How do decibels scale with wave amplitude?

Question

I know the decibel scale is logarithmic in nature.
Given that at 1V amplitude the power of the sound generated by this wave is 80dB, what would the dB be for the wave with amplitude 0.4V?
I'm trying to figure out how the dB scales with wave amplitude. In my application, the "voltage" varies between 1 and -1 (audio signal).
I found this voltage-db equation:
$$N_text{dB}=10log_{10}left( frac{V^2_2}{V^2_1}right)\
=20log_{10}left( frac{V_2}{V_1}right)$$
However, I don't think this equation is directly applicable to my situation.
Is there a formula I can use for converting voltage to db (given the db at a particular voltage, like above).

Nathan Huchon · Answer

Regarding your problem it seems that there is a misunderstanding between dB and dBm. When we talk about the power ratios between signals we will express the difference in dB while when we will express the power of the signal we will be interested in the power in dBm which is none other than the power of the signal in question compared to 1mW.
Which leads me to say that you better calculate the power in dBm for your full scale signal and come and rely on this result to calculate your other values.
Full scale power : begin{equation} P_{full} (dBm) = 30 + 10log_{10}(frac{V_{RMS}I_{RMS}}{1W})end{equation}
For the second one :
begin{equation} P_2 (dBm)= P_{full} - 10log_{10} (0.4^2) approx P_{full} - 7.958end{equation}

How do decibels scale with wave amplitude?

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