Waves

     As we discuss simple harmonic motions, or SHM, we will be dealing with many wave functions. Below is an example of a wave function. We first need to understand key points about this graph until we can fully understand SHM.

Here is a graph that we can assume is a distance-time graph.

    First, recognize that wave functions repeat themselves over equally spaced intervals. In the graph above, for example, we can see that it takes the same difference in the x-value to reach the first highest point of the graph to the second highest point. Conversely, it takes the same x-value for the first point of the lowest point to the second lowest point. The interval of which it takes the wave function to fully undergo one repeat is called a period, T. In SHM, it is measured in seconds. 

When we talk about periods, they are not to be confused with frequency. Frequency is the inverse of periods, and describes how many times something happens in a second. So, the unit is 1/seconds, which are Hertz (Hz). Frequency is extremely important, as it plays a role in how we hear and see, thanks to the electromagnetic spectrum. It is important to understand the difference between the two units. When we add formulas on to our lessons later, using the wrong unit can be costly. An easy way to remember the difference between the two is by seeing how they are used. Frequency, of course, refers to how frequent something occurs, while period refers to how long it takes. We can denote the two as follows:

T = 1/f

f = 1/T

    Another property to be recognized here is that the distance from the midline to the highest point is the same as the distance from the midline to the lowest point. In other words, the graph is symmetrical via the x axis. We call the distance from the high or the low to the midline, the amplitude. 

    In terms of SHM, the midline is also called the equilibrium. This term is extremely important to know as we move forward in future lessons with SHM. Having learned about forces before, we know that equilibrium happens when net forces are zero and are cancelled out. Oscillations are what happen in a simple harmonic motion. An oscillation is when something moves back and forth from an equilibrium position. They will travel a distance A before returning back towards the equilibrium point. In the graph above, we can see the phenomenon in which the SHM returns to equilibrium at various points. All these properties play important roles.

    Let's quickly sum up the basics of wave functions in a visual diagram.

    Now, let's imagine a mass-spring system. We know the formula for such is F=-kx. When we leave the spring as is, it is in its equilibrium position. If we stretch the spring the maximum displacement it can go, that is a representation of amplitude, where x = A. The period will be how long it takes until the spring, when released, will return to its initial displacement of A. 

    Let's do some practice problems.

    1) What is the frequency and period of a girl who completes 20 cycles on a swing in 25 seconds?

    2) If a speaker can produce a note of frequency 440 Hz, how long does it take for the speaker to complete one full vibration?

Answers:

1)

f = 20/25 = .80 Hz

T = 1/f = 1/.80 = 1.25 sec

2)

T = 1/f = 1/440 Hz = .0023 sec

    Next blog, we will cover linear restoring forces.