If several sine waves were combined but each had a separate frequency, for example 10 Hz, 20 Hz and and 30 Hz, what would the required Nyquist rate be for analysis? From my research, it suggests that Nyquist rate should be double the maximum frequency and for this case it should be 60 Hz. Is this correct or does the fact that it is a combined signal change this concept?
According to the lowpass sampling theorem, the required minimum sampling-rate (a.k.a Nyquist rate) is twice the highest frequency present in the frequeqncy spectrum of the bandlimited signal.
Note however that, the lowpass sampling theorem assumes there is no impulse (no pure sinusoidal component) at the highest frequency (a.k.a Nyquist frequency) of the signal being sampled. If there's an impulse, then you should choose a rate arbitrarily larger than the Nyquist rate.
Therefore, for your combined (linearly superposed) signal, the highest frequqency is 30 Hz; and thus the Nyquist rate is 60 Hz, however, since there's a pure sinusoidal at 30 Hz, then your sampling rate should be arbitrarily larger than 60 Hz.
In practice a 10% tolerance is already applied for various reasons. Hence you may use about 66 Hz for sampling your combined signal.