What Every Wave Has in Common

Waves are a disturbance that transmits energy, and every wave has a period. One common misconception to avoid: not all waves need a medium — this course's exam explicitly flags "they all need a medium" as the false statement (electromagnetic waves, covered later, travel through a vacuum).

Key Terms and Their Units

Term	What it means	Unit
Wavelength (λ)	Length of one complete wave cycle	m
Period (T)	Time for one complete oscillation	s
Frequency (f)	Oscillations per second (f = 1/T)	Hz (= "per second")
Wave number (k)	2π / λ	m⁻¹

A wavelength is the length of the wave — it is not itself "a complete oscillation" (that's what a period/cycle describes); the period is what produces repeated wavelengths as the wave travels.

Wave Speed — Three Equivalent Formulas

$v = \lambda f = \dfrac{\omega}{k}$

Note: $k/\omega$ is NOT a valid expression for wave speed — it's the reciprocal of the correct $\omega/k$, a common wrong-answer trap.

A Fully Worked Example

A transverse wave of amplitude 20cm travels a 30m rope in 6 seconds, at frequency 4Hz.

• Wave speed = distance/time = 30/6 = 5 m/s.
• Period = 1/f = 1/4 = 0.25 s.
• Wavelength = v/f = 5/4 = 1.25 m (equivalently, v×T).

The Wave Equation (Calculus View)

For a wave $y(x,t) = A\sin(kx - \omega t + \varphi)$:

• Slope $\left(\dfrac{\partial y}{\partial x}\right) = Ak\cos(kx-\omega t+\varphi)$

The ratio of acceleration to curvature gives v² exactly (specifically, $\text{acceleration}/\text{curvature} = \omega^2/k^2 = v^2$) — this is the compact form of the wave equation linking a wave's shape to its motion.

Waves on a String

Wave speed on a string depends on the string's tension and its linear density (mass per unit length) — not on the wave's own frequency or wavelength. The formula: $v = \sqrt{F/\mu}$, where μ is linear density.

Worked example: tension 56.4N produces a speed of 427.23 m/s. Rearranging: μ = F/v² = 56.4/427.23² ≈ 3.09×10⁻⁴ kg/m.

Why This Matters for Your Exams

Memorise the v=λf=ω/k triangle of relationships — nearly every numeric question here gives you two of (v, λ, f, T) and asks for a third, direct substitution.