We present numerical results from three-dimensional evolutions of scalar perturbations of Kerr black holes .
Our simulations make use of a high-order accurate multi-block code which naturally allows for fixed adaptivity and smooth inner ( excision ) and outer boundaries .
We focus on the quasinormal ringing phase , presenting a systematic method for extraction of the quasinormal mode frequencies and amplitudes and comparing our results against perturbation theory .

The detection of a single mode in a ringdown waveform allows for a measurement of the mass and spin of a black hole ; a multimode detection would allow a test of the Kerr nature of the source .
Since the possibility of a multimode detection depends on the relative mode amplitude , we study this topic in some detail .
The amplitude of each mode depends exponentially on the starting time of the quasinormal regime , which is not defined unambiguously .
We show that this time-shift problem can be circumvented by looking at appropriately chosen relative mode amplitudes .
From our simulations we extract the quasinormal frequencies and the relative and absolute amplitudes of corotating and counterrotating modes ( including overtones in the corotating case ) .
We study the dependence of these amplitudes on the shape of the initial perturbation , the angular dependence of the mode and the black hole spin , comparing against results from perturbation theory in the so-called asymptotic approximation .
We also compare the quasinormal frequencies from our numerical simulations with predictions from perturbation theory , finding excellent agreement .
For rapidly rotating black holes ( of spin j = 0.98 ) we can extract the quasinormal frequencies of not only the fundamental mode , but also of the first two overtones .
Finally we study under what conditions the relative amplitude between given pairs of modes gets maximally excited and present a quantitative analysis of rotational mode–mode coupling .
The main conclusions and techniques of our analysis are quite general and , as such , should be of interest in the study of ringdown gravitational waves produced by astrophysical gravitational wave sources .