Doppler Effect and Sound Intensity for the MCAT: Formula and Decibels

The Doppler effect is the change in perceived frequency when a sound source and a detector move relative to each other. When they move toward each other, the perceived frequency is higher; when they move apart, it is lower. The source's actual frequency does not change.

The formula

f' = f x (v +/- v_detector) / (v -/+ v_source)

Here v is the speed of sound in the medium, f is the emitted frequency, and f' is the perceived frequency. Choose signs by asking whether the motion raises or lowers the frequency.

• Detector moving toward the source: plus in the numerator. Away: minus.
• Source moving toward the detector: minus in the denominator. Away: plus.

If the final result must be higher, the signs must be set to make the fraction larger than 1.

Quick conceptual rules

• Toward each other: f' > f.
• Away from each other: f' < f.
• Only the component of motion along the line between them matters.
• Wavelength shortens in front of a moving source and lengthens behind it.

Sound basics

• Sound is a longitudinal pressure wave and needs a medium.
• Speed is greatest in solids, then liquids, then gases. In air at room temperature it is about 340 m/s.
• Pitch corresponds to frequency, and loudness corresponds to intensity.

Intensity and decibels

Intensity is power per unit area (W/m^2). For a point source, it falls with the square of distance: doubling the distance cuts intensity to one quarter.

The decibel level is beta = 10 log (I / I0), where I0 = 10^-12 W/m^2 is the threshold of hearing. Every factor of 10 in intensity adds 10 dB, and doubling intensity adds about 3 dB.

Common traps

• Thinking the source frequency changes. Only the perceived frequency changes.
• Using the wrong signs. Check against the intuition that approaching raises pitch.
• Treating decibels as linear. The scale is logarithmic.

One-minute recap

• Approaching raises the perceived frequency; receding lowers it.
• Doppler: f' = f (v +/- v_d) / (v -/+ v_s), signs chosen by direction.
• Beta = 10 log (I / I0); 10 times the intensity is +10 dB.