= Solution
Use the flat two-layer model, with thickness $h=30\,\mathrm{km}$, crustal shear <speed> $\beta_1=3.5\,\mathrm{km\,s^{-1}}$ and <mantle> <speed> $\beta_2=4.5\,\mathrm{km\,s^{-1}}$. For <SH-wave travel times in a crustal waveguide>, unfolding each reflected path gives all <reflection> orders at once:
$$
\boxed{t_D=\frac x{3.5},\qquad
t_m=\frac{\sqrt{x^2+(60m)^2}}{3.5},\quad m=1,2,\ldots,}
$$
with $x$ in kilometres and times in seconds. The direct wave begins at the origin; reflected orders have intercepts $60m/3.5=17.14m\,\mathrm s$ and approach the direct-wave slope at large distance.
The critical angle, measured from the vertical, is $i_c=\arcsin(3.5/4.5)\simeq51.1^\circ$. Associated critical <refraction> and its multiples give
$$
\boxed{t_{H,m}=\frac x{4.5}+2mh\sqrt{\beta_1^{-2}-\beta_2^{-2}}
=\frac x{4.5}+10.77m,\qquad
x\geq x_{c,m}=2mh\tan i_c\simeq74.25m\,\mathrm{km}.}
$$
Each head-wave line is tangent to its <reflection> curve at the critical distance. In particular the primary head wave exists before it becomes the first arrival; it overtakes the direct wave only around $170\,\mathrm{km}$.
The slope of a reflected travel-time curve is $p=dt_m/dx=\sin i/\beta_1$. For $x<x_{c,m}$, $p<1/\beta_2$, so the <mantle> supports outgoing shear radiation and successive reverberations leak <energy>. For $x>x_{c,m}$, $1/\beta_2<p<1/\beta_1$: the <mantle> field is evanescent and ideal interface <reflection> is total. These <leaking and trapped SH-wave reverberations> explain the annotations in the original travel-time drawing below. A reflected ray is not itself a separate Love mode: constructive interference of the trapped contributions, including <reflection> phase, selects the <normal modes>.
For a horizontally propagating mode with $\beta_1<c<\beta_2$, define vertical <wavenumbers>
$$
q=\omega\sqrt{\beta_1^{-2}-c^{-2}},\qquad
s=\omega\sqrt{c^{-2}-\beta_2^{-2}}.
$$
The free upper surface and welded interface impose the Love-wave <dispersion> condition
$$
\tan(qh)=\frac{\mu_2s}{\mu_1q},\qquad
qh=n\pi+\arctan\frac{\mu_2s}{\mu_1q},\quad n=0,1,2,\ldots,
$$
where $\mu_j=\rho_j\beta_j^2$ are shear moduli. Each phase-velocity branch falls towards $3.5\,\mathrm{km\,s^{-1}}$ with increasing <frequency>. Higher branches join $4.5\,\mathrm{km\,s^{-1}}$ at their low-frequency cutoffs. At cutoff $s\to0$, so
$$
\boxed{f_{c,n}=\frac{n}{2h\sqrt{\beta_1^{-2}-\beta_2^{-2}}}
\simeq0.0928n\,\mathrm{Hz}.}
$$
Thus the first three <overtones> have maximum <periods> about $10.77$, $5.39$ and $3.59\,\mathrm s$. The <fundamental mode> has $n=0$ and \b[no positive-frequency cutoff in this flat layer-over-half-space model]: it approaches $c=\beta_2$ as $f\to0$, with ever-deeper penetration. The phase-velocity sketch uses an explicitly illustrative equal-density choice, because the question gives <speeds> but no <density> or rigidity ratio. That missing ratio changes detailed branch curvature, not the cutoff values or the requested branch topology.
For the actual finite spherical <Earth>, the very-long-period continuation must instead be described by discrete toroidal <normal modes>, not a flat infinite <waveguide> at arbitrary <wavelength>. The flat-model answer therefore does not assert a continuum of arbitrarily low-frequency global nonrigid oscillations. A global lowest oscillation <frequency> cannot be computed from the supplied crust-and-half-space data alone.<Essentials of Geophysics: Seismology by R. D. van der Hilst>[§4.18, Love-wave trapping; §4.23, finite spherical Earth and discrete normal modes, especially printed pp. 184–187]{ref}
\Image[/past-exam-of-the-physics-of-the-earth-as-a-planet-course-of-the-university-of-cambridge/2013/paper-1-pep-sh-waveguide.png]
{title=Accurate direct, reflected and critical-head travel times, with illustrative fundamental and first three overtone Love-wave branches}
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