Physics 20 - STANote Booklet

Unit 4 – Oscillatory Motion and WavesChapter 8 – Mechanical Waves

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Contents

8.1 Properties and Types of Waves

8.2 Reflection, Superposition, and Interference

8.3 Resonance and the Doppler Effect

8.1 PROPERTIES AND TYPES OF WAVES

Introduction

Energy can be transmitted from one place to another by either particles or waves.

To transfer energy with a particle, it must move and physically strike something.

A wave transfers energy without transferring matter.

A MECHANICAL WAVE is a wave that needs a medium to travel through.

The particles of the medium vibrate with simple harmonic motion as the energy travels through the medium.

*A wave front is an imaginary line that connects all the points reached by the wave at the same time.

*A wave front moving out for the point of origin toward a barrier is called an incident wave.

*A wave front moving away from the barrier is called a reflected wave.

* A ray is a line (or an arrow) that indicates the direction of the motion of the wave

*A series of waves connected together is called a wave train. This is a regular repetition of the motion of the medium through which the wave travels.

Transverse Waves

A transverse wave causes the particles of the medium to vibrate perpendicularly to the direction of wave propagation (ie. the direction the wave travels).

The bobbing of a float on a pond is like the bobbing of a mass on the end of a spring. It is in simple harmonic motion.

Transverse waves may occur along a string, on the surface of a liquid and through a solid.

Each point in the medium vibrates regularly in response to the travelling wave.

Longitudinal Waves

A longitudinal wave causes the particles of the medium to vibrate parallel to the direction of wave propagation (ie. the direction the wave travels).

crest

crest

compression

compression

compression

rarefaction

rarefaction

1Tf

1fT

and

A

A

A

A

The best example of longitudinal waves is sound waves in air.

Properties

The wavelength of a wave is the shortest distance between two points on a wave where the wave pattern repeats itself.

i) Transverse wave

ii) Longitudinal wave

For longitudinal waves, the wavelength is measured as the distance between the middle of compressions, or the middle of rarefactions.

The frequency of a wave is the number of cycles per second measured at a fixed location, or, the number of wavelengths that pass a point in one second. The period of a wave is the time it takes (in seconds) for a wave to travel a distance of one wavelength, or, the shortest time interval during which the motion repeats itself. Remember that frequency and period are the inverse of each other:

The amplitude of a wave is the maximum displacement from the equilibrium position. It is an indication of the amount of energy that the wave is carrying.

λ

λ

λ

λ

λ

The universal wave equation is used to calculate the speed of travelling waves.

The speed of a wave depends only on the medium that it is travelling through. For example, a sound wave will move much more quickly through liquid than through the air.

Examples:

1) An ocean wave has a wavelength of 12.0 m and is moving at 4.40 m/s. How long does it take the wave to travel a distance of one wavelength? i.e. what is the period of the wave?

2) Power 92 (now Joe FM) broadcasts bad music at a frequency of 92.5 MHz. If radio waves travel at the speed of light (v = 3.00 x 108 m/s), what is the wavelength of the radio signal?

3) You are watching a beach ball bob up and down on the surface of a lake as waves pass by. You notice that it bobs up and down 50 times in 3.0 minutes. You estimate the distance between wave crests is 2.5 m. What is the speed of the waves?

uniform motion:

dvt

but for a wave . . .

vT

and since

1fT

v f

v = speed (m/s)f = frequency (Hz)λ = wavelength (m)

8.2 REFLECTION, SUPERPOSITION, AND INTERFERENCE

Interference

When two or more waves travel through the same point in a medium, they each affect the medium independently.

The principle of superposition states:

“The displacement of a medium caused by two or more waves is the algebraic sum of the

displacement of the waves.”

The superposition of two or more waves is called

interference. Interference can either be

constructive or destructive.

When the waves are in phase, the displacements of the medium are in the same direction. Constructive interference results when a crest meets a crest, or when a trough meets a trough.

1)

2)

3)

4) The pulses are not changed by

their interaction.

They are still two separate waves, they just happen to be in the same spot. These two pulses are superimposed.

1)

2)

3)

When the waves are out of phase, the displacements of the medium are in opposite directions.

Destructive interference results when a crest meets a trough.

Reflection

Total destructive interference : occurs when a crest and a trough completely cancel one another out.

Total constructive interference : occurs when a crest meets another crest and adds together to make a bigger crest

When a wave passes from one medium into another (eg. from air into water), some of the energy is transmitted, and some is reflected from the boundary between the two media.

• incident wave: the original wave that came moving in through the first medium

• transmitted wave: the wave that continues into the new medium

• reflected wave: the wave that bounces back

Different things will happen, depending on the densities of the two media, relative to each other.

1) From a less dense medium to a more dense medium: (example: air to water)

normal line

i r

incident ray

reflected ray

i = angle of incidence

r = angle of reflection

If a line is normal to another line, they are perpendicular to one another.

The law of reflection states:“The angle of

incidence equals the angle of

reflection.”

All angles are measured with respect to the normal, NEVER WITH THE SURFACE.

2) from a more dense medium to a less dense medium (example: water to air)

incident pulse

transmitted pulse

When the incident pulse reaches the boundary the transmitted pulse is right side up (erect) and the reflected pulse is inverted.

