We will study diffraction, in which waves bend around and reach the shadow of an obstacle, and the Doppler effect, in which the frequency changes because of the motion of a source or observer.
Basic Knowledge
Diffraction: The phenomenon in which waves spread into a shadowed region when they pass around the edge of an obstacle or through an opening. Diffraction is more pronounced when the wavelength is longer and the opening is smaller.
Doppler effect: The phenomenon in which the observed frequency changes when the source or observer moves.
When the speed of sound is V and the source frequency is f₀, the frequency f heard by the observer is:
f = f₀ × (V + vo) / (V − vs)
（vo: positive in the direction in which the observer moves toward the source, vs: positive in the direction in which the source moves toward the observer）
・When the source approaches: f > f₀（it sounds higher）
・When the source recedes: f < f₀（it sounds lower）
Diffraction（The phenomenon in which waves spread at obstacles or openings. More pronounced for larger λ）
Doppler effect（The observed frequency changes because of the relative motion of the source and observer）
Redshift（The phenomenon in which light from a receding galaxy shifts toward lower frequencies and longer wavelengths）
Doppler radar（A device that measures the speed of an object from the Doppler shift of reflected waves）
Mach number（The speed of an object / the speed of sound. A shock wave occurs when it exceeds Mach 1）
Further Study (Background・Significance)
Derivation of the Doppler-effect formula: When the source approaches the observer, the waves are compressed in front of the source, making the wavelength shorter (and f higher). This corresponds to the denominator V − vs becoming smaller.
In astronomy, cosmic expansion (Hubble’s law) was discovered from the redshifts of spectral lines from galaxies. The Doppler effect shows that all galaxies in the universe are moving away from one another.
The siren of an ambulance (sounding higher before it passes and lower afterward) is a typical example. In diffraction, the fact that AM radio (with its longer wavelength) is easier to receive behind mountains than FM radio is a familiar example.
- Diffraction is more pronounced for longer wavelengths and smaller openings
- Doppler formula: f = f₀(V + vo)/(V − vs)
- The positive directions of vo and vs are “the directions in which the source and observer approach each other”
- When they approach, f increases (higher); when they recede, f decreases (lower)
- Redshift = evidence of cosmic expansion (application in astronomy)
Points to Note (Easily Confused)
① Sign convention in the Doppler formula: vo is the observer’s velocity, positive in the direction of approach, and vs is the source’s velocity, positive in the direction of approach (subtracted in the denominator). An incorrect sign reverses the relative size of f. ② Diffraction is not that “waves bend,” but that “waves spread.” Light also diffracts, but because its λ is short, this is difficult to see in daily life. ③ The Doppler effect uses the speed of sound V. When vs ≥ V (supersonic speed), the formula cannot be used (shock wave).
Practice
- Find the sound frequency emitted by a source approaching a stationary observer at vs = 20 m/s (f₀ = 500 Hz) (speed of sound V = 340 m/s).
- The source is stationary and the observer is moving away from the source (vo = 40 m/s). Express the change in frequency using the formula.
- Explain why diffraction is “more pronounced for longer wavelengths,” using an analogy with water waves.