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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 [link] . Similarly, [link] 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 cos 2 πt T , size 12{x left (t right )=X"cos" { {2π`t} over {T} } } {}

where X size 12{X} {} is amplitude. At t = 0 size 12{t=0} {} , the initial position is x 0 = X size 12{x rSub { size 8{0} } =X} {} , and the displacement oscillates back and forth with a period T . (When t = T , we get x = X size 12{x=X} {} again because cos = 1 .). Furthermore, from this expression for x size 12{x} {} , the velocity v size 12{v} {} as a function of time is given by:

v ( t ) = v max sin t T , size 12{v \( t \) = - v rSub { size 8{"max"} } "sin" left ( { {2π`t} over {T} } right )} {}

where v max = X / T = X k / m size 12{v rSub { size 8{"max"} } =2πX/T=X sqrt {k/m} } {} . The object has zero velocity at maximum displacement—for example, v = 0 size 12{v=0} {} when t = 0 size 12{t=0} {} , and at that time x = X size 12{x=X} {} . The minus sign in the first equation for v ( t ) size 12{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 ) , v ( t ) , t , size 12{x \( t \) ,v \( t \) ,t} {} and a ( t ) size 12{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 = kx / m size 12{a=F/m= ital "kx"/m} {} . So, a ( t ) size 12{a \( t \) } {} is also a cosine function:

a ( t ) = kX m cos t T . size 12{a \( t \) = - { { ital "kX"} over {m} } " cos " { {2π t} over {T} } } {}

Hence, a ( t ) size 12{a \( t \) } {} is directly proportional to and in the opposite direction to x ( t ) .

[link] shows the simple harmonic motion of an object on a spring and presents graphs of x ( t ) , v ( t ), size 12{x \( t \) ,v \( t \) `} {} and a ( t ) size 12{`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 ) , v ( t ) , size 12{x \( t \) ,v \( t \) `} {} and a ( t ) size 12{`a \( t \) } {} versus t size 12{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 size 12{X} {} ; v size 12{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.

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.

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

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
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Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
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David Reply
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David
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emma Reply
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what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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Adjanou
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Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
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Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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