Unit 3 - Foundations of Waves - Chapter 5 - Properties of Waves · 2019-05-13 · Vibrations Vibrations are the periodic or repeated motion of a particle. There are two types of vibrations

Unit 3 - Foundations of WavesChapter 5 - Properties of Waves

Mr. Palmarin Chapter 5 - Properties of Waves 1 / 34

Section 5.1 - Characteristics of Waves

Waves transmit energy - A wave moves and carries energy.

Waves travel in a medium - Waves generally need a medium suchas water, springs or air to travel in, but this is not always the case(light, radio, x-rays don’t)

Waves have to be started - the medium has to be disturbed to geta wave started. Whatever starts the wave is called the wave source.

Waves pass through one another - this property is called thePrinciple of Superposition. This characteristic distinguishes wavesfrom particles.

Speed of the Wave - The speed of the wave depends upon the typeof medium the wave is moving through.


Vibrations

Vibrations are the periodic or repeated motion of a particle. There are twotypes of vibrations.

1 Transverse - vibrations are perpendicular to its axis.

2 Longitudinal - vibrations are parallel to its axis. For example, whenyou push a spring back and forth.


Pulse Waves

A pulse wave is a single disturbance that travels through a medium. Awave pulse may change its size and shape as it moves. If the disturbancecontinues in a periodic way, a regular series of pulses can be generated.This series of pulses is referred to as a train of waves.


Transverse Wave Characteristics

1 Wavelength - In a transverse wave a wavelength is generallyconsidered the distance from crest to crest. A wavelength would alsobe the distance from trough to trough, or any two corresponding pairsof points that are in-phase.

The amplitude of the wave is the maximum displacement of thewave.

Note: A wavelength is defined as the distance travelled by a wave duringone complete cycle of a vibrating source. It is represented by the Greekletter lambda, λ.


Example: Using the diagram below, determine:

a) The pairs of points that are in-phase:

“In-phase” means the points that are separated by exactly onewavelength. So, we have: (ab), (cd), (ef), (fg), (gh), (hi), and (ij).

b) The velocity of the waves if they take 2.5 s to travel from a to e. Thewavelength is exactly 3.0 m.

Recall from Science 10: v =∆d

∆t

If λ = 3.0 m then we must travel four wavelengths to go from “a” to“e”, which makes ∆d = (3.0)(4) = 12 m. Also, ∆t = 2.5 s so then,

v =12 m

2.5 s= 4.8 m/s


2 Frequency - The frequency of a wave is the number of completevibrations or wavelengths which pass a given point in one second.The derived unit of frequency is the hertz (Hz).

1Hz = 1 cycles/s

3 Period - The period is the time, T, required for a single wave to passa given point. That is,

Period = 1/frequency or more simply: T =1

f


Examples:

1) A mass hanging on a slow spring vibrates vertically 15 times in1.0× 101 s. Calculate:

a) Frequency

Always repeat to yourself: “Frequency is cycles per second”.

Therefore, “15 times in 10 s” =⇒ f =15 cycles

10 s=

1.5 cycles/s = 1.5 Hz

b) Period

Period(T)=1

f=⇒ T =

1

1.5 Hz= 0.67 s/cycle. The unit for period

typically omits the word “cycle”. So then we have: 0.67 s

2) A piano sound-wave has a period of 9.1× 10−3 s. What is itsfrequency?

T =1

f=⇒ f =

1

T=

1

0.0091 s= 110 Hz


4 Velocity - The velocity of any transverse wave can be determined byusing the universal wave equation which is stated as follows:

velocity =wavelength

period

Symbolically, we have:

v =λ

T, where T = 1/f

∴ v = λf


Examples:

1) What is the velocity of a wave with a frequency of 2.5 Hz and awavelength of 0.60 m?

v = λf and λ = 0.60 m; f = 2.5 Hz

=⇒ v = (0.60 m)(2.5 Hz) = (0.60 m/��cycle)(2.5��cycles/s) = 1.5 m/s

Note: It is very uncommon to show the cancellation step for the word“cycle”. You can certainly omit that step.

