Manipulating sine waves...

I'm trying to determine how sine waves are generated.

Here is an image link for the examples.

http://firebelow2009.webs.com/phaseA...ng_illusrt.png

The first graph has 3 sine waves.

The blue sine wave is given by the equation: 5 * sine(5x)

The red sine wave is given by the equation: sine(50x)

The black wave is the sum of the two equations: (5 * sine(5x)) + sine(50x)

The second graph has 3 waves.

The dotted blue sine wave is given by the equation: 5 * sine(5x)

The red sine wave is given by the equation: sine(50x)

My question is, on the second graph, how do I achieve the black line? A lab mate of mine said it is the product of blue and red, however this is not correct as the resulting black wave has twice as many bursts as it should. Perhaps I'm not multiplying it correctly. Help on this problem would be appreciated.

Thanks,

Shawn