You are an acoustic engineer. You have recorded a pure note of music that is described mathematically as f ( t ) = 20 sin (880  t ), where t is time in seconds, and the output is the strength of the sound given in microbels. This sound wave is shown below.

If you, as an engineer, need to modify the properties of this sound wave, making the sound louder or softer, or to change the tone, you must be able to manipulate the mathematical model of this wave. The volume of the sound wave is determined by the height of the wave. The height is called the amplitude of the wave. The tone of the sound wave is determined by the speed of the wave. The speed is determined by the period of the wave, which is the time it takes for the wave to go through a full cycle.

 The unmodified sine function f ( x ) = sin x starts at 0, goes up to 1, goes back to 0, goes to -1, and back to 0.  The whole cycle takes place in the domain 0 < x < 2 , and repeats continuously.  The function never goes above 1 or below -1.

 How do we mathematically change the height (amplitude) of a sine wave?  How do we mathematically change the length (period) of a sine wave?

 Think about it: how would you double the height of a sine wave?  What do you know about doubling the height of another graph? Think about the graph of f ( x ) = x; how would you double this graph? From:To:

 To double the height of the graph of f ( x ) = x, you would multiply the input by 2: f ( x ) = 2 x.  To double the height of the graph of f ( x ) = sin x, multiply the input by 2: f ( x ) = 2 sin x  The amplitude is whatever value is multiplied by the sine function. For the function f ( t ) = 20 sin (880  t ), the amplitude is 20. Therefore, this function will go up to 20 and down to -20, instead of going up to 1 and down to -1

 Consider the cycle of a standard sine wave. The function peaks at sin (  /2) = 1  Now consider the function f ( x ) = sin 2 x. When x =  /2, f ( x ) = sin 2(  /2) = sin  = 0.  When x =  /4, f ( x ) = sin 2(  /4) = sin  /2 = 1.  When x = 3  /4, f ( x ) = sin 3  /2 = -1.  Finally, when x = , f ( x ) = sin 2  = 0.  When x is multiplied by 2, the sine wave cycles twice as fast. The whole cycle is complete in half the time.

 The period is the amount of time it takes for a sine wave to complete a whole cycle. Because there are 2  radians in a whole circle, a standard sine wave completes a whole cycle in 2  radians.  A sine wave that is twice as fast [sin (2 x )] has half the period of a standard sine wave, thus the period is 2  /2 =  radians.  The period of any sine wave is 2  / b, where b is the value multiplied by the variable. For the function f ( t ) = 20 sin (880  t ), b is 880 . The period of this function is 2  /880  = 1/440.

We can use the mathematical properties of sine waves to graph waves and compare them. If we calculate the amplitude and period of a sine wave, we can use these properties to help graph a sine wave. Remember: if our function has the format f ( x ) = a sin bx, then the amplitude is a, and the period is 2  / b.

 The amplitude a defines the height of the peaks of the sine wave.  The period 2  / b defines the length of a cycle; for a wave that begins at 0, this value also marks the endpoint of the wave.

Use the graph at the website spx?ID=174 to help you answer the following questions: spx?ID=174  What happens to the graph of a sine wave if the amplitude a is negative?  What happens to the graph of a sine wave if you divide the variable by 2 instead of multiplying it by 2? [such as f ( x ) = sin ( x/2 )]  What happens to the graph of if you add 2 to the function? [such as f ( x ) = sin ( x ) + 2]