Wave Equation/Examples
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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:
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.