Question

what factor controls wave speed

Answer 1

Wave speed is set by the properties of the medium the wave travels in (and sometimes by the wave’s wavelength/frequency). In general the two physical ingredients are:

- how stiff or restoring the medium is (elasticity, tension, gravity, surface tension, electromagnetic permittivity/permeability)
- how much inertia the medium has (mass density or equivalent)

Common formulas/examples
- Transverse wave on a stretched string: v = sqrt(T/μ) where T is tension and μ is linear mass density.
- Sound in a fluid: v = sqrt(B/ρ) where B (bulk modulus) measures stiffness and ρ is density. For an ideal gas, v = sqrt(γRT/M).
- Shallow-water surface waves: v = sqrt(g h) where g is gravity and h is depth.
- Deep-water surface waves (gravity waves): v depends on wavelength λ: v ≈ sqrt(gλ/2π) (so these are dispersive).
- Electromagnetic waves in a medium: v = 1/√(εμ) = c/n where ε and μ are permittivity and permeability and n is refractive index (in vacuum v = c).

Note: dispersive media cause wave speed to depend on frequency (phase vs group velocity). If you tell me which kind of wave you mean (sound, string, water, light…), I can give the exact relation and a brief derivation.