04 Waves & Wavelike motion

WAVES AND WAVELIKE MOTION

WHY WAVES ARE A SPECIAL KIND OF MOTION?

Vibrational Motion:

*Things wiggle.
*They do the back and forth.
*They vibrate;
*They shake;
*They oscillate.

These phrases describe the motion of a variety of objects. They even describe the motion of matter at the atomic level. Even atoms wiggle - they do the back and forth. Wiggles, vibrations, and oscillations are an inseparable part of nature.

What is the relationships between vibrational motion and waves?

Waves occur frequently in nature.

The most obvious examples are waves in water, on a dam, in the ocean, or in a bucket. We are most interested in the properties that waves have. All waves have the same properties so if we study waves in water then we can transfer our knowledge to predict how other examples of waves will behave.

What are waves?

*Waves are disturbances which propagate (move) through a medium.
*Waves can be viewed as a transfer energy rather than the movement of a particle.
*Particles form the medium through which waves propagate but they are not the wave. This will become clearer later.

01 WAVES MODEL

We can no see them, but we can describe them with a model

Crest:
*Region of maximum amplitude.
*The crest of a wave is the point on the medium that exhibits the maximum amount of positive or upward displacement from the rest position

Trough (Valle):
*Region of minimum amplitude.
*The trough of a wave is the point on the medium that exhibits the maximum amount of negative or downward displacement from the rest position.
*Points C and J on the diagram represent the troughs of this wave.

Node:
A standing wave is a pattern which results from the interference of two or more waves traveling in the same medium.  All standing waves are characterized by positions along the medium which are standing still.  Such positions are referred to as nodes. 

Amplitude (A):

*The characteristic height of a peak and depth of a trough is called the amplitude of the wave.
*The amplitude is defined as the maximum displacement of an object from its resting position.
*Normally the letter A is used for the amplitude of a wave. The units of amplitude are meters (m).

EXERCISE

The amount of energy carried by a wave is related to the amplitude of the wave. A high energy wave is characterized by a high amplitude; a low energy wave is characterized by a low amplitude.

A mass is tied to a spring and begins vibrating periodically.
The distance between its highest and its lowest position is 38 cm. What is the amplitude of the vibrations?

Answer: 19 cm
The distance that is described is the distance from the high position to the low position.
The amplitude is from the middle position to either the high or the low position.

Wavelength λ:

*The distance between two adjacent (next to each other) peaks is the same no matter which two adjacent peaks you choose. So there is a fixed distance between the peaks.
*Distance between one wave peak to the next.
*The units are meters (m).
*The wavelength of a wave is simply the length of one complete wave cycle.

Frequency (f):

*The frequency is the number of waves occurring within a unit of time (1 second).
*The frequency is defined as the number of complete cycles occurring per period of time
*Since the standard metric unit of time is the second, frequency has units of cycles/second, Hertz.

Period (T):

Now imagine you are sitting next to a pond and you watch the waves going past you. First one peak, then a trough, and then another peak. If you measure the time between two adjacent peaks you'll find that it is the same.
Waves have a characteristic time interval which we call the period of the wave and denote with the symbol T
The period is equal the time elapsing during the passage of one full wavelength and is measured in seconds.

This reciprocal relationship is easy to understand. After all, the two quantities are conceptual reciprocals (a phrase I made up). Consider their definitions as restated below:
period = the time for one full cycle to complete itself; i.e., seconds/cycle
frequency = the number of cycles that are completed per time; i.e., cycles/second


Example:


According to Wikipedia (and as of this writing), Tim Ahlstrom of Oconomowoc, WI holds the record for hand clapping.
He is reported to have clapped his hands 793 times in 60.0 seconds.
What is the frequency and what is the period of Mr. Ahlstrom's hand clapping during this 60.0-second period?

Answer:
In this problem, the event that is repeating itself is the clapping of hands; one hand clap is equivalent to a cycle.
Frequency = cycles per second
= 793 cycles/60.0 seconds = 13.2 cycles/s
= 13.2 Hz
Period = seconds per cycle
= 60.0 s/793 cycles = 0.0757 seconds

Exercise 01:
A pendulum is observed to complete 23 full cycles in 58 seconds.
Determine the period and the frequency of the pendulum.

Answer:
The frequency can be thought of as the number of cycles per second. Calculating frequency involves dividing the stated number of cycles by the corresponding amount of time required to complete these cycles. In contrast, the period is the time to complete a cycle. Period is calculated by dividing the given time by the number of cycles completed in this amount of time.
Frequency = 23 cycles/58 seconds
= 0.39655 Hz = ~0.40 Hz
Period = 58 seconds/23 cycles
= 2.5217 sec = ~2.5 s

Speed:

Now if you are watching a wave go by you will notice that they move at a constant velocity.
The speed is the distance you travel divided by the time you take to travel that distance.
This is excellent because we know that the waves travel a distance λ in a time T.

