(II) If a speaker mounted on an automobile broadcasts a...

(II) If a speaker mounted on an automobile broadcasts a song, with what speed $(\mathrm{km} / \mathrm{h})$ does the automobile have to move toward a stationary listener so that the listener hears the song with each musical note shifted un by one note in comparison to the song heard by the automobile's driver? On the equally tempered chromatic scale, the ratio of frequencies of neighboring notes is $2^{1 / 12}$ .

(II) If a speaker mounted on an automobile broadcasts a song, with what speed $(\mathrm{km} / \mathrm{h})$ does the automobile have to move toward a stationary listener so that the listener hears
the song with each musical note shifted un by one note in comparison to the song heard by the automobile's driver? On the equally tempered chromatic scale, the ratio of frequencies of neighboring notes is $2^{1 / 12}$ .

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Key Concepts

Equally Tempered Scale

The equally tempered scale is a system of musical tuning in which the octave is divided into equal frequency ratios, typically by the twelfth root of 2. This means that each semitone (the interval between two adjacent notes) has the same ratio, ensuring consistent pitch intervals across the scale. This concept is essential in quantifying precise pitch shifts, such as those resulting from the Doppler Effect when notes are perceived to shift by consistent intervals.

Doppler Effect

The Doppler Effect is a phenomenon that describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. In acoustics, when either the source or the listener (or both) is moving, the perceived pitch of the sound changes. This principle is used to calculate how the observed frequency of sound differs from the emitted frequency when relative motion is involved.

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