4.4.1describe a wave pulse and a continuous progressive (traveling) wave. 4.4.2state that...

4.4.1 Describe a wave pulse and a continuous progressive (traveling) wave.

4.4.2 State that progressive (traveling) waves transfer energy.

4.4.3 Describe and give examples of transverse and of longitudinal waves.

4.4.4 Describe waves in two dimensions, including concepts of wavefronts and of rays.

4.4.5 Describe the terms crest, trough, compression and rarefaction.

4.4.6 Define the terms displacement, amplitude, frequency, period, wavelength, wave speed, and intensity.

Topic 4: Oscillations and waves4.4 Wave characteristics

Describe a wave pulse and a continuous progressive (traveling) wave.

Consider a taught rope which is anchored securely to a table:

You can send a single wave pulse through the rope by moving your hand up and then down exactly once:

Or you can repeat the motion to produce a continuous traveling wave:


FYI

Note that the rope doesn’t travel to the right. The rope particles vibrate up and down.

Describe a wave pulse and a continuous progressive (traveling) wave.

Instead of using a rope to transmit a wave let’s use a spring:

You can send a single wave pulse through the spring by moving your hand forward and backward exactly once (push and pull):

Or you can repeat the motion to produce a continuous traveling wave:


FYI

Note that the spring doesn’t travel to the right. The spring particles vibrate left and right.

State that progressive (traveling) waves transfer energy.

Consider a taught rope which is anchored securely to a table:

If we send a single wave pulse through the rope we see that when it reaches the end, it can do work on the mass:

Note that in this case work was done against gravity in the form of an increase in the gravitational potential energy of the mass.

You can think of the energy being transferred from your hand to the mass via the momentum or EK of the particles while they vibrate.


∆EP = mg∆h

∆h

Describe and give examples of transverse and of longitudinal waves.

Both the rope and the spring were examples of traveling waves, and both traveled in the positive x-direction.

We call the material through which a wave propagates the medium. So far we have seen examples of two mediums: rope and spring steel.

The rope transferred its wave pulses by vibrations which were perpendicular to the direction of the wave velocity.

Any wave produced by vibrations perpendicular to the wave direction is called a transverse wave.


v


The spring transferred its wave pulses by vibrations which were parallel to the direction of the wave velocity.

Any wave produced by vibrations parallel to the wave direction is called a longitudinal wave.


v



PRACTICE:Categorize a water wave as transverse, or as longitudinal.If you have ever been fishing and used a bobber you should know the answer:Firstly, the wave velocity is to the left.Secondly, the bobber vibrates up and down.Thus the water particles vibrate up and down.Thus water waves are transverse waves.

vTransverse waves are perpendicular to the

wave velocity.

EXAMPLE:

Consider a speaker cone which is vibrat- ing due to electrical input in the form of music.

As the cone pushes outward, it squishes the air molecules together in a process called compression.

As the cone retracts, it separates the air molecules in a process called rarefaction.

Since the vibrations are parallel to the wave velocity, sound is a longitudinal wave.



v


A “microscopic” view of a sound wave may help:


Pul

se G

ener

ator

FYI

As you watch the simplified animation observe…

there is a pulse velocity v.there is a compression or condensation.there is a decompression or rarefaction.the particles are displaced parallel to v.

Suppose the distance from the generator to the wall is 5 m and the pulse took 22 s to reach the wall. Then the speed of sound in this medium is v = 5 m / 22 s = 0.23 m s-1.

As you watch this animation look at the circular wave fronts as they travel through space from the sound source.

Observe further that the waves in the red sectors are out of phase with the waves in the blue sector. By how much?

Describe waves in two dimensions, including concepts of wavefronts and of rays.

Looking at a snapshot of the previous 2D animation we can label various parts:

The wavefronts are located at the compressions.

The rays are drawn from the source outward, and show the direction of the wave velocity.


FYI

Rays and wavefronts are perpendicular.

Describe the terms crest, trough, compression and rarefaction.

Compare the waves traveling through the mediums of rope and spring.