Speed = Wavelength / Period
Speed = Wavelength • Frequency v = (λ)(f)


A Wave Transports Energy and Not Matter


When a wave is present in a medium (that is, when there is a disturbance moving through a medium), the individual particles of the medium are only temporarily displaced from their rest position. There is always a force acting upon the particles that restores them to their original position. In a slinky wave, each coil of the slinky ultimately returns to its original position. In a water wave, each molecule of the water ultimately returns to its original position. And in a stadium wave, each fan in the bleacher ultimately returns to its original position. It is for this reason, that a wave is said to involve the movement of a disturbance without the movement of matter. The particles of the medium (water molecules, slinky coils, stadium fans) simply vibrate about a fixed position as the pattern of the disturbance moves from one location to another location.

Exercises


1. A medium is able to transport a wave from one location to another because the particles of the medium are ____.
a. frictionless
b. isolated from one another
c. able to interact
d. very light 

Answer: C
For a wave to be transmitted through a medium, the individual particles of the medium must be able to interact so that they can exert a push and/or pull on each other; this is the mechanism by which disturbances are transmitted through a medium.

2. In order for John to hear Jill, air molecules must move from the lips of Jill to the ears of John.
Answer ==> False.
A sound wave involves the movement of energy from one location to another, not the movement of material.
The air molecules are the particles of the medium, and they are only temporarily displaced, always returning to their original position.

LONGITUDINAL VS TRANSVERSE WAVES

Transverse waves

A transverse wave is a wave in which particles of the medium move in a direction perpendicular to the direction that the wave moves. Suppose that a slinky is stretched out in a horizontal direction across the classroom and that a pulse is introduced into the slinky on the left end by vibrating the first coil up and down. Energy will begin to be transported through the slinky from left to right.

As the energy is transported from left to right, the individual coils of the medium will be displaced upwards and downwards. In this case, the particles of the medium move perpendicular to the direction that the pulse moves. This type of wave is a transverse wave. Transverse waves are always characterized by particle motion being perpendicular to wave motion.

Exercises:

1. Mac and Tosh are experimenting with pulses on a rope.
They vibrate an end up and down to create the pulse and observe it moving from end to end.
How does the position of a point on the rope, before the pulse comes, compare to the position after the pulse has passed?

Answer:

The point returns to its original position.
Waves (and pulses) do not permanently displace particles from their rest position.

2. A transverse wave is transporting energy from east to west.
The particles of the medium will move_____.
a. east to west only
b. both eastward and westward
c. north to south only
d. both northward and southward

Answer: D
*The particles would be moving back and forth in a direction perpendicular to energy transport.
*The waves are moving westward, so the particles move northward and southward.

Longitudinal waves:

A longitudinal wave is a wave in which particles of the medium move in a direction parallel to the direction that the wave moves. Suppose that a slinky is stretched out in a horizontal direction across the classroom and that a pulse is introduced into the slinky on the left end by vibrating the first coil left and right. Energy will begin to be transported through the slinky from left to right.

As the energy is transported from left to right, the individual coils of the medium will be displaced leftwards and rightwards. In this case, the particles of the medium move parallel to the direction that the pulse moves. This type of wave is a longitudinal wave. Longitudinal waves are always characterized by particle motion being parallel to wave motion.

Exercises

1. Describe how the fans in a stadium must move in order to produce a longitudinal stadium wave.

Answer:
The fans will need to sway side to side.
Thus, as the wave travels around the stadium they would be moving parallel to its direction of motion.
If they rise up and sit down, then they would be creating a transverse wave.

Sound is a longitudinal wave

A sound wave traveling through air is a classic example of a longitudinal wave. As a sound wave moves from the lips of a speaker to the ear of a listener, particles of air vibrate back and forth in the same direction and the opposite direction of energy transport. Each individual particle pushes on its neighboring particle so as to push it forward. The collision of particle #1 with its neighbor serves to restore particle #1 to its original position and displace particle #2 in a forward direction.

This back and forth motion of particles in the direction of energy transport creates regions within the medium where the particles are pressed together and other regions where the particles are spread apart. Longitudinal waves can always be quickly identified by the presence of such regions. This process continues along the chain of particles until the sound wave reaches the ear of the listener.

Exercise:

Curly and Moe are conducting a wave experiment using a slinky. Curly introduces a disturbance into the slinky by giving it a quick back and forth jerk. Moe places his cheek (facial) at the opposite end of the slinky. Using the terminology of this unit, describe what Moe experiences as the pulse reaches the other end of the slinky.

Answer:
When the slinky reaches the end of the slinky and hits Moe in the cheek, Moe experiences a pulse of energy. The energy originated on Curly's end and is transported through the medium to Moe's end. The last particle on Moe's end transports that energy to Moe's cheek.

Exercise:

A wave is transporting energy from left to right. The particles of the medium are moving back and forth in a leftward and rightward direction.
This type of wave is known as a ____.
a. mechanical
b. electromagnetic
c. transverse
d. longitudinal

Answer: D
The particles are moving parallel to the direction that the wave is moving.
This must be a longitudinal wave.

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