Equation of Harmonic Wave

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:

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

$\phi$ can also be expressed in the following forms:

In Terms of Wavelength and Velocity

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

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

In Terms of Wavelength and Period

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

where:

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

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

In Terms of Wavelength and Frequency

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

where:

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

$\nu$ is the frequency of $\phi$.

In Terms of Wave Number and Frequency

$\map \phi {x, t} = a \map \cos {2 \pi \paren {k x - \nu t} }$

where:

$k$ is the wave number of $\phi$

$\nu$ is the frequency of $\phi$.

Proof

From Wave Profile of Harmonic Wave:

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

From Equation of Wave with Constant Velocity:

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

Hence the result.

$\blacksquare$

Sources

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