Equation of Harmonic Wave/Wavelength and Period

Theorem

Let $\phi$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$.

Then the disturbance of $\phi$ at point $x$ and time $t$ can be expressed using the equation:

$\map \phi {x, t} = a \map \cos {2 \pi \paren {\dfrac x \lambda - \dfrac t \tau} }$

where:

$\lambda$ is the wavelength of $\phi$

$\tau$ is the period of $\phi$.

Proof

\(\ds \map \phi {x, t}\)

\(=\)

\(\ds a \map \cos {\dfrac {2 \pi} \lambda \paren {x - c t} }\)

Equation of Harmonic Wave in terms of Wavelength and Velocity

\(\ds \)

\(=\)

\(\ds a \map \cos {\dfrac {2 \pi} \lambda \paren {x - \dfrac \lambda \tau t} }\)

Period of Harmonic Wave: $\tau = \dfrac \lambda c$

\(\ds \)

\(=\)

\(\ds a \map \cos {2 \pi \paren {\dfrac x \lambda - \dfrac t \tau} }\)

rearranging

$\blacksquare$

Sources

• 1955: C.A. Coulson: Waves (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Equation of Wave Motion: $\S 3$: $(8)$