Chapter 11
The Giancoli onLine Tutor

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Vibrations and Waves

I. Periodic Motion

periodic Motion is any motion that repeats itself over a fixed period of time, T, called the period. Circular motion, which we studied in an earlier chapter, is an example of periodic motion.

Harmonic Motion

In this chapter we are interested in a particular type of " one-dimensional" back-and-forth motion called Harmonic Motion. Two examples of this are a mass on a spring and the motion of a simple pendulum, a mass bob on a string. In both cases the restoring force when the mass is moved away from equilibrium is proportional to the displacement of the mass from its equilibrium position. (See the figure on the left.) This is only true for a pendulum providing that its swing is less than the one shown in the figure. Under that case a pendulum behaves as if it were a mass on a spring with a spring constant equal to (mg/l). Thus everything we say and find out about spring motion will be immediately extended to the pendulum by replacing "k" by "mg/l". The motion of a mass on a spring is described by

x=Acos(wt) and v=-w Asin(w t),

where "A", called the amplitude, is the initial distance the string is stretched or compressed and "x" is the position of the mass at any given time "t" and "v" is its velocity at that time. The symbol w is meant to represent the lower case greek letter omega. The value for w, called the angular frequency, will always be given by

w=√(k/m).

The spring constant "k" commonly given in the units of N/cm must always be converted to N/m by multiplying by 100. The angular frequency is related to the actual frequency, f by the relationship

w=2p*f,

The actual frequency is normally what is of interest. You find it by using k/m to determine w and then the above relationship to find f. The units of f are called Hz which stands for cycles per second. Thus an answer of 4 Hz would mean that the mass oscillates back-and-forth 4 times a second. That also means that the period, T (the time for one complete cycle), is 1/4 of a second. In general frequency and period are always the reciprocal of each other! Combining the various relationships given above we write:

f=(1/2p)*√(k/m) and T=2p*√(m/k).

For the simple pendulum, substituting mg/l for k, these equations become:

f=(1/2p)*√(g/l) and T=2p*√(l/g).

This last relationship says that the period of a pendulum is independent of its mass and the size of its swing, this made pendulums very useful in clock mechanisms.

Interactive Example
A particular 2 kg harmonic oscillator has a velocity given by
v=-12sin(4t)
all in metric units.

What is its amplitude? (Enter your answer and click outside)

What is its angular frequency? (Enter your answer and click outside)

To the nearest hundredth of a hertz what is its frequency? (Enter your answer and click outside)

What is the spring constant, k? (Enter your answer and click outside)

If this were a pendulum what would be its length to the nearest cm, use 10 for g? (Enter your answer and click outside)

To the nearest hundredth of a second what is its period, use 10 for g? (Enter your answer and click outside)

II. Wave Motion

Periodic motions and vibrations, in general, produce waves. Your vocal coils produce sound waves; vibrating molecules produce microwaves, and so on. All waves, except Electromagnetic waves, covered elsewhere are disturbances, travelling through a medium. Water waves, sound waves, earthquakes all fall in this category. We will limit ourselves to simple waves also called harmonic waves. These waves are divided into three categories:

Transverse Waves (The waves on a violin string.)
Longitudinal Waves (sound waves)
and
Combinations (water waves).

Transverse Waves

Transverse waves are very much like the "wave" seen at football games. There is up and down motion which appears to move left to right. If you stretch a long wire and wiggle one end up and down you will produce such a wave. A longitudinal waves can be thought of as like a multiple car pileup on a freeway where the last car hits the car in front of it knocking it into the car in front of it, and so on and on, until the front car is finally struck. The collision at the back is passed to the front by a series of successive collisions. This is very much how sound waves work except that the air (or whatever) molecules undergo elastic collisions with the molecules in front of them and bounce back rather than coming to a stop as a car would. Combination waves are a combination of these. Simple waves have the following properties:

At any point in space each particle will vibrate up and down just like a mass on a spring with a fixed frequency f and repeating with a fixed period T. Its maximum motion up or down is called the wave Amplitude
At any given time the wave pattern will repeat its shape in space with a fixed distance, called the wavelength.
The wave velocity will always equal the product of the wavelength and frequency.

Usually the wave velocity is known and you use the relationship to find either the wavelength or frequency. The velocity of waves are determined by the material properties of the medium through which it moves. For example, the velocity of a wave in a rope or wire is given by,

v=√(T/(m/l)),

where T is the tension and m/l is the mass density per unit length. For a longitudinal wave travelling through a fluid, the velocity is given by,

v=√(B/(m/V)),

where B is the bulk modulus and m/V is the normal mass density. (If the wave had been moving through a solid such as rock B would be replaced by E, the elastic modulus. Both B and E along with the density will normally be givens.)

