Chapter Twenty Five Notes:

Vibrations and Waves

Pendulums swing to and fro with regularity. A complete to-and-fro oscillation is one vibration. The time of a to-and-fro vibration (or swing) is called the

period of the pendulum. The period depends only on the length of a pendulum and the acceleration of gravity.

A long pendulum has a longer period than a short pendulum; that is, it swings to and fro less frequently than a short pendulum.

A repeating back-and-forth motion about an equilibrium position in a VIBRATION. A vibration cannot exist in one instant. It needs time to move back and forth.

A disturbance that is transmitted progressively from one place to the next with no actual transport of matter is a WAVE. A wave cannot exist in one place but must extend from one place to another.

The back and forth vibratory motion (often called oscillatory motion) of a swinging pendulum is called simple harmonic motion (SHM). If you fill the pendulum with sand, and allow a conveyor belt to move at constant speed beneath the pendulum, the sand traces out a special curve known as a sine curve. A sine curve is a pictorial representation of a wave. The source of all waves is something that vibrates.

Vibration of a pendulum. The to-and-fro vibratory motion is also called oscillatory motion (or oscillation).

Wave description (parts of wave)

Crests and troughs: the high points of a wave are called crests, and the low points are called troughs.

Midpoint of vibration: the straight dashed line represents the “home” position, or midpoint of the vibration. The midpoint position is also the equilibrium position.

Amplitude: the distance from the midpoint to the crest (or trough). The amplitude equals the maximum displacement from equilibrium.

Wavelength: the distance from the top of one crest to the top of the next one. Or equivalently, the wavelength is the distance between any successive identical parts of the wave.

Frequency: how frequently a vibration occurs is described by its frequency. For example, the frequency of a vibrating pendulum specifies the number of complete to-and-fro vibrations it makes in a given time (usually one second).

More about frequency • The unit of frequency is called the hertz (Hz). For example, if a pendulum makes two vibrations in one second,

its frequency is 2 Hz. • The source of all waves is something that vibrates. • The frequency of the vibrating source and the frequency of the

wave it produces are the same. • Relationship between frequency and period:

Frequency = 1/ Period or Period = 1/ Frequency.

– For example, suppose that a pendulum makes two vibrations in one second. Its frequency is 2 Hz. Its period, that is, the time needed to complete one vibration is 1/2 second.

Electrons in the transmitting antenna vibrate 940,000 times each second and produce 940-kHz radio waves.

Most information about our surroundings comes to us in some form of waves. It is through wave motion that sounds come to ears, light to our eyes, and electromagnetic signals to our radios and television sets.

• Through wave motion, energy can be transferred from a source to a receiver without the transfer of matter between the two points.

• In wave motion, what is transported from one place to another is a disturbance (vibration) in a medium, not the medium itself.

If one end of a rope is shaken up and down, a rhythmic disturbance travels along the rope. Each particle of the rope moves up and down, while at the same time the disturbance moves along the length of the rope. The medium, rope or whatever, returns to its initial condition after the disturbance has passed. What is propagated is the disturbance, not the medium itself.

A wave is a disturbance which moves along a medium from one end to the other. If one watches an ocean wave moving along the medium (the ocean water), one can observe that the crest of the wave is moving from one location to another over a given interval of time. The crest is observed to cover distance. The speed of an object refers to how fast an object is moving and is usually expressed as the distance traveled per time of travel. In the case of a wave, the speed is the distance traveled by a given point on the wave (such as a crest) in a given interval of time. In equation form,

If the crest of an ocean wave moves a distance of 20 meters in 10 seconds, then the speed of the ocean wave is 2 m/s. On the other hand, if the crest of an ocean wave moves a distance of 25 meters in 10 seconds (the same amount of time), then the speed of this ocean wave is 2.5 m/s. The faster wave travels a greater distance in the same amount of time.

Sometimes a wave encounters the end of a medium and the presence of a different medium. For example, a wave introduced by a person into one end of a slinky will travel through the slinky and eventually reach the end of the slinky and the presence of the hand of a second person. One behavior which waves undergo at the end of a medium is reflection. The wave will reflect or bounce off the person's hand. When a wave undergoes reflection, it remains within the medium and merely reverses its direction of travel. In the case of a slinky wave, the disturbance can be seen traveling back to the original end. A slinky wave which travels to the end of a slinky and back has doubled its distance. That is, by reflecting back to the original location, the wave has traveled a distance which is equal to twice the length of the slinky.

