The Link Between Simple Harmonic Motion and Waves

The Link between Simple Harmonic Motion and Waves

If a time-exposure photograph of the bouncing car were taken as it drove by, the headlight would make a wavelike streak, as shown in this figure. Similarly, this figure shows an object bouncing on a spring as it leaves a wavelike "trace of its position on a moving strip of paper. Both waves are sine functions. All simple harmonic motion is intimately related to sine and cosine waves.

The figure shows the front right side of a running car on an uneven rough surface which also shows the driver in the driving seat. There is an oscillating sine wave drawn from left to the right side horizontally throughout the figure.

The bouncing car makes a wavelike motion. If the restoring force in the suspension system can be described only by Hooke’s law, then the wave is a sine function. (The wave is the trace produced by the headlight as the car moves to the right.)

There are two iron paper roll bars standing vertically with a paper strip stitched from one bar to the other. There is a vertical hanging spring just over the middle of the two bars, perpendicular to the strip of the paper, having an object with mass m tied to it. There is a line graph with amplitude scale as X, zero and negative X on the left side of the paper strip, vertically over each other with their points marked. A perpendicular line is drawn through this amplitude scale toward the right with a point T marked over it, showing the time duration of the amplitude. This line has an oscillating wave drawn through it.

The vertical position of an object bouncing on a spring is recorded on a strip of moving paper, leaving a sine wave.

The displacement as a function of time t in any simple harmonic motion—that is, one in which the net restoring force can be described by Hooke’s law, is given by

\(x(t)=X\phantom{\rule{0.25em}{0ex}}\text{cos}\cfrac{2\mathrm{\pi t}}{T},\)

where \(X\) is amplitude. At \(t=0\), the initial position is \({x}_{0}=X\), and the displacement oscillates back and forth with a period \(T\). (When \(t=T\), we get \(x=X\) again because \(\text{cos}\phantom{\rule{0.25em}{0ex}}2\pi =1\).). Furthermore, from this expression for \(x\), the velocity \(v\) as a function of time is given by:

\(v(t)=-{v}_{\text{max}}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}(\cfrac{2\pi t}{T}),\)

where \({v}_{\text{max}}=2\pi X/T=X\sqrt{k/m}\). The object has zero velocity at maximum displacement—for example, \(v=0\) when \(t=0\), and at that time \(x=X\). The minus sign in the first equation for \(v(t)\) gives the correct direction for the velocity. Just after the start of the motion, for instance, the velocity is negative because the system is moving back toward the equilibrium point. Finally, we can get an expression for acceleration using Newton’s second law. [Then we have \(x(t),\phantom{\rule{0.25em}{0ex}}v(t),\phantom{\rule{0.25em}{0ex}}t,\) and \(a(t)\), the quantities needed for kinematics and a description of simple harmonic motion.] According to Newton’s second law, the acceleration is \(a=F/m=\text{kx}/m\). So, \(a(t)\) is also a cosine function:

\(a(t)=-\cfrac{\text{kX}}{m}\text{cos}\cfrac{2\pi t}{T}.\)

Hence, \(a(t)\) is directly proportional to and in the opposite direction to \(x(t)\).

This figure shows the simple harmonic motion of an object on a spring and presents graphs of \(x(t),v(t),\) and \(a(t)\) versus time.

In the figure at the top there are ten springboards with objects of different mass values tied to them. This makes some springs highly compressed some as loosely stretched and some at equilibrium, which are shown as red spherical shaped. Alongside the figure there is a scale given for different amplitude values as x equal to positive X, zero and negative X. the upward and downward pointing arrows are shown with a few springboards. In the second figure there are three graphs. The first graph shows distance covered in form of a sine wave starting from a point x units on positive y-axis. The height of the wave above x-axis is marked as amplitude. The gap between two consecutive crests is marked as T. Below first graph there is another graph showing velocity in form of a sine wave starting from the origin downward. In the third graph below the second one, acceleration is shown in the form of sine wave starting from x units on the negative y-axis upward. In the last figure three position of a spring are shown. The first position shows the unstretched length of a spring pendulum. A hand is holding the bob of the pendulum. In the second position the equilibrium position of the spring and bob is shown. This position is lower the first one. In the third case the up and down oscillations of the spring pendulum are shown. The bob is moving x units in upward and downward directions alternatively.

Graphs of \(x(t),\phantom{\rule{0.25em}{0ex}}v(t),\) and \(a(t)\) versus \(t\) for the motion of an object on a spring. The net force on the object can be described by Hooke’s law, and so the object undergoes simple harmonic motion. Note that the initial position has the vertical displacement at its maximum value \(X\); \(v\) is initially zero and then negative as the object moves down; and the initial acceleration is negative, back toward the equilibrium position and becomes zero at that point.

The most important point here is that these equations are mathematically straightforward and are valid for all simple harmonic motion. They are very useful in visualizing waves associated with simple harmonic motion, including visualizing how waves add with one another.

Check Your Understanding

Suppose you pluck a banjo string. You hear a single note that starts out loud and slowly quiets over time. Describe what happens to the sound waves in terms of period, frequency and amplitude as the sound decreases in volume.

Solution

Frequency and period remain essentially unchanged. Only amplitude decreases as volume decreases.

Check Your Understanding

A babysitter is pushing a child on a swing. At the point where the swing reaches \(x\), where would the corresponding point on a wave of this motion be located?

Solution

\(x\) is the maximum deformation, which corresponds to the amplitude of the wave. The point on the wave would either be at the very top or the very bottom of the curve.

PhET Explorations: Masses and Springs

A realistic mass and spring laboratory. Hang masses from springs and adjust the spring stiffness and damping. You can even slow time. Transport the lab to different planets. A chart shows the kinetic, potential, and thermal energy for each spring.

The Link Between Simple Harmonic Motion and Waves