Velocity of Periodic Wave is Wavelength times Frequency
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Theorem
Let $\phi$ be a periodic wave expressed as:
- $\forall x, t \in \R: \map \phi {x, t} = \map f {x - c t}$
where $c$ is the velocity of $\phi$.
Then:
- $c = \nu \lambda$
where:
- $\nu$ is the frequency of $\phi$
- $\lambda$ is the wavelength of $\phi$.
Proof
\(\ds \tau\) | \(=\) | \(\ds \dfrac \lambda c\) | Period of Periodic Wave, where $\tau$ is the period of $\phi$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds c\) | \(=\) | \(\ds \dfrac 1 \tau \times \lambda\) | algebra | ||||||||||
\(\ds \) | \(=\) | \(\ds \nu \lambda\) | Frequency of Periodic Wave |
$\blacksquare$
Sources
- 1955: C.A. Coulson: Waves (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Equation of Wave Motion: $\S 3$