Reflection phenomenon are commonly observed with sound waves. When you let out a holler within a canyon, you often hear the echo of the holler. The sound wave travels through the medium (air in this case), reflects off the canyon wall and returns to its origin (you). The result is that you hear the echo (the reflected sound wave) of your holler. A classic physics problem goes like this:

In this instance, the sound wave travels 340 meters in 1 second, so the speed of the wave is 340 m/s. Remember, when there is a reflection, the wave doubles its distance. In other words, the distance traveled by the sound wave in 1 second is equivalent to the 170 meters down to the canyon wall plus the 170 meters back from the canyon wall.

Since the distance a single wave moves is measured in Wavelength, and the time it takes to move this distance is measured as period, it can be said that the speed of a wave is also the wavelength/period.

Noah stands 170 meters away from a steep canyon wall. He shouts and hears the echo of his voice one second later. What is the speed of the wave?

Since the period is the reciprocal of the frequency, the expression 1/f can be substituted into the above equation for period. Rearranging the equation yields a new equation of the form:

Speed = Wavelength • Frequency

The above equation is known as the wave equation. It states the mathematical relationship between the speed (v) of a wave and its wavelength (λ) and frequency (f). Using the symbols v, λ, and f, the equation can be rewritten as

v = λ • f

• Shake the rope with a regular continuing up-and-down motion, the series of pulses will produce a wave.

  

• Since the motion of the medium (up and down arrows in the rope in this case) is at right angles to the direction the wave travels, this type of wave is called a transverse wave.

• Examples of transverse waves:◦ Waves in the stretched strings of musical instruments.◦ Waves upon the surfaces of liquids.◦ Electromagnetic waves, which make up radio waves and light.

Not all waves are transverse. Sometimes parts that make up a medium move to and fro in the

same direction in which the wave travels. In this case, motion is along the direction of the wave rather than

at right angles to it. This produces a longitudinal wave. Sound waves are longitudinal waves.

What happens when two waves meet while they travel through the same medium? What affect will the meeting of the waves have upon the appearance of the medium? Will the two waves bounce off each other upon meeting (much like two billiard balls would) or will the two waves pass through each other? These questions involving the meeting of two or more waves along the same medium pertain to the topic of wave interference.

Wave interference is the phenomenon which occurs when two waves meet while traveling along the same medium. The interference of waves causes the medium to take on a shape which results from the net effect of the two individual waves upon the particles of the medium. To begin our exploration of wave interference, consider two pulses of the same amplitude traveling in different directions along the same medium. Let's suppose that each displaced upward 1 unit at its crest and has the shape of a sine wave. As the sine pulses move towards each other, there will eventually be a moment in time when they are completely overlapped. At that moment, the resulting shape of the medium would be an upward displaced sine pulse with an amplitude of 2 units. The diagrams below depict the before and during interference snapshots of the medium for two such pulses. The individual sine pulses are drawn in red and blue and the resulting displacement of the medium is drawn in green.

This type of interference is sometimes called constructive interference. Constructive interference is a type of interference which occurs at any location along the medium where the two interfering waves have a displacement in the same direction. In this case, both waves have an upward displacement; consequently, the medium has an upward displacement which is greater than the displacement of the two interfering pulses. .

Constructive interference is observed at any location where the two interfering waves are displaced upward. But it is also observed when both interfering waves are displaced downward. This is shown in the diagram below for two downward displaced pulses.

In this case, a sine pulse with a maximum displacement of -1 unit (negative means a downward displacement) interferes with a sine pulse with a maximum displacement of -1 unit. These two pulses are drawn in red and blue. The resulting shape of the medium is a sine pulse with a maximum displacement of -2 units.

Destructive interference is a type of interference which occurs at any location along the medium where the two interfering waves have a displacement in the opposite direction. For instance, when a sine pulse with a maximum displacement of +1 unit meets a sine pulse with a maximum displacement of -1 unit, destructive interference occurs. This is depicted in the diagram below.

In the diagram above, the interfering pulses have the same maximum displacement but in opposite directions. The result is that the two pulses completely destroy each other when they are completely overlapped. At the instant of complete overlap, there is no resulting displacement of the particles of the medium. This "destruction" is not a permanent condition. In fact, to say that the two waves destroy each other can be partially misleading. When it is said that the two pulses destroy each other, what is meant is that when overlapped, the affect of one of the pulses on the displacement of a given particle of the medium is destroyed or

canceled by the affect of the other pulse.