Interactive Example
A 4 meter long wire has a mass of 100 gm and is under a tension of 100 N, what is its wave velocity to the nearest m/s? (Enter your answer and click outside)

A 10 meter . 50 gm wire has a wave velocity of 100 m/s, what is its tension? (Enter your answer and click outside)

Superposition and Interference

When multiple waves of the same kind travel through a common medium the result at any point is the algebraic sum of the multiple waves (The Principle of Superposition). In other words, if a particle at the given point is told to go up 2 cms by one wave and 4 cms by the other wave, it will move up 6 cms. If the first wave says to go down 2 cms and the second to move up 4 cms, it will move up 2 cm. If one says "down 4 cms" and the other "up 4 cms,ä the particle will move at all. This is called Destructive Interference.

If two sources of waves are emitting identical waves in phase*, but are a different distance from a given point, it is possible that the two waves will completely cancel out at that point. This will happen when the different distance is exactly half of a wavelength. In that case the one wave will always be "telling" the point to do the opposite, for example when a crest from one wave reaches that point, it will be joined by a trough from the other producing complete destructive Interference. The Amplitude at that point will always be zero. Conversely, if the different distance is a multiple of a wavelength, the two waves will always line up at that point, trough to trough and crest to crest. The point will oscillate up and down with twice the amplitude of either wave. This is called Constructive Interference.

Standing Waves

When waves reflect off a surface they can set up a Standing Wave pattern. This can occur with any type of wave. Water waves reflecting off shoals commonly set up such standing wave patterns wherein the waves pulse up and down but do not otherwise move.
Transverse standing waves form the basis of musical string instruments. A transverse wave travelling left to right on a piano wire and encountering a fixed end will be reflected backward on the same wire but upside down from the incoming wave. The energy in a wave is proportional to the square of the amplitude so that you now have two waves carrying the same energy but in the opposite directions. Thus there is no net transfer of energy in either direction and hence no back and forth motion just an up and down pulsing. In some places the wire will be moving up and down with a maximum amplitude (these are called antinodes) in other places it will not be moving at all (these are called nodes). The antinodes will always lay half way between two nodes and there must always be nodes at both ends (- because they are tied down.)

Additional nodes, if there are any, will always lie in the middle between two nodes and the distance between a pair of nodes will always be half a wavelength (l).
If we have a wire of length L with the only nodes at the ends and one antinode in the middle then

L=l/2.
If you have an additional node in the middle (and antinodes at L/4 and 3L/4) then
L/2= l/2 or L=l.
If you continue with this pattern you will discover a relationship between the allowed wavelengths of a standing wave in a wire and the actual length of the wire, specifically

l= 2*L/n,

where n= 1,2,3,... Since we know that frequency(f) times wavelength(l) is equal to the wave velocity(v),
(and that for a wave in a string the velocity is given by v= √(T/(m/l)) )
we now know that

f_n=n*(v/2L)
where n=1,2,3,... and is generally known or easily found. The quantity f₁= v/2L is called the fundamental frequency (or first harmonic); f₂ is the first overtone (or second harmonic), f₃ the second overtone (or third harmonic), and so on.

Wave Refraction

When waves move from one region to another they bend or refract if the wave speed changes in the move.

You need Quicktime 3 or better to use the following.

Interactive Activity
Red Man Blue Man and Green Man are having an argument. They have to run a race from point 1 to point 2 over concrete and then soft sand. They can each run at 9 m/s on the concrete but only 5.2 m/s in the sand.
Red Man says "You need to maximize the distance over the concrete surface. You should run to the boundary point above point 2 and then run straight down."
Green Man says "Nonsense! You should minimize the distance. Run straight to point 2."
Blue Man (being a New Age type of guy) says "Nada. You must follow the path of light"
Fermat's Principle states that light travels from point 1 to point 2 in the least possible time. This gives rise to snell's law of refraction for light:
v₂sin(q₁)= v₁sin(q₂)
where q₁ is the angle of incidence (with the perpendicular) at a surface , q₂ is the angle of refraction in the material, such as, glass, on the other side of the surface. Actually this law applies to all waves going from 1 medium to another where the speed changes, such as, water waves entering shallow water. The application below is designed to help you understand this.
You are runner whose task it is to run from point 1 to point 2 in the shortest possible time.

Click on each man to line them up at the start.
Click anywhere to start the race.
Click anywhere to line them up for a new race.

As you can see when waves move from a region of higher speed to that of lower speed they bend towards to the normal, that is, its exit angle is less than its entrance angle.

That explains why water waves, if any, at the beach always appear to be coming in at the beach no matter the direction of the wind. In the figure at the left the dark lines are the crests of water waves and the red arrows show the direction of motion . Wave speed decreases as water depth decreases. The lighter the blue in the figure, the shallower the water and the slower the wave. Notice that as the wave fronts enter progressively shallower water they are bent towards the bottom or shore. Also note that the wave lengths decrease as this occurs and the waves tend to 'pile up'.

Longitudinal Standing Waves will be covered in the next chapter.

*For the waves to be identical they must have the same wavelength and amplitude. For the sources to be in phase they must emitting the waves such that when one source is emitting a maximum crest so is the other. BACK.