Hadron spectroscopy is predominantly the study of resonances that decay via the strong interaction into a multitude of stable hadrons, such as the pion. The vast majority of resonances decay via an intermediate hadron with non-zero intrinsic spin. In this thesis, I will present the results of scattering calculations featuring mesons with non-zero intrinsic spin. Before doing so, I will first give a brief introduction to QCD and review the framework necessary to perform lattice QCD calculations in Chapters 1 and 2.

In Chapter 3, I present the first lattice calculation of $\rho \pi$ scattering in isospin-2. Here, $\rho \pi$ features in dynamically-coupled $3S1$ and $3D1$ partial-waves with $JP=1+$. No resonance enhancement is anticipated in the flavour exotic isospin-2 channel and as such it provides an ideal testing ground for this first calculation. I work at heavier than physical quark masses at the $SU(3)F$ point where the up, down and strange quarks are mass degenerate. Finite-volume spectra are calculated and, utilising the relationship between the discrete energy spectrum and the infinite-volume scattering amplitudes, partial-wave amplitudes with $J\le 3$ and the degree of dynamical mixing between the coupled $3S1$ and $3D1$ channels are determined.

In Chapter 4, I investigate $\rho \pi$ in isospin-1 where the $a1$ axial-vector resonance is expected to feature. Here, I present a discussion on $G$-parity and Bose-symmetry at the $SU(3)F$ point. Working at heavier than physical quark masses, the resulting finite volume spectrum suggests that the $a1$ is a bound-state and that the $3S1$- and $3D1$-wave, $\rho \pi$ scattering amplitudes are similar to those in isospin-2.

I present the first calculation of coupled $\pi \omega$ and $\pi \varphi$ scattering in Chapter 5 where resonant enhancement is seen experimentally in the $JP=1+$ channel. Working at a somewhat lighter pion mass than in previous chapters, the finite-volume spectra are determined and the scattering amplitudes are calculated. Analytically continuing the amplitudes into the complex energy plane, a resonance pole is found, interpreted as the analogue of the $b1$ axial-vector, which couples dominantly to $3S1$-wave $\pi \omega$, with a much-suppressed coupling to $3D1$-wave $\pi \omega$, and a negligible coupling to $\pi \varphi$.

In Chapter 6, the exotic $JPC=1-+$ channel is studied. These quantum numbers are not allowed in the quark model but can be obtained, for example, through a gluonic excitation coupled to a quark-antiquark pair. In this exploratory calculation, performed at the $SU(3)F$ point, the finite-volume spectra and coupled-channel scattering amplitudes are presented. A single resonance pole is found, interpreted as the exotic $\pi 1$, and couplings to meson-meson channels, including for example $\pi \eta \{1P1\}$, $\pi \eta \prime \{1P1\}$ and $\rho \pi \{3P1\}$, are calculated for the first time in lattice QCD.

In order to minimally present the contents of a unitary $n$-channel scattering matrix, I introduce, in Chapter 7, an $n$-channel generalisation of the traditional two-channel Stapp parameterisation.