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I recently came across a question whereby it is required to show that the solutions to the following equations are equivalent for some condition on the wavelength, $\lambda$, of the wave field $u(x,t)$: \begin{align} \frac{\partial^2 u}{\partial t^2} &= c_s^2\frac{\partial^2 u}{\partial x^2} - g\gamma\frac{\partial u}{\partial x}\tag{1}\\ \frac{\partial^2 u}{\partial t^2}& = c_s^2\frac{\partial^2 u}{\partial x^2}\tag{2} \end{align} where $\gamma$ is the ratio of constant pressure specific heat to constant volume specific heat, and $$c_s = \sqrt{\frac{\gamma P(x)}{\rho(x)}}$$ is the speed of sound in the medium.

By observation, the solutions of (1) and (2) and equivalent if the term $g\gamma\frac{\partial u}{\partial x}$ in (1) is small enough to be negligible compared to the other term $c_s^2\frac{\partial^2 u}{\partial x^2}$. The problem solution obtains the necessary condition for this by approximating the following: $$g\gamma\frac{\partial u}{\partial x} \sim \frac{g\gamma}{\lambda}$$ and $$c_s^2\frac{\partial^2 u}{\partial x^2}\sim \frac{c_s^2}{\lambda^2}$$

so that $$\frac{g\gamma}{\lambda}\ll \frac{c_s^2}{\lambda^2}$$ gives the necessary condition for $\lambda$. However, I am confused as to how these approximations were made as I'm not able to see the connection between the spacial derivatives of $u(x,t)$ and its wavelength.

I would greatly appreciate it if anyone could help explain this to me.

Qmechanic
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ethanolex
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1 Answers1

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You can see this if you write things out in terms of complex wave notation. If we assume a plane wave solution of the form $u = A e^{i(kx - \omega t)}$, then $$ \frac{\partial u}{\partial x} = i k u = \frac{2 \pi i}{\lambda} u \qquad \frac{\partial^2 u}{\partial x^2} = - k^2 u = - \frac{4 \pi^2}{\lambda^2} u $$ So to within an order of magnitude or so, the amplitude of the first derivative is approximately $u/\lambda$, and the amplitude of the second derivative is approximately $u / \lambda^2$. This then determines the relative sizes of the terms in Eq. (1), and in particular gives you a limit in which the second term is negligible.

Michael Seifert
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