radar gun

658.47 nm

hydrogen-alpha seen from 1,000 km/s away

What moves, and what the wave does about it

Sound needs something to travel through, so it matters who is moving relative to the air. A source moving toward you packs each new wave crest closer to the last, and the wavelength in front of it shrinks: a 700 Hz siren has crests 49.03 cm apart when it stands still and 45.2 cm apart ahead of it at 60 mph. A listener moving toward a still source does not change the wavelength at all; the listener simply meets the crests more often. Both raise the pitch, by different amounts, which is why the calculator asks for the two speeds and the two directions separately.

The formula is f = f₀ × (v + v_o) / (v − v_s), with v the speed of sound and both motions counted positive when they close the gap. In dry air v follows the temperature, 331.3 m/s × √(T / 273.15 K), which is 767.7 mph at 68 °F and 741.1 mph at 32 °F.

A radar gun is its own case. The beam goes out, the moving car receives it at a shifted frequency and sends that back, and the gun sees a second shift on the way home. For road speeds the result is Δf ≈ 2 u f₀ / c, exactly twice what a one-way formula gives. Light needs no medium, so only the relative speed counts, and above a few percent of the speed of light the square-root formula f = f₀ × √((1 + β) / (1 − β)) takes over from the everyday one.

Setting up a Doppler problem in the form

Kind of wave - sound, a radar reflection, or light and radio.
What to find - the frequency that arrives, or the speed behind a frequency you measured.
Source frequency and unit - hertz to terahertz; for light you can enter a wavelength in nanometers instead, such as 656.28 for hydrogen-alpha.
Speed unit - mph, km/h, m/s or km/s, used for every speed on the form and in the answer.
For sound - the medium and, for air, the temperature in °F; then the source speed and direction and the listener speed and direction, with 0 for whichever stands still.
For radar and light - the speed along the line of sight and whether the two are closing or separating; in reverse mode, the measured shift in hertz or the observed frequency or wavelength.
Read the result - the received frequency, the shift in hertz and percent, semitones for sound, z and β for light, and the working written out.

A 700 Hz siren going by at five speeds

Listener standing at the roadside, air at 68 °F. The first two columns are what you hear while the vehicle approaches and after it has passed; the last is the size of the drop in musical terms.

Vehicle speed	Approaching	Receding	Drop as it passes
15 mph	713.9 Hz	686.6 Hz	0.68 semitones
30 mph	728.5 Hz	673.7 Hz	1.35 semitones
45 mph	743.6 Hz	661.2 Hz	2.03 semitones
60 mph	759.3 Hz	649.3 Hz	2.71 semitones
75 mph	775.8 Hz	637.7 Hz	3.39 semitones

The drop grows almost in step with speed: 0.68 semitones at 15 mph, about a whole tone at 45 mph and more than three semitones at 75 mph. A 400 Hz train horn at 80 mph falls from 446.5 Hz to 362.3 Hz, a drop of 3.62 semitones.

Radar bands and the shift they read

US traffic radar works in three licensed bands. The shift per mph grows in proportion to the beam frequency, so the same car produces a Ka-band shift more than three times the X-band one. Figures for a car closing at 65 mph, computed at a representative frequency inside each band.

Band	Licensed range	Frequency used	Shift per mph	Shift at 65 mph	Interpretation
X	10.50 to 10.55 GHz	10.525 GHz	31.39 Hz	2.04 kHz	Oldest of the three
K	24.05 to 24.25 GHz	24.15 GHz	72.02 Hz	4.682 kHz	Used by many handheld guns
Ka	33.4 to 36.0 GHz	34.7 GHz	103.49 Hz	6.727 kHz	Newest, now widespread

Reverse mode does the gun's job: a 4,682 Hz shift on a 24.15 GHz beam comes back as 65.01 mph, the tiny excess over 65 being nothing more than rounding in the shift you typed.

Hydrogen-alpha at four speeds, and when the simple formula fails

The red hydrogen line at 656.28 nm is the one astronomers use most for velocities. The classical estimate z ≈ β is shown alongside the relativistic answer.

