Wave Equation/Examples

Examples of Use of the Wave Equation

Wave with Constant Velocity

Let $\phi$ be a wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$ and without change of shape.

From Equation of Wave with Constant Velocity, the disturbance of $\phi$ at point $x$ and time $t$ can be expressed using the equation:

$(1): \quad \map \phi {x, t} = \map f {x - c t}$

where:

$x$ denotes the distance from the origin along the $x$-axis

$t$ denotes the time.

This equation satisfies the wave equation.

Harmonic Wave

Let $\phi$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$ and without change of shape.

From Equation of Harmonic Wave, the disturbance of $\phi$ at point $x$ and time $t$ can be expressed using the equation:

$(1): \quad \map \phi {x, t} = a \map \cos {2 \pi \paren {\dfrac x \lambda - \dfrac t \tau} }$

where:

$x$ denotes the distance from the origin along the $x$-axis

$t$ denotes the time

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

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

This equation satisfies the wave equation.