= Solution
The displacement is along $x$, transverse to propagation in the membrane's $yz$ plane. Write $A=|A|e^{i\phi}$; then $d=|A|\cos(\omega t-k_yy-k_zz+\phi)$. Here $|A|$ is the displacement <amplitude>, $\phi$ the initial phase, $\omega$ the <angular frequency> and $\mathbf k$ the <wavevector> normal to constant-phase lines. Its magnitude is $2\pi/\lambda$, and its direction specifies phase propagation for the chosen sign convention.
For <equal-phase two-wave membrane modes>, two equal <complex amplitudes> with opposite $k_y$ sum to
$$
d=2\operatorname{Re}\{A\cos(k_yy)e^{i(\omega t-k_zz)}\}.
$$
The edge condition gives $\cos(k_ya/2)=0$, hence $k_y=(2j+1)\pi/a$, $j=0,1,\ldots$, for this even-in-$y$ family. Opposite relative phase gives the <sine> family; together they yield all Dirichlet transverse modes $k_y=n\pi/a$, $n\ge1$. For the equal-phase family,
$$
\boxed{k_z=\sqrt{(\omega/c)^2-[(2j+1)\pi/a]^2}.}
$$
The lowest mode has $k_y=\pi/a$. Therefore
$$
\boxed{v_z=\frac\omega{k_z}=\frac c{\sqrt{1-(\lambda/2a)^2}},\qquad0<\lambda<2a.}
$$
It approaches $c$ for $\lambda\to0$ and diverges as $\lambda\to2a^-$; there is no real travelling-mode $k_z$ beyond the cutoff. This is an axial <phase velocity>, not the <group velocity>: $v_g=c\sqrt{1-(\lambda/2a)^2}<c$. Here $c$ denotes the membrane wave <speed>, not the <speed of light>.
\Image[/past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1-ia2010-membrane.png]
{title=Membrane cutoff and evanescent response}
{description=Original lowest-mode phase-velocity curve, with group velocity distinguished, and a schematic below-cutoff displacement envelope of exponential form. All lengths and amplitudes are dimensionless choices; no source apparatus drawing is copied.}
For the <evanescent membrane response below cutoff>, below $f_c=c/(2a)$, put $k_z=-i\kappa$ for decay along positive $z$, with $\kappa=\sqrt{(\pi/a)^2-(\omega/c)^2}$. A localized central drive creates an evanescent, sinusoidally oscillating deformation decaying away in both axial directions. Far enough from the drive the leading profile is proportional to $\cos(\pi y/a)e^{-\kappa|z-z_0|}\cos(\omega t+\phi_0)$. Higher evanescent modes contribute near the point source. The pattern has no steadily advancing axial phase front or far-field propagating wave; decreasing <frequency> increases its spatial localization.
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