Recall from an earlier lesson that waves transport energy through a medium by means of each individual particle pulling upon its nearest neighbor. When two pulses with opposite displacements (i.e., one pulse displaced up and the other down) meet at a given location, the upward pull of one pulse is balanced (canceled or destroyed) by the downward pull of the other pulse. Once the two pulses pass through each other, there is still an upward displaced pulse and a downward displaced pulse heading in the same direction which they were heading before the interference. Destructive interference leads to only a momentary condition in which the medium's displacement is less than the displacement of the largest-amplitude wave.

The two interfering waves do not need to have equal amplitudes in opposite directions for destructive interference to occur. For example, a pulse with a maximum displacement of +1 unit could meet a pulse with a maximum displacement of -2 units. The resulting displacement of the medium during complete overlap is -1 unit.

This is still destructive interference since the two interfering pulses have opposite displacements. In this case, the destructive nature of the interference does not lead to complete cancellation.

Interestingly, the meeting of two waves along a medium does not alter the individual waves or even deviate them from their path. This only becomes an astounding behavior when it is compared to what happens when two billiard balls meet or two football players meet. Billiard balls might crash and bounce off each other and football players might crash and come to a stop. Yet two waves will meet, produce a net resulting shape of the medium, and then continue on doing what they were doing before the interference.

 

The task of determining the shape of the resultant demands that the principle of superposition is applied. The principle of superposition is sometimes stated as follows:

When two waves interfere, the resulting displacement of the medium at any location is the algebraic sum of the displacements of the individual waves at that same location.

The animation below depicts two waves moving through a medium in opposite directions. The blue wave is moving to the right and the green wave is moving to the left. As is the case in any situation in which two waves meet while moving along the same medium, interference occurs. The blue wave and the green wave interfere to form a new wave pattern known as the resultant. The resultant in the animation below is shown in black. The resultant is merely the result of the two individual waves - the blue wave and the green wave - added together in accordance with the principle of superposition.

The result of the interference of the two waves above is a new wave pattern known as a standing wave pattern. Standing waves are produced whenever two waves of identical frequency interfere with one another while traveling opposite directions along the same medium. Standing wave patterns are characterized by certain fixed points along the medium which undergo no displacement. These points of no displacement are called nodes (nodes can be remembered as points of no displacement). The nodal positions are labeled by an N in the animation above. The nodes are always located at the same location along the medium, giving the entire pattern an appearance of standing still (thus the name "standing waves"). A careful inspection of the above animation will reveal that the nodes are the result of the destructive interference of the two interfering waves. At all times and at all nodal points, the blue wave and the green wave interfere to completely destroy each other, thus producing a node.

Midway between every consecutive nodal point are points which undergo maximum displacement. These points are called antinodes; the anti-nodal nodal positions are labeled by an AN. Antinodes are points along the medium which oscillate back and forth between a large positive displacement and a large negative displacement. A careful inspection of the above animation will reveal that the antinodes are the result of the constructive interference of the two interfering waves.

In conclusion, standing wave patterns are produced as the result of the repeated interference of two waves of identical frequency while moving in opposite directions along the same medium. All standing wave patterns consist of nodes and antinodes. The nodes are points of no displacement caused by the destructive interference of the two waves. The antinodes result from the constructive interference of the two waves and thus undergo maximum displacement from the rest position.

Suppose that there is a happy bug in the center of a circular water puddle. The bug is periodically shaking its legs in order to produce disturbances that travel through the water. If these disturbances originate at a point, then they would travel outward from that point in all directions. Since each disturbance is traveling in the same medium, they would all travel in every direction at the same speed.

The pattern produced by the bug's shaking would be a series of concentric circles as shown in the diagram at the right. These circles would reach the edges of the water puddle at the same frequency. An observer at point A (the left edge of the puddle) would observe the disturbances to strike the puddle's edge at the same frequency that would be observed by an observer at point B (at the right edge of the puddle). In fact, the frequency at which disturbances reach the edge of the puddle would be the same as the frequency at which the bug produces the disturbances. If the bug produces disturbances at a frequency of 2 per second, then each observer would observe them approaching at a frequency of 2 per second.

