The speed of tsunami waves is typically calculated using the shallow-water approximation over a rigid-body Earth. Recent comparisons of tsunami arrival times from the 11 March 2011 tsunami suggest, however, that the standard formulation has errors around the 1% level, and it has been suggested that the elasticity of the Earth can explain the discrepancy. While previous work has indeed shown that such elastic deformation can modify tsunami speeds, the effect has been neglected partly due to the difficulty in understanding how large this elastic effect is. Here, we remedy this by providing a new derivation and expression for how to incorporate the first-order effect that solid Earth elasticity and ocean water compressibility have on tsunami speeds. This result is shown to agree approximately with previous theory and helps to explain observed timing discrepancies from the 11 March 2011 tsunami. The dispersive elastic correction and the non-dispersive compressibility correction together may account for the majority of the observed discrepancy.