Worksheet on Sine Waves & Sound

Worksheet on Sine Waves & Sound

Objective: To explore the concepts of amplitude, frequency, and phase shift of sine waves by making connections with their effects on sound.

Sound waves are formed through the process of compressing and expanding air. The variations in pressure that you feel in your eardrum are what the body translates into sound. (This is why you would not be able to hear someone speak in a place devoid of air – in space, for example.) A diagram of this is shown below.[1]

Speaker membrane expands, creating a region
where the air molecules are packed closely
together, a "condensation". The air pressure
in a condensation is higher than normal. /
As the membrane moves back, a region
is left behind where few molecules are
located, a "rarefaction". Meanwhile, the
condensation moves forward.

The pictures we see of “sound waves” are essentially graphs of pressure vs. time:

Now you’ll explore the ways sine waves can be transformed and the effects that has on the resulting sound. (As you might guess, all of this could be done with cosine waves, too. But we’ve already seen that sine waves and cosine waves are just transformations of each other.)

1. The intensity or volume of a sound is determined by the amplitude of the sine wave. The amplitude is given by the coefficient A in . Practice with this by sketching graphs of the following functions.

a)

Effect:

b)

Effect:

c)

Effect:

1. The pitch of a sound is determined by the period of the sine wave. (This is related to the frequency of the sound – how many times it goes through a complete cycle in one unit of time.) The period is controlled by the coefficient B in . Since the period of a basic sine wave is 2, that means the sine function will repeat every time Bx reaches another multiple of 2. In other words, sine repeats when Bx = 2, so . This fraction is the period. Use it to graph

a)

Effect:

b)

Effect:

c)

Effect:

As you know, most of the sounds we hear are not just one pure tone. This means that an individual sound is made up of many different sound waves combined together – called “superposition.” Because each sound has a different frequency, sometimes the effects support each other, and sometimes they cancel. A “beat” is the difference in the sounds’ frequencies.

1. Since you need two sine waves to talk about a beat, in math we focus on a related feature of a single wave called the phase shift. This essentially describes the “delay” from the standard wave (think of this as the delay between groups of people singing a “round” like Row, row, row your boat). This is related to the coefficient C in . As we did with the frequency, compute the phase shift by recognizing that the standard sine wave “starts” (i.e. hits zero) when the formula inside the sine equals zero. So we solve and get that the graph is shifted by units. (When B is positive, this slides the graph in the direction of the sign – though this is the OPPOSITE of the sign of C.)

a)

Effect:

b)

Effect:

c)

Effect:

A single sine wave represents one tone, or note. As was mentioned above, complex sound comes from the combination of numerous tones. The specific tones included are what give a sound its “texture.” For example, the two waves shown below represent the same note on different instruments; the overall shape of the sound wave is clearly the same in both cases, but slight variations are how we know one instrument from the other.

1. Now you’ll experiment with combining (superposing) different waves to create new ones.

a)Start with the “tone” given by the standard sine wave, .

b)Create the sine wave with twice the frequency in order to get the same tone one octave higher (an “overtone”). The formula for this new wave is:

c)Use your calculator to graph both curves on the same grid. Do the peaks of the two graphs match?

d)Create a new graph by adding your two sine curves together. Describe how the overall shape of the new graph could be predicted from your previous graphs. This is most easily done by graphing the two sine curves separately on the first set of axes below, and then the “sum” graph on the second set of axes.

e)Create a new version of your answer to (b) that is more “in phase” with the original. This means you need to calculate the phase shift so the peaks of the original sine graph match some of the peaks of your new graph. The formula for this new wave is:

f)Again, create the sum of the basic sine wave and your wave from part (e). Describe the basic pattern, and predict the differences you’d hear in the sound.

[1] The text in the above table and all images related to sound waves are taken from Dr. Joseph F. Alward’s online physics text. These materials can be found at