Now suppose that our bug is moving to the right across the puddle of water and producing disturbances at the same frequency of 2 disturbances per second. Since the bug is moving towards the right, each consecutive disturbance originates from a position which is closer to observer B and farther from observer A. Subsequently, each consecutive disturbance has a shorter distance to travel before reaching observer B and thus takes less time to reach

observer B. Thus, observer B observes that the frequency of arrival of the disturbances is higher than the frequency at which disturbances are produced. On the other hand, each consecutive disturbance has a further distance to travel before reaching observer A. For this reason, observer A observes a frequency of arrival which is less than the frequency at which the disturbances are produced. The net effect of the motion of the bug (the source of waves) is that the observer towards whom the bug is moving observes a frequency which is higher than 2 disturbances/second; and the observer away from whom the bug is moving observes a frequency which is less than 2 disturbances/second. This effect is known as the Doppler effect.

The Doppler effect is observed whenever the source of waves is moving with respect to an observer. The Doppler effect can be described as the effect produced by a moving source of waves in which there is an apparent upward shift in frequency for observers towards whom the source is approaching and an apparent downward shift in frequency for observers from whom the source is receding. It is important to note that the effect does not result because of an actual change in the frequency of the source. Using the example above, the bug is still producing disturbances at a rate of 2 disturbances per second; it just appears to the observer whom the bug is approaching that the disturbances are being produced at a frequency greater than 2 disturbances/second. The effect is only observed because the distance between observer B and the bug is decreasing and the distance between observer A and the bug is increasing.

The Doppler effect can be observed for any type of wave - water wave, sound wave, light wave, etc. We are most familiar with the Doppler effect because of our experiences with sound waves. Perhaps you recall an instance in which a police car or emergency vehicle was traveling towards you on the highway. As the car approached with its siren blasting, the pitch of the siren sound

(a measure of the siren's frequency) was high; and then suddenly after the car passed by, the pitch of the siren sound was low. That was the Doppler effect - an apparent shift in frequency for a sound wave produced by a moving source.

Light: The Doppler effect is of intense interest to astronomers who use the information about the shift in frequency of electromagnetic waves produced by moving stars in our galaxy and beyond in order to derive information about those stars and galaxies. The belief that the universe is expanding is based in part upon observations of electromagnetic waves emitted by stars in distant galaxies. Furthermore, specific information about stars within galaxies can be determined by application of the Doppler effect. Galaxies are clusters of stars which typically rotate about some center of mass point. Electromagnetic radiation emitted by such stars in a distant galaxy would appear to be shifted downward in frequency (a red shift) if the star is rotating in its cluster in a direction which is away from the Earth. On the other hand, there is an upward shift in frequency (a blue shift) of such observed radiation if the star is rotating in a direction that is towards the Earth.

When the source moves at a speed equal to the speed of the wave, a barrier wave is produced in front of the source as each successive wave front piles on top of the previous one. This regions of constructive interference pattern is a physical reality that must be overcome if the source is to move any faster. 

Boats, after they break through the barrier wave that is produced when their speeds equal the speed of the water waves in that region, start trailing a two-dimensional bow wave. Down the center of the bow wave is a region of destructive interference while the edges, or wake, are regions of high amplitude constructive interference. As a general rule, the speed of a water wave is principally determined by the water's depth and its temperature.

 

When planes fly through the "sound barrier," they are doing more than just traveling in excess of 340 m/sec. They have traveled also through a region of high amplitude constructive interference which, for sound waves, is a region of high compression followed by a region of low rarefaction. The pressure differential is tremendous. After breaking through the barrier wave, the plane then trails a three-dimensional bow wave, or a shock cone, and is said to be traveling supersonic. The width of the bow wave or shock cone depends on the speed of the source; the faster the source travels, the narrower the wave becomes.

When a shock wave, trailing a supersonic plane, passes through your position, you heard a sonic boom. It is actually a "double boom." When the leading edge of the cone passes, it raises equilibrium pressure rapidly upwards (compression), followed by a rapid drop in pressure (rarefaction), followed by a return to equilibrium. We hear a sonic boom whenever the Shuttle's glide path passes over Daytona for its landing at the Cape.

 

  Doppler Effect Barrier Wave Bow Wave (2-D)

Shock Wave (3-D)

a stationary wave source

a wave source moving to the right at a speed less

than the wave speed

a wave source moving to the right at a speed equal

to the wave speed

wave source moving to the right at a speed in excess of the wave speed

To determine the "Mach" number for a supersonic plane you compare the ratio of the plane's radius to the ratio of the sound's radius.

The Doppler Effect

Use this physlet animation by Wolfgang Christian at Davidson College to watch the wave patterns as the red particle travels faster and faster.