Motion	Observed line	Redshift z	Classical z	Interpretation
Approaching at 30 km/s	656.21 nm	−1.001e-4	−1.001e-4	Formulas agree
Receding at 1,000 km/s	658.47 nm	0.00334	0.00334	Formulas agree
Receding at 0.1c	725.54 nm	0.10554	0.1	Classical 5.3% low
Receding at 0.5c	1.137 µm	0.73205	0.5	Classical 31.7% low, line now infrared

Going the other way, a line observed at 658.47 nm gives a recession speed of 998.7 km/s. The 1.3 km/s gap from 1,000 is again only the rounding of the wavelength to two decimals.

Relationships hiding in the formula

Moving yourself is not the same as the source moving. A 440 Hz tone heard while you drive toward it at 30 mph arrives at 457.2 Hz; the same tone on a vehicle driving toward you at 30 mph arrives at 457.9 Hz. At 300 mph the gap is dramatic: 611.9 Hz for a moving listener against 722.2 Hz for a moving source.

Cold air makes the effect slightly stronger. Sound is slower at 32 °F, so a 60 mph siren is a larger fraction of it: 761.7 Hz approaching in freezing air against 757.8 Hz at 95 °F.

Two pitches are enough to get the speed. If you heard the approach and the retreat but do not know the rest pitch, v_s = v × (f_a − f_r) / (f_a + f_r). With 759.3 Hz and 649.3 Hz that returns 59.95 mph, within rounding of the 60 mph that produced them.

A radar gun at an angle reads low, never high. It sees only the part of the speed along its beam, the speed times the cosine of the angle. At 10° that costs 1.5%, at 30° it costs 13.4%.

At the speed of sound the formula runs out. A source at 900 mph in 68 °F air is at Mach 1.17. Nothing is heard ahead of it; the waves pile into a cone with a half-angle of 58.5° and arrive as one sonic boom.

Reading your own result

For sound, the semitone figure translates the shift into what an ear knows: under 0.5 is less than a quarter of a whole tone, 1 is one step on a piano, 2 is a whole tone, and 12 would be a full octave. For light, a positive z means the source is receding and its light is stretched toward the red, a negative z that it is approaching. Every result assumes motion straight along the line between the two ends; motion at an angle shrinks the shift by the cosine of that angle. A radar reading worked out here is a physics check, not evidence about a speeding ticket.

Doppler questions people actually search

Why does a siren sound higher coming toward you?

Each wave crest leaves the siren from a point a little closer to you than the last, so the crests arrive bunched up. A 700 Hz siren at 60 mph reaches you at 759.3 Hz, 8.48% higher. Once it has passed, the crests are spread out and you hear 649.3 Hz.

How does a police radar gun measure speed with the Doppler effect?

It sends a steady microwave beam, receives the reflection from the car and measures the difference in frequency. On a 24.15 GHz K-band gun every mph adds 72.02 Hz, so 65 mph shows up as about 4.68 kHz. The shift counts twice because the car both receives and re-radiates the beam.

Does wind change the pitch you hear?

Not when the source and the listener are both standing still: the wind speeds the waves up on the way, but crests still leave and arrive at the same rate, so the frequency is unchanged. It only matters in combination with motion, and even then by a small amount.

Is redshift the same thing as the Doppler effect?

For stars in our galaxy and nearby galaxies, yes: the redshift measures motion along the line of sight. For distant galaxies most of the redshift comes from the expansion of space during the light's journey, and above about z = 0.1 the speed from this calculator should not be read as motion through space.

What happens when the source reaches the speed of sound?

The denominator of the formula goes to zero and no steady pitch reaches anyone ahead of it. The calculator reports the Mach number and the cone angle instead, for example Mach 1.17 and 58.5° for 900 mph. A supersonic source moving away still has a lower pitch behind it.

Can I get a car's speed from its sound on a recording?

Yes, if you can read the pitch before and after it passes. The two-pitch formula above needs no rest frequency, or use reverse mode if you know it: 700 Hz heard as 760 Hz means the source is closing at 60.61 mph.

Does the Earth's own motion shift starlight?

Yes, and astronomers correct for it. The Earth circles the Sun at roughly 30 km/s, so over a year a star near the plane of the orbit is approached and receded from at up to that speed. Its hydrogen-alpha line swings between about 656.21 nm and 656.35 nm, which is why measured velocities are quoted after subtracting the Earth's motion.

Why does the calculator want the air temperature?

Because the speed of sound depends on it: 741.1 mph at 32 °F and 787.1 mph at 95 °F. The Doppler shift is the vehicle speed as a fraction of the sound speed, so a colder day gives a slightly larger shift for the same vehicle.

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