Ex. A transverse wave pulse passes from a light string to a heavy rope.

A pulse reaching the end of a medium becomes inverted whenever it either:

reflects off a fixed end

or is moving in a less dense medium and reflects off a more dense medium.

8.3 RESONANCE AND THE DOPPLER EFFECT

Resonance

incident pulse

transmitted pulse

reflected pulse

When the incident pulse reaches the boundary , both the transmitted pulse and the reflected pulse are erect.

A transverse wave pulse passes from a heavy rope to a light string.

A pulse reaching the end of a medium does not become inverted whenever it either

reflects off a free end,

or is moving in a more dense medium and reflects off a less dense medium.

The amplitude of any vibrating object can be greatly increased by applying small external forces at specific regular intervals of time. The time interval between applied forces must be equal to the period of the oscillations.

This effect is called mechanical resonance.

In sound applications, a resonant frequency is a natural frequency of vibration determined by the physical parameters of the vibrating object.

Standing waves consist of a series of stationary nodes ( N ) and antinodes ( A ).

Nodes are regions which always appear to be standing still. They are regions with no displacement. They occur when the reflected wave destructively interferes with the incident wave.

Antinodes are regions where the medium has the maximum displacement. They occur when the reflected wave constructively interferes with the incident wave.

Resonance and Musical Instruments

-musical instruments produce sound from air resonating through the tubes.

- All instruments like this can be divided into two categories, open ended or closed ended.

*An open ended instrument has both ends open to the air.

*A closed ended instrument has one end closed off, and the other end open.

• The sounds the instrument columns make depends on how much of a wavelength can fit into the column.

• Different wavelengths in the column produce different frequencies being produced. (AKA sound pitch)

• The frequencies that are produced by these waves are the natural frequencies and we call them resonant frequencies.

• In music these are referred to as harmonics.

• Pushing different keys or controlling how hard you blow into the instrument contributes to the differences in notes. Ex. High C or Low C

• The lowest note you can play is the minimum wavelength that can fit into the column. This note is called the Fundamental Note.

● The air at the closed end of the column must be a node (not moving), since the air is not free to move there and must be able to be reflected back.● There must also be an antinode where the opening is, since that is where there ismaximum movement of the air.

The length of the wave above is ¼ of a wavelength. (even though it might not look like it.) Draw a wavelength and see.

• As more waves fit into the column more harmonics are played. They are all multiples of the fundamental wavelength.

a) Closed ended columns

All of the harmonics in closed end columns are going to be odd numbers!

b) Open ended columns

Open end columns can have any number harmonic they want, odd or even.

● The fundamental (first harmonic) for an open end column needs to be an antinode at both ends, since the air can move at both ends.

A tunning fork is struck and produces a frequency of 256 Hz. Determine the wavelength of the sound wavelength produced if the speed of sound is 343 m/s.

A transverse water wave has an amplitude of 48 mm and a wavelength of 86mm. What is the vertical displacement from the top of a crest to the bottom of a trough?What is the length of a trough?

(0.3943 m)

The Doppler Effect

The Doppler effect is the perceived change in frequency of sound emitted by a source moving relative to the observer: as a plane flies overhead, the note of the engine becomes noticeably lower, as does the siren noise from a fast-moving emergency vehicle as it passes.

The effect is widely used to measure velocities, usually by reflection of a transmitted wave from the moving object, ultrasound for blood in arteries, radar for speeding cars and thunderstorms. The velocities of distant galaxies are measured using the Doppler effect (the red shift).

The circles are separated by one wavelength λ and they travel outwards at the speed of sound v.

The Doppler effect arises because once a moving source emits a circular wave (and provided the source is moving at less than the speed of the wave) the circular wave crest emitted continues its outward expansion centered on where the source was when it was emitted, independent of any subsequent motion of the source. Therefore, if the source is moving at a steady speed, the centers of the emitted circles of waves will be equally spaced along its path, indicating its recent history. In particular, if the source is moving steadily to the left, the wave crests will form a pattern:

The wavelengths are closer in front of the moving source (higher frequency) and further apart behind the moving source (lower frequency).

To determine the frequency the observer hears, we derive a formula using the wave equation and the uniform motion equation. Basically we are finding the distance the source moves and either subtracting from wavelength (moving toward you) or adding it to the wavelength (moving away from you). For a full derivation, see pages 430 and 431 in your textbook. The derived formula (found in your data booklet) is:

Where:f is the apparent frequency (frequency observer hears)

v is speed of sound in medium (usually air ~ 330 m/s)

vs is speed of source

fs is frequency of source

Remember to subtract vs if the source is moving toward the observer or add vs if the source is moving away from the observer.

Example 1: You are crossing the street when a car (the driver) blows its horn. If the true frequency of the horn is 264 Hz and the car is approaching you at a speed of 60.0 km/h, what is the apparent frequency of the horn? Assume the speed of sound in air is 340 m/s. [278 Hz]

Example 2: As a train moves away from you, the frequency of its whistle is determined to be 475 Hz. If the actual frequency of the whistle is 500 Hz and the speed of sound in air is 350 m/s, what is the train’s speed? [18.4 m/s]