Which time series prediction techniques are useful given harmonic properties?
I would suggest to make the correlation plot and observe the dependencies. Yes, ARIMA, is so far the best forecasting model for time series and it should work in this data set too. 
For the implementation, in R, there is auto ARIMA, which automatically takes the best suited valued for the parameters( 
    p: The number of lag observations included in the model, also called the lag order.
    d: The number of times that the raw observations are differenced, also called the degree of differencing.
    q: The size of the moving average window, also called the order of moving average.)

But as far as I am concerned, in Python, you have to manually tune the parameters by yourself by analyzing the correlation plot. This is a very well explained beginner level tutorial for the implementation in Python.  
Let me know in the comments if it works (or not).