Oscillations and Waves

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What do waves, oscillations, and rotations have in common?

Big Ideas: 
  • Cosine and sine functions are used to describe periodic motion
  • The math describing oscillations, waves, and rotations is similar
  • Adding two or more waves ('superposition') can lead to cool wave effects like interference

Main Article

            Oscillatory motion is an important physical occurrence that will often come up in both physics classes and everyday life. The most common example of oscillatory motion is a mass attached to a spring. When the mass stretches the spring past its starting point – its equilibrium – the spring exerts a force on the mass causing it to travel past its starting position. The mass will oscillate around the spring’s equilibrium until some other force acts to damp out this motion.

Fig. 1: Snapshot from PhET - Masses and Springs. 1

            When in motion, the displacement of the mass from equilibrium is described by a sinusoidal function. In particular, a cosine function is often used to describe the displacement x, where A is the amplitude of the oscillation, T is the period of oscillation, and t is the time.

$ x(t) = A cos(\dfrac {2 \pi} {T} t) $

 

            Sinusoidal motion often occurs in everyday situations as well. When riding a Ferris wheel each compartment goes around and around in a periodic fashion and when tracking the height of the compartment the motion is clearly sinusoidal! Or, for example, consider the shocks in your car. They act very similar to the mass and spring previously described but the restoring forces act very quickly. But the principle behind their sinusoidal motion is the same: a periodic motion about equilibrium.

            In this first video, we explore the various parameters involved in sinusoidal motion and visualize their effects on the motion. To investigate sinusoidal motion we will use Excel spreadsheets.

For a quick refresher of Excel visit http://c21.phas.ubc.ca/article/using-spreadsheets and watch the first few videos.

           We can also connect sinusoidal motion to circular motion. When a particle traverses a circular path the x and y components of its motion are sinusoidal. This may be an explanation you are familiar with from trigonometry when discussing the unit circle. In fact, we can describe the x component of our motion using a cosine function and the y component using a sine function

            Think back to our example of a Ferris wheel exhibiting sinusoidal motion. The height goes up and down in a periodic fashion around the middle of the wheel. Similarly, the horizontal component of the motion goes side to side in a periodic fashion about the middle of the wheel. Separately both the x and y components of the motion are sinusoidal. Together the two components create circular motion! This kind of circular motion can also be seen in the motion of the gears in a watch or the motion of a helicopter blade.

            One final ingredient in our discussion of sinusoidal functions is the phase constant: an additive constant in the argument of the sinusoidal function. To include the phase constant we modify our original cosine function

$ x(t) = A cos(\dfrac {2 \pi} {T} t + \phi_0) $

where $ \phi_0 $ is the phase constant. Often, the argument of the cosine function,

$ \dfrac {2 \pi} {T} t + \phi_0 $

is referred to as the phase of the motion. The phase of the motion represents what angle we are at in the sinusoidal motion. The phase constant represents the initial angle of the motion at zero time. Think of it as an initial condition for the motion.

            The following video is split into two parts. The first part connects sinusoidal motion and circular motion. The second part introduces the phase constant into our cosine function and discusses the various effects the phase constant has on sinusoidal motion. For simplicity, the Excel spreadsheet used in the first video will be expanded upon and modified in the following video.

In the remaining videos we extend the established knowledge of sinusoidal functions to a discussion of waves. Mechanical waves are due to oscillations and the interactions between many oscillating particles. Consider a string made up of many particles that can interact with their neighbors. When the first particle in the string is displaced it interacts with its neighbor so that its neighbor is also displaced. The displacement of the first particle is transmitted down the string as in the figure below.

 

Fig. 2: Snapshot from PhET - Wave on a String. 2

Associated with each particle is a displacement from equilibrium that exhibits sinusoidal motion which we can write as

$ D(x) = A cos(\dfrac {2 \pi} {\lambda} t + \phi_0) $

            Here we have introduced a new parameter, $ \lambda $, called the wavelength. Where before the period represented the amount of time required before the sinusoidal function repeated itself, the wavelength represents the distance required before the sinusoidal function repeats itself.

            This equation works well to describe waves standing still in space. A more interesting topic arises when we allow the waves to move with velocity, v. Now, after time t the wave will have moved a distance vt. We can then write a formula for a wave travelling to the right as

$ D(x) = A cos(\dfrac {2 \pi} {\lambda} (x - v t)) $

           If you look at waves such as water waves closely, you will see that the water moves mainly up and down. The same is true for a wave on a string. The wave pattern travels because the medium oscillates as mentioned above. Have another look at the PhET simulation 'Wave on a String', (shown in Fig. 2 above) to convince yourself that this is true. There is a unique connection between the travelling wave pattern and the oscillating medium: The wave travels a distance x = $ \lambda $ in the time T it takes for one full oscillation of a particle in the medium, which is the period. We know that the velocity is distance divided by time, so the wave speed is

$ v = \dfrac {\lambda} {T} = \lambda f = \dfrac {1} {2 \pi} \lambda \omega $

This equation is applicable to all kinds of waves. If we substitute the wave speed v in our wave equation, we can bring it into a form that is often used in textbooks and in our videos below

$ D(x) = A cos(\dfrac {2 \pi} {\lambda} x - \omega t) = A cos(k x - \omega t) $

with $ k = \dfrac {2 \pi} {\lambda} $

The next video will introduce the topic of sinusoidal waves and illustrate their motion in time. Further, the topic of superposition will be broached.

            The principle of superposition is a common occurrence in both classical and modern physics and is often explained in introductory physics courses via the superposition of sinusoidal waves. Unlike particles, which cannot occupy the same space, waves can pass directly through each other and when two waves occupy the same space they will interfere. The displacement of the medium will then be the sum of the displacement due to each individual wave.

            In more formal terms, the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have been produced by the individual waves separately. In the next video we will explore the principle of superposition using sinusoidal waves. For this video a new Excel spreadsheet will be created.

 

          While the previous video introduced a special case of superposition, called a standing wave, we also wish to be able to understand more general types of superposition. In particular, the next video details what happens as we vary the different parameters present in our equations.

This video carries on from the previous one and continues to use the same spreadsheet.

 

 

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