WAV-04

Acoustic Doppler shift

f' = f (v ± vo) / (v ± vs).

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DopplerDoppler 1842

Governing equation

f'=f\dfrac{v+v_o}{v+v_s}

$v_s \in \left[-50, 50\right]$

where

f: Source frequency (Hz)
v: Sound speed (m/s)
v_o: Observer speed (+ toward) (m/s)
v_s: Source speed (+ away) (m/s)
f': Observed frequency (Hz)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

f'=f\dfrac{v+v_o}{v+v_s}

$v_s \in \left[-50, 50\right]$

$v_s = -50$ $f' = 515.1$

Along the curve

②

Domain of the variable — blue

v_s_{\min} = -50,\quad v_s_{\max} = 50

$v_s \in \left[-50, 50\right]$

$v_s = -50$ $v_s = -50$

Along the curve

③

Response curve — green

f'=f\dfrac{v+v_o}{v+v_s}

$v_s \in \left[-50, 50\right]$

$v_s = -50$ $f' = 515.1$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}f'}{\mathrm{d}v_s}

$v_s \in \left[-50, 50\right]$

$v_s = -49.37$ $f'^{\prime} = -1.75$

Along the curve

Lecture brief

Historical brief

d’Alembert’s wave equation, Snell, the thin-lens maker, Doppler and Bragg interference are the classical optics-and-sound toolkit. The lab is propagation, image and shift. This sheet (WAV-04 — Acoustic Doppler shift) is the form associated with Doppler 1842. Working symbols: $f$ , $v$ , $v_o$ , $v_s$ $\rightarrow$ $f'$ . A moving source shortens the wavelength; a moving observer changes the encounter rate. Combined they give the acoustic Doppler formula.

Purpose

Purpose: compute

f'

from

f

v

v_o

v_s

in Waves & optics via

f'=f\dfrac{v+v_o}{v+v_s}

f' = f (v ± vo) / (v ± vs). Use it when a real waves & optics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

f = 440.000\,\mathrm{Hz}

v = 343.000\,\mathrm{m/s}

v_o = 0.000\,\mathrm{m/s}

v_s = 0.000\,\mathrm{m/s}

, the governing relation

f'=f\dfrac{v+v_o}{v+v_s}

yields

f' = 440.000\,\mathrm{Hz}

. Signs: +vo toward source, −vs toward observer (usual textbook). Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Source frequency f440 HzSound speed v343 m/sObserver speed (+ toward) v_o0 m/sSource speed (+ away) v_s0 m/s

Outputs

Observed frequency f'440.000 Hz

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Narration of this film

Signs: +vo toward source, −vs toward observer (usual textbook).

A moving source shortens the wavelength; a moving observer changes the encounter rate. Combined they give the acoustic Doppler formula.

Reading speed

Watch on YouTube