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3.2: A Classical Wave Equation

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  • The easiest way to find a differential equation that will provide wavefunctions as solutions is to start with a wavefunction and work backwards. We will consider a sine wave, take its first and second derivatives, and then examine the results. The amplitude of a sine wave can depend upon position, \(x\), in space,

    \[ A (x) = A_0 \sin \left ( \frac {2 \pi x}{\lambda} \right ) \label {3-3}\]

    or upon time, \(t\),

    \[A(t) = A_0\sin(2\pi \nu t) \label {3-4}\]

    or upon both space and time,

    \[ A (x, t) = A_0 \sin \left ( \frac {2 \pi x}{\lambda} - 2\pi \nu t \right ) \label {3-5}\]

    We can simplify the notation by using the definitions of a wave vector, \(k = \frac {2\pi}{\lambda}\), and the angular frequency, \(\omega = 2\pi \nu\) to get

    \[A(x,t) = A_0\sin(kx − \omega t) \label {3-6}\]

    When we take partial derivatives of A(x,t) with respect to both \(x\) and \(t\), we find that the second derivatives are remarkably simple and similar.

    \[ \frac {\partial ^2 A (x, t)}{\partial x^2} = -k^2 A_0 \sin (kx -\omega t ) = -k^2 A (x, t) \label {3-7}\]

    \[ \frac {\partial ^2 A (x, t)}{\partial x^2} = -\omega ^2 A_0 \sin (kx -\omega t ) = -\omega ^2 A (x, t) \label {3-8}\]

    By looking for relationships between the second derivatives, we find that both involve \(A(x,t)\); consequently an equality is revealed.

    \[ k^{-2} \frac {\partial ^2 A (x, t)}{ \partial x^2} = - A (x, t) = - \omega \frac {\partial ^2 A (x, t)}{\partial x^2} \label {3-9}\]

    Recall that \(\nu\) and \(λ\) are related; their product gives the velocity of the wave, \(\nu \lambda = v\). Be careful to distinguish between the similar but different symbols for frequency \(\nu\) and the velocity v. If in ω = 2πν we replace ν with v/λ, then

    \[ \omega = \frac {2 \pi \nu}{\lambda} = \nu k \label {3-10}\]

    and Equation \(\ref{3-11}\) can be rewritten to give what is known as the classical wave equation in one dimension. This equation is very important. It is a differential equation whose solution describes all waves in one dimension that move with a constant velocity (e.g. the vibrations of strings in musical instruments) and it can be generalized to three dimensions. The classical wave equation in one-dimension is

    \[\frac {\partial ^2 A (x, t)}{\partial x^2} = \nu ^{-2} \frac {\partial ^2 A (x, t)}{\partial t^2} \label {3-11}\]

    Example \(\PageIndex{1}\)

    Complete the steps leading from Equation \(\ref{3-5}\) to Equations \(\ref{3-7}\) and \(\ref{3-8}\) and then to Equation \(\ref{3-11}\).

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