Velocity of Harmonic Wave is Wavelength times Frequency

Theorem

Let $\phi$ be a harmonic wave expressed as:

$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$

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 1

By definition, a harmonic wave is an instance of a periodic wave.

Hence Velocity of Periodic Wave is Wavelength times Frequency can be used directly.

Proof 2

\(\ds \tau\)

\(=\)

\(\ds \dfrac \lambda c\)

Period of Harmonic 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 Harmonic Wave

$\blacksquare$

Sources

• 1955: C.A. Coulson: Waves (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Equation of Wave Motion: $\S 3$: $(7)$