Key ideas
A wave carries energy without carrying matter. Its speed, frequency and wavelength are tied together, and waves in strings and pipes form standing waves at particular frequencies.
- Wave speed is v = fλ. The speed is set by the medium, so a change of frequency changes the wavelength. Sound in air at 20 °C travels at about 343 m/s.
- A string fixed at both ends or a pipe open at both ends has harmonics f_n = nv ÷ 2L. A pipe closed at one end has only odd harmonics, f_n = nv ÷ 4L.
- Two close frequencies produce beats at the difference frequency |f₁ − f₂|.
- Sound level in decibels is logarithmic: every +10 dB is ten times the intensity, and intensity falls with the square of the distance from a point source.
Key equations
v = fλf_n = nv ÷ 2L (string, open pipe)f_n = nv ÷ 4L, n odd (closed pipe)f_beat = |f₁ − f₂|β = 10 log₁₀(I ÷ I₀)f′ = f(v ± v_o) ÷ (v ∓ v_s)d sin θ = mλ
Worked example
A pipe closed at one end is 0.50 m long (v = 343 m/s). Its fundamental is f₁ = v ÷ 4L = 343 ÷ 2.0 = 172 Hz. The next resonance is the third harmonic, 3 × 172 = 515 Hz, because even harmonics are missing in a closed pipe.
Common mistakes
- Using the open-pipe formula for a pipe that is closed at one end.
- Treating decibels as a linear scale. 60 dB is not twice as intense as 30 dB; it is 1000 times as intense.
- Getting the signs wrong in the Doppler formula. A source moving toward the listener raises the frequency.
Practice
Test yourself with practice questions on this topic: Waves and sound