July 2004, Issue 168 Test Your EQ:

Answer 4—The sum (or difference) of two sine waves, the second with a different amplitude and phase from the first, is expressed as:

 

 

Using standard trigonometric identities, the sine of a sum of two angles can be expressed as:

 

 

This allows the original expression to be written:

 

 

If the assertion is true, then the sum can also be expressed as:

 

 

which, using the same identity, can be written as:

 

 

This simply requires that you can find an amplitude (B) and angle () that meets the following requirements:

 

 

Dividing one by the other gives:

 

which allows to be solved, and then this value can be substituted into the above pair of equations to get B. Therefore, the assertion that the sum of the original two sine waves is also a sine wave at the same frequency with a different phase and amplitude is proved.

Contributor: David Tweed