Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 3 / 37C / Solution

The acoustic pressure satisfies $p_{tt}=c_0^2\Delta p$. Rigid walls impose vanishing normal velocity, equivalently the Neumann boundary condition on pressure. Separation of variables gives transverse modes $\cos (m\pi x/a)\cos (n\pi y/a)$ with $m,n\ge 0$. With axial factor $e^{ikz-i\omega t}$, their dispersion relation isThe $(0,0)$ mode is nondispersive. Every other mode propagates only above its waveguide cutoff frequency ${\omega }_{mn}=\pi c_0\sqrt{m^2+n^2}/a$. Thus ${\omega }_{\min }=\pi c_0/a$.

Expand the prescribed entrance pressure in this cosine basis, with coefficients $A_{mn}$. Impose outgoing propagation and decay at infinity, with no incoming disturbance. At $\omega ={\omega }_{\min }/2$, only the constant transverse mode propagates, with $k_{00}=\pi /(2a)$. All other modes have $k=i{\chi }_{mn}$, where ${\chi }_{mn}=\pi \sqrt{m^2+n^2-1/4}/a$. The exact modal field is the outgoing constant mode plus ${\sum }_{m+n>0}{A}_{mn}\cos (m\pi x/a)\cos (n\pi y/a)e^{-{\chi }_{mn}z-i\omega t}$.

Since the slowest evanescent decay rate is $\sqrt{3}\pi /(2a)$, the far field isIf that area average is zero, there is no propagating far-field component, and the field decays exponentially instead. For sufficiently regular entrance data the omitted terms are $O(e^{-\sqrt{3}\pi z/(2a)})$.