Wavelength of Harmonic Wave

Theorem

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

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

The wavelength $\lambda$ of $\phi$ can be expressed as:

$\lambda = \dfrac {2 \pi} \omega$

Proof

By definition, $\lambda$ is the period of the wave profile of $\phi$.

From Wave Profile of Harmonic Wave, the wave profile of $\phi$ is given by:

$\paren {\map \phi x}_{t \mathop = 0} = a \cos \omega x$

From Period of Real Cosine Function:

$\paren {\map \phi x}_{t \mathop = 0} = a \cos {\omega x + 2 \pi}$

So the period of $\paren {\map \phi x}_{t \mathop = 0}$ is $\dfrac {2 \pi} \omega$.

Hence the result by definition of wavelength.

$\blacksquare$

Sources

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