If a wave's period doubles, how does the wave's frequency change?

Frequency is 1 over the period, or the reciprocol of the period. If the period is p, the frequency is 1/p. If the period doubles, the period is 2p, and the frequency is 1/2p. As you can see, multiplying the period by 2 multiplies the frequency by 1/2 since (1/p) * (1/2) = 1/2p

So, the wave's frequency is halved.