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Is there a built-in function to convert decibels to amplitude in Mathematica? I tried with:

UnitConvert[Quantity[40, "decibels"], "amplitude"]

but it does not work.

Online I found only a widget for Wolfram Alpha:

http://www.wolframalpha.com/widgets/view.jsp?id=21e1ea77bd91aaa0fc4d01a943a654e

There is nothing in the documentation, and bellow you can see what you can find in the internal functions, of which nothing that seems to do the job. So the answer to your question is:

No, there is no built-in function to convert decibels to amplitude in Mathematica.

Speculation on "why"

We can only guess why the developers didn't implement this, a possible narrative may be similar to the case of the now deprecated Units Package. There the documentation said:

"The conversion of temperature units is different from most other unit
conversions because it is not multiplicative. This is simply because
the zeros of various systems are set at different values. For example,
zero degrees Centigrade is the same as 32 degrees Fahrenheit."

So you had to use ConvertTemperature instead of Convert. How is that similar? It's similar because both cases are somehow special. Wikipedia says

The decibel (symbol: dB) is a unit of measurement used to express the
ratio of one value of a physical property to another on a logarithmic
scale.

dB is NOT an amount of a physical quantity, but a the logarithm of a ratio of two quantities in the same units, therefore unitless and non-linear. Also depending if the you are talking about amplitudes or power (amplitude squared) the factors are 20 or 10.

Therefore, non-linear relationship of something that arguably may not be even a unit with a potentially ambiguous definition... better leave it to the users to define their own, rather simple, solutions.

A solution

dB2lin[x_] := N[Power[10, x/20]];
lin2dB[x_] := N[20 Log10[x]];

Internal functions

Some functions with matching names

Names["*`*Decibel*", IgnoreCase -> True]
(* "Signal`Utils`decibelQ", "Signal`Utils`decibelQ$", 
"CalculateUnits`UnitCommonSymbols`DecibelsMuUnit", 
"CalculateUnits`UnitCommonSymbols`CalculateUnits`UnitCommonSymbols`
DecibelsMuUnit", "CalculateUnits`UnitCommonSymbols`DecibelsRUnit", 
"CalculateUnits`UnitCommonSymbols`DecibelsVUnit", 
"CalculateUnits`UnitCommonSymbols`DecibelsZUnit" *)

Names["*`db*"]
(* "Image`ColorOperationsDump`db", 
"Image`ColorOperationsDump`Image`ColorOperationsDump`db", 
"Predictions`Private`db", 
"Predictions`Private`Predictions`Private`db", 
"Statistics`Library`Dump`dbag", "Statistics`Library`Dump`dbag$", 
"PacletManager`LayoutDocsCollection`Private`dbFile", 
"System`FourierTransformDump`dbgPrintFT", 
"Charting`ChartLabelingDump`dbgstyle", 
"Charting`ChartLabelingDump`dbgstyle$", 
"Charting`ChartLabelingDump`dbox", "Charting`dbPrint", 
"Image`ColorOperationsDump`dbPrint", "Image`HumanDump`dbPrint", 
"Image`SpatialOperationsDump`dbPrint", 
"Visualization`VectorFields`VectorFieldsDump`dbPrint", 
"Wavelets`WaveletUtilities`dbPrint", 
"PacletManager`LayoutDocsCollection`Private`dbStrm", 
"PacletManager`LayoutDocsCollection`Private`dbStrm$", 
"Charting`CommonDump`dbstyle", 
"Charting`ChartLabelingDump`dbTimingReap", 
"Charting`ChartLabelingDump`dbVpp", "Charting`ParserDump`dbVpp", 
"System`ListPointPlot3DDump`dbVpp", 
"System`ListPointPlot3DDump`System`ListPointPlot3DDump`dbVpp" *)