Mechanical Waves and Sound
Waves are this course's only unit built around a single governing relationship (v = fλ) rather than a force law, so the emphasis shifts to recognizing which quantities a change in medium, frequency, or boundary condition actually affects. Superposition and standing waves on strings and in air columns are where that relationship stops being abstract -- they're the direct explanation for how real instruments produce the pitches they do, with the Doppler effect as the unit's one genuinely separate idea.
What you'll learn
- Distinguish transverse from longitudinal waves and identify examples of each.
- Relate wave speed, wavelength, and frequency via the wave equation.
- Apply superposition to analyze constructive and destructive interference.
- Analyze standing waves on strings and in air columns, identifying nodes and antinodes.
- Calculate resonant frequencies for strings and open/closed pipes.
- Apply the Doppler effect qualitatively and quantitatively.
- Interpret wave diagrams and graphs (displacement vs. position, displacement vs. time).
1. Wave Properties and the Wave Equation
A wave's amplitude is its maximum displacement from equilibrium; its wavelength \(\lambda\) is the distance between successive identical points (crest to crest); its frequency \(f\) is how many complete cycles pass a point per second; and its period \(T = 1/f\). These combine in the wave equation \(v = f\lambda\).
Wave speed \(v\) is set entirely by the properties of the medium (its tension, density, stiffness, and so on) — not by the frequency of the source. When a source vibrates faster, the wavelength shrinks to compensate, keeping the wave speed in that medium exactly the same.
2. Transverse vs. Longitudinal Waves
In a transverse wave, particles of the medium move perpendicular to the direction the wave travels — a wave on a string, or light, are transverse. In a longitudinal wave, particles move parallel to the direction of travel, oscillating back and forth as compressions and rarefactions — sound is the standard example. The wave speed formula on a stretched string, \(v = \sqrt{F_T/\mu}\) (tension over linear mass density), is a common companion calculation to this topic.
3. Superposition and Interference
When two or more waves overlap, their displacements simply add together at every point — the superposition principle. If two waves arrive in phase (path difference equal to a whole number of wavelengths), they add constructively, producing a larger amplitude. If they arrive exactly out of phase (path difference equal to a half-integer number of wavelengths), they cancel destructively. Overall energy is always conserved; destructive interference at one location is always accompanied by extra constructive interference somewhere else in the pattern.
4. Standing Waves on Strings
A standing wave forms when a wave and its reflection interfere on a string of fixed length, creating stationary nodes (no motion) and antinodes (maximum motion). For a string fixed at both ends, only certain wavelengths "fit": \(\lambda_n = \dfrac{2L}{n}\) for \(n = 1, 2, 3, \ldots\). The fundamental (first harmonic) frequency is \(f_1 = v/(2L)\), and higher harmonics are integer multiples: \(f_n = nf_1\).
5. Standing Waves in Air Columns
An open pipe (open at both ends) behaves like a string fixed at both ends for allowed wavelengths — \(\lambda_n = 2L/n\) — and supports every harmonic. A closed pipe (closed at one end, open at the other) instead only supports odd harmonics: \(\lambda_n = \dfrac{4L}{n}\) for \(n = 1, 3, 5, \ldots\).
6. The Doppler Effect
When a source and observer are in relative motion, the observed frequency shifts: \(f' = f\dfrac{v \pm v_o}{v \mp v_s}\), where \(v\) is the wave speed in the medium, \(v_o\) the observer's speed, and \(v_s\) the source's speed. Rather than memorizing which sign goes where, use the physical rule directly: motion that brings source and observer *closer together* raises the observed frequency (choose signs to make the fraction larger); motion that moves them *apart* lowers it (choose signs to make the fraction smaller).
Key equations
- v = fλ — Any basic wave relationship
- String, both ends fixed: λn = 2L/n — String resonance
- Closed pipe: λn = 4L/n (odd n); Open pipe: λn = 2L/n — Air-column resonance
- f' = f(v ± vo)/(v ∓ vs) — Doppler effect
- v (string) = √(FT/μ) — Wave speed on a string