Real World Examples of Periodic Functions:

Real World Examples of Periodic Functions:

Blood Pressure

Equation:

The equation for blood pressure can be represented as follows: P(t) = 22cos(2.5πt) + 98, where t represents time in seconds.

As our hearts go through systoles, cycles of contraction and relaxation, they are pumping blood throughout our body. This process repeats itself 70 times in a single minute, representing a periodic function. As the heart contracts pressure rises, as the heart relaxes, pressure drops. Mathematicians have discovered that when blood flows through arteries its pattern is similar to a sine or cosine wave. In these cases, the amplitude represents the the maximum blood flow and the frequency represents the heart rate.

Sound Waves

Sound waves, like any other waveform, consist of wavelength, frequency, velocity, and amplitude. These numbers vary based on the air temperature; however, at sea level (68 degrees,) the speed of sound is 767 miles per hour. Sound waves are created through vibrations and travel through many materials including water and steel but they are most commonly found traveling through air. Sound waves are characterized as mechanical waves because they are a disturbance that is transported through a medium. Sound waves are incapable of traveling through vacuums (an area of empty matter).

Equation:

Tidal Waves

In order to calculate the speed of sound in other temperatures in air the following equation can be used: V(sound in air) = 331.4 + 0.6Tc m/s (Tc is the Celsius temperature)

Example Problems:

Both tidal waves and tides, created by the gravitational pull between the moon and the sun, represent periodic functions. Tidal waves are a result of daily tides. As the tides come in and out each day, they produce a rise and fall in sea level. High tides and low tides follow a periodic pattern that you can model with the sine function. For example, on a particular winter day, the high tide in Boston, Massachusetts, occurred at midnight. The equation below can be used to determine the height of the water in the harbor.

1. Sound Waves

• Find the speed of sound in air at 24 degrees Celsius

2. Tidal Waves

• A tsunami is a fast-moving ocean wave caused by an underwater earthquake. The water first goes down from its normal level, then rises an equal distance above its normal level, and finally returns to its normal level. The period is about 15 minutes.
• Suppose that a tsunami with an amplitude of 10 meters approaches the pier at Honolulu, where the normal depth of the water is 9 meters.

3. Blood Pressure

• A doctor is monitoring his patient's blood pressure, which has a maximum of 130 mmHg, and a minimum of 70 mmHg. The heart rate is 60 beats per minute (60 cycles/minute).
• 1. Assuming the graph is a sine graph, write an equation to monitor the patient's blood pressure assuming it starts from the maximum, in terms of blood pressure and time in seconds.
• 2. If the heart rate increases to 70 beats per minute, how would you change the equation?

Equation:

H(t)=4.8sinπ/ 6(t+3)+5.1