Waves and their properties
A wave carries energy from place to place without carrying matter along with it. In a transverse wave, such as a wave on a string, the medium moves at right angles to the direction of travel. In a longitudinal wave, such as sound in air, the medium moves back and forth along the direction of travel in compressions and rarefactions. Four properties describe a periodic wave: amplitude (the maximum displacement), wavelength λ (the length of one cycle), frequency f (cycles per second, in hertz) and period T = 1 / f. The speed of any wave obeys v = f λ.
Wave speed in a medium
The speed depends on the medium, not on the frequency. For a stretched string, v = √(T / μ), where T is the tension and μ the mass per unit length. Sound in air at room temperature travels at about 343 m/s, faster in water and faster still in solids. When a wave passes into a new medium its frequency stays fixed, so its wavelength changes along with its speed.
Superposition, beats and standing waves
Waves that overlap add together. Two sounds of slightly different frequencies f₁ and f₂ produce beats at a rate |f₁ − f₂|. A wave reflected from a fixed end interferes with itself to make a standing wave. For a string of length L fixed at both ends, the allowed wavelengths are λ = 2L / n, so the fundamental has λ = 2L. The same holds for a pipe open at both ends, with f = n v / (2L). A pipe closed at one end only allows odd multiples: f = n v / (4L) with n = 1, 3, 5.
Sound intensity and the Doppler effect
Loudness is measured by the intensity level in decibels. Every time the intensity is multiplied by 10, the level goes up by 10 dB, so a factor of 100 adds 20 dB. The Doppler effect is the change of observed frequency when source and observer move relative to each other. For a source moving toward a stationary listener, f' = f v / (v − v_s); for one moving away, f' = f v / (v + v_s).
Worked example
Take the speed of sound as 343 m/s.
- A 256 Hz tone has wavelength λ = 343 / 256 = 1.34 m.
- A pipe closed at one end and 0.33 m long has fundamental f₁ = 343 / (4 × 0.33) = 260 Hz.
- The next resonance is the third harmonic at 3 × 260 = 780 Hz, since only odd multiples are allowed.
- A source at 256 Hz moving toward a listener at 30 m/s is heard at 256 × 343 / (343 − 30) = 281 Hz.
Common mistakes
- Thinking a faster wave must have a higher frequency. Frequency is set by the source.
- Using the Doppler formula with the wrong sign for approaching versus receding.
- Treating a closed pipe like an open pipe.
- Adding decibels as if they were linear.