2) A train of waves moves with a velocity of 7.3 m/s. Determine thefrequency and the period of the source, if the wavelength is exactly11 cm.

v = λf =⇒ f =v

λ=⇒ f =

7.3 m/s

0.11 m= 66.36... =⇒ f = 66 Hz

T =1

f=⇒ T =

1

66.3 Hz=⇒ T = 0.015 s


Pendulums

As a pendulum swings it repeats the same motions in the same timeintervals. This is called periodic motion. The distance from rest tomaximum displacement is called the amplitude.

Two pendulums are said to be in-phase when the two pendulums are atrest or moving in the same direction at the same time.


Example: A pendulum swings back and forth exactly 20 times in 15 s.Calculate its period and its frequency. (Use 2 s.f.)

Repeat the frequency mantra: “Frequency is cycles per second”.

Therefore, f =20 cycles

15 s= 1.3 Hz

You have two options to compute period. First, you can use the definition:

T = second/cycle. Then, T =15 s

20 cycles= 0.75 s

Now if you choose to use T =1

f=, then you must take one extra s.f. if

you have computed f yourself. In this case, f = 1.3 Hz, but that isrounded from 1.33. So we must compute:

T =1

1.33 Hz= 0.75 s


Practice: A metronome is a device used to keep time for musicians. Itclicks every time it passes the centre (at 0◦). If a metronome (pendulum)clicks exactly 80 times in 20 s, what is its frequency and period?(Use 2 s.f.) [See Section 5.1 Video]


Section 5.2 - Study of Waves Formed by a Spring

Transmission and Reflection

Waves in a long spring travel at a uniform velocity as long as the mediumdoes not change.

Fixed end - When a wave reflects from fixed-end reflections (held byhand), the wave is inverted.

Open end - If the reflection occurs from an open end (hanging) where themedium is allowed to move, there is no inversion.


Wave Interference

When two waves traveling in opposite directions meet they pass throughone another unaffected. Wave Interference occurs when two waves actsimultaneously on the same particles of a medium.

Principle of Superposition - The displacement of a given particle of themedium is equal to the sum of the displacements that would have beenproduced by each wave independently.

Constructive Interference versus Destructive Interference


Constructive Interference - In general, the amplitude of the wave will bethe sum of the amplitudes of both waves. The point where the maximumamplitude occurs is called an antinode.

Destructive Interference - In general, the amplitude of the wave will bethe difference of the individual amplitudes. When these two waves meet apoint exists where no motion takes place. This point is called a node.


Examples:

Using the diagrams below, determine the resultant displacement using thePrinciple of Superposition.

See next slide for steps.


1 First, mark where the two waves overlap.

2 Next, place a reasonable amount of points on each wave in the regionof overlap.

3 Label each point with some value in relation to the centre axis. Here,I go by the fraction of “square” (from the grid) that a point is. Abovethe axis is positive, below is negative.

4 Now, add the corresponding points to give you where the resultantpoints will be.

5 You should now have a pattern of new points. Connect them,completing the resultant displacement wave.

See the next slide for the solution.


Zoom in to see the values of the points. The yellow line in Step 5 is thefinal answer. That is the wave that would be formed.


Recall: The yellow line is the final answer. That is the wave that would beformed when these two waves collide in this fashion.

I omitted labelling the points on the graph. The values were as follows (adding the

red dots from left to right to form the green ones):

(0.2+1.6=1.8), (0.4+1=1.4), (0.8+0.8=1.6), (0.4+1=1.4), (0.2+1.6=1.8)


Practice:

Using the diagrams below, determine the resultant displacement using thePrinciple of Superposition. [See Section 5.2 Video]


Standing Waves

Waves that appear to not move but in fact are continuously beinggenerated are called standing waves. By adjusting the frequency of thewaves you can create additional nodes in the waves where the wavescancel each other out.


Practice: Using measurements taken directly from the diagram below of astanding wave pattern, determine the speed of the waves if they movefrom the start to the end in 3.0 s. (Assume each square is 1 cm).

[See Section 5.2 Video]


Section 5.3 - Study of Waves Formed in Water

Shapes of Waves

Can be generalized as either circular or straight. Every wave has adefinite velocity. The speed of a wave depends mainly on the medium, butin water it depends upon the depth of the water.


Wave Reflection

Key points:1 When a wave strikes a barrier which is parallel to the wave source,

the wave strikes the barrier and is reflected back in the direction fromwhich it came.

2 When a wave strikes a barrier at an angle the wave will strike thebarrier and project itself at an opposite angle from its originaldirection.


3 When a circular wave strikes a straight barrier an arc is formed.

4 When a straight wave strikes a curved barrier the reflected wave isprojected towards a central point called a focus and then spreads outwith an increasing radius. This occurs because the incident andreflection angles are equal, causing destructive interference. Theythen pass through each other.

On the other hand when circular waves strike a curved barrier thereflected waves are straight.


Wave Refraction

The bending of a wave front when it passes from one medium to anotheris called wave refraction. Two conditions will be used in explainingrefraction of water waves.

1 The boundary between the media is parallel to the incident wavesproduced.

Since the waves travel from deep water to shallow water straight on, nobending occurs. The wavelength is less in the shallow water than in deepwater. The velocity must be less in the shallow region since v = f λ.


Therefore, we get the following generalization:

v1v2

=λ1λ2

Example: The speed and the wavelength of water waves in deep water are15.0 cm/s and 2.2 cm, respectively. If the velocity in shallow water is10.0 cm/s, what is the wavelength?

v1 = 15.0 cm/s; λ1 = 2.2 cm (deep water)v2 = 10.0 cm/s; λ2 =? (shallow water)

v1v2

=λ1λ2

=⇒ 15.0 cm/s

10.0 cm/s=

2.2 cm

λ2

=⇒ λ2 =(2.2 cm)(10.0 cm/s)

15.0 cm/s

λ2 = 1.5 cm


Practice: Waves travel 0.75 times as fast in shallow water as they do indeep water. What will be the wavelength of the waves in deep water, iftheir wavelength is 2.0 cm in shallow water? [See Section 5.3 Video]


2 The boundary between the media is not parallel to the incident wavesproduced. Or, the waves being produced are not parallel to theboundary.

Since the water wave is traveling at an angle to the shallow water thewave will change direction when it reaches this change of depth. This iscalled refraction.

In general, when a wave travels at an angle into a medium in which itsvelocity decreases, the refracted wave ray is bent towards the normal andif the wave velocity increases, the refracted wave is bent away from thenormal.


Wave Diffraction

The bending of waves when they pass through small openings or edges iscalled wave diffraction. How much they are diffracted depends mainly ontheir wavelength. Shorter wavelengths are diffracted slightly and longerwavelengths are diffracted a greater extent as shown in the diagramsbelow.


As the size of the opening decreases, the amount of diffraction increases asshown in the diagrams below.


Interference of Waves

Occurs when two or more waves act simultaneously on the same particlesof the medium. The diagram below shows the effect of constructive anddestructive interference on the waves.

Watch: ”The Slit Experiment”Mr. Palmarin Chapter 5 - Properties of Waves 33 / 34

https://www.youtube.com/watch?v=GDwRvv5NZLA&ab_channel=RogerLinsell

Practice: Sketch the waves in the diagram below after they have struck orpassed through or around the following barriers.

[See Section 5.3 Video]


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Unit 3 - Foundations of Waves - Chapter 5 - Properties of Waves · 2019-05-13 · Vibrations Vibrations are the periodic or repeated motion of a particle. There are two types of vibrations