Wave Equation/Examples/Wave with Constant Velocity

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Examples of Use of the Wave Equation

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.


Proof

The wave equation is expressible as:

$\dfrac 1 {c^2} \dfrac {\partial^2 \phi} {\partial t^2} = \dfrac {\partial^2 \phi} {\partial x^2} + \dfrac {\partial^2 \phi} {\partial y^2} + \dfrac {\partial^2 \phi} {\partial z^2}$


We have by partial differentiation of $(1)$:

\(\ds \map {\dfrac \partial {\partial t} } {\map \phi {x, t} }\) \(=\) \(\ds \map {\dfrac \partial {\partial t} } {\map f {x - c t} }\)
\(\ds \) \(=\) \(\ds -c \map {\dfrac \partial {\map \partial {x - c t} } } {\map f {x - c t} }\) Derivative of Function of Constant Multiple: Corollary
\(\ds \leadsto \ \ \) \(\ds \map {\dfrac {\partial^2} {\partial t^2} } {\map \phi {x, t} }\) \(=\) \(\ds \map {\dfrac {\partial^2} {\partial t^2} } {\map f {x - c t} }\)
\(\ds \) \(=\) \(\ds \paren {-c}^2 \map {\dfrac {\partial^2} {\map {\partial^2} {x - c t} } } {\map f {x - c t} }\)
\(\ds \) \(=\) \(\ds c^2 \map {\dfrac {\partial^2} {\map {\partial^2} {x - c t} } } {\map f {x - c t} }\)

and:

\(\ds \map {\dfrac \partial {\partial x} } {\map \phi {x, t} }\) \(=\) \(\ds \map {\dfrac \partial {\partial x} } {\map f {x - c t} }\)
\(\ds \) \(=\) \(\ds \map {\dfrac \partial {\map \partial {x - c t} } } {\map f {x - c t} }\) Derivative of Function of Constant Multiple: Corollary
\(\ds \leadsto \ \ \) \(\ds \map {\dfrac {\partial^2} {\partial x^2} } {\map \phi {x, t} }\) \(=\) \(\ds \map {\dfrac {\partial^2} {\partial x^2} } {\map f {x - c t} }\)
\(\ds \) \(=\) \(\ds \map {\dfrac {\partial^2} {\map {\partial^2} {x - c t} } } {\map f {x - c t} }\)


As $y$ and $z$ do not appear in $(1)$, the partial derivative of $(1)$ with respect to $y$ and $z$ is identically zero.


Hence we have:

$\dfrac {\partial^2 \phi} {\partial t^2} = c^2 \paren {\dfrac {\partial^2 \phi} {\partial x^2} + \dfrac {\partial^2 \phi} {\partial y^2} + \dfrac {\partial^2 \phi} {\partial z^2} }$

and the result follows.


Sources

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