Oscillations & waves (advanced)
The idea
At this level a wave is a function of two variables, y(x, t), and the sinusoidal traveling wave y = A sin(kx − ωt) is the workhorse. The wave number k = 2π/λ counts radians per meter, the angular frequency ω = 2πf counts radians per second, and the pattern moves at v = ω/k = fλ. On a stretched string the medium sets that speed: v = √(T/μ) for tension T and linear mass density μ. Everything you learned about amplitude, wavelength, and frequency still holds — now packaged in one formula you can differentiate.
Keep two velocities firmly separate. The wave speed v is how fast the pattern translates along x and is fixed by the medium. The transverse particle velocity is the partial derivative of y with respect to t at fixed x — each piece of string bobs up and down with maximum speed ωA, which depends on amplitude while the wave speed does not. Conflating these two is the most common error in wave problems.
This framework also covers superposition phenomena: two identical waves traveling opposite ways add to a standing wave with fixed nodes, and damping or driving terms generalize simple harmonic motion to the resonant systems that dominate engineering practice.
Worked example
A transverse wave on a string is described by y(x, t) = 0.020 sin(40x − 800t), in SI units. The string has linear mass density 5.0 g/m. Find the wavelength, frequency, wave speed, maximum transverse particle speed, and the string tension.
- Read off the structure: comparing with A sin(kx − ωt) gives amplitude A = 0.020 m, wave number k = 40 rad/m, and angular frequency ω = 800 rad/s; the minus sign means the wave travels in the +x direction.
- Convert to wavelength and frequency: λ = 2π/k = 2π/40 ≈ 0.157 m and f = ω/2π = 800/2π ≈ 127 Hz.
- Compute the wave speed: v = ω/k = 800/40 = 20 m/s (equivalently fλ ≈ 127 × 0.157 ≈ 20 m/s — consistent).
- Differentiate y with respect to time at fixed x to get the particle velocity; its amplitude is ωA = 800 × 0.020 = 16 m/s, slightly less than the 20 m/s wave speed but a completely different motion — vertical, not horizontal.
- Use the medium relation to find tension: v = √(T/μ) means T = μv² = 0.0050 × 20² = 2.0 N.
Answer. λ ≈ 0.157 m, f ≈ 127 Hz, wave speed 20 m/s, maximum particle speed 16 m/s, and the string tension is 2.0 N.
Check your understanding
- Why does the maximum particle speed depend on amplitude while the wave speed does not?
- What physically distinguishes a standing wave from a traveling wave, and how does superposition produce the nodes?
- How would doubling the string tension change λ, f, and v for a wave driven at the same frequency?
- Why does the argument kx − ωt describe motion in the +x direction — what stays constant for a point riding a crest?
Build the foundations first
Oscillations & waves (advanced) builds on these concepts. If any feel shaky, start there.