  We investigate an acoustical analog of circuit quantum electrodynamics that
facilitates compact high-Q (${>}20,000$) microwave-frequency cavities with
dense spectra. We fabricate and characterize a device that comprises a flux
tunable transmon coupled to a $300\,\mathrm{\mu m}$ long surface acoustic wave
resonator. For some modes, the qubit-cavity coupling reaches
$6.5\,\mathrm{MHz}$, exceeding the cavity loss rate ($200\,\mathrm{kHz}$),
qubit linewidth ($1.1\,\mathrm{MHz}$), and the cavity free spectral range
($4.8\,\mathrm{MHz}$), placing the device in both the strong coupling and
strong multimode regimes. With the qubit detuned from the cavity, we show that
the dispersive shift behaves according to predictions from a generalized
Jaynes-Cummings Hamiltonian. Finally, we observe that the qubit linewidth
strongly depends on its frequency, as expected for spontaneous emission of
phonons, and we identify operating frequencies where this emission rate is
suppressed.