CREST

TROUGH

COMPRESSION

TRANSVERSE WAVE

LONGITUDINAL WAVE

RAREFACTION

Displacement y

x

Define displacement, amplitude, frequency, period, wavelength, wave speed, and intensity.

Here is an animation of transverse wave motion created by placing each of the blue particles of the medium in simple harmonic motion.

As you watch the animation note-each particle has the same period T.

-each particle is slightly out of phase.

-the wave crest appears to be moving left.



Consider a snapshot of the following identical mass/spring systems, each of which is oscillating at the same period as the system to the right.

Note that they are all out of phase in such a way that they form a wave as you move in the x-direction.

At each position x we have a different value y.The systems at x1 and x2 are ¼ cycle out of phase.


x

y

x1 x2


Now we see the same system a short time later:The mass at x1 has gone lower.

The mass at x2 has gone lower.

Which way does it appear the wave is traveling? Left or right?


x

y

x1 x2

x

y

x1x2

t1 (from last slide)

t2 (a short time later)


If we look at either of the graphs we can define various wave characteristics:

The signed distance from the equilibrium position is called the displacement. In this graph it would be the y value.

At a horizontal coordinate of x1 along the length of the wave train we see that its displacement y is (-), whereas at x2 we see that y is (+).

The amplitude is the maximum displacement. The amplitude is just the distance from crest to the equilibrium position.


x

y

x1x2



If we look at either of the graphs we can define various wave characteristics:

The length in the horizontal dimension over which a wave repeats itself is called a wavelength represented with the symbol (the Greek lambda).

The wavelength is the distance from crest to crest (or trough to trough).

The period T is the time it takes a wave crest to travel exactly one wavelength.


x

y

x1x2



The speed at which a crest is moving is called the wave speed. This is really a measure of the rate at which a disturbance can travel through a medium.

Since the time it takes a crest to move one complete wavelength () is one period (T), the relation between v, and T is

Finally frequency f measures how many wave crests per second pass a given point and is measured in cycles per second or Hz. Again, f = 1/T.


v = /T relation between v, and T

f = 1/T relation between f and T



PRACTICE: A spring is moved in SHM by the hand as shown. The hand moves through 1.0 complete cycle in 0.25 s. A metric ruler is placed beside the waveform.

(a) What is the wavelength?

= 4.7 cm = 0.047 m.(b) What is the period?

T = 0.25 s.(c) What is the wave speed?

v = /T = 0.047/0.25 = 0.19 m s-1.

1234567891011121314 CM


Intensity is the rate energy is being transmitted per unit area and is measured in (W m-2).


I = power/area definition of intensity

EXAMPLE:

A 200. watt speaker projects sound in a spherical wave. Find the intensity of the sound at a distance of 1.0 m and 2.0 m.

Note that whatever power is in the first wavefront is also in all the subsequent ones.

The area of a sphere of radius r is A = 4r2.

For r = 1 m: I = P/(4r2) = 200/412 = 16 W m-2.

For r = 2 m: I = P/(4r2) = 200/422 = 4.0 W m-2.

Doubling your distance reduces intensity by 75%!

4.4.7 Draw and explain displacement-time graphs and displacement-position graphs for transverse and longitudinal waves.

4.4.8 Derive and apply the relationship between wave speed, wavelength and frequency.

4.4.9 State that all electromagnetic waves travel with the same speed in free space.


EXAMPLE:

Graph 1 shows the variation with time t of the displacement d of a traveling wave. Graph 2 shows the variation with distance x along the same wave of its displacement d.

(a) Use the graphs to determine the amplitude of the wave motion.

Amplitude (maximum displacement) is 0.004 m.

Draw and explain displacement-time graphs and displacement-position graphs for transverse and longitudinal waves.


Either graph gives the correct

amplitude.

EXAMPLE:


(b) Use the graphs to determine the wavelength.

Wavelength is measured in meters and is the length of a complete wave. = 2.4 cm = 0.024 m.



Graph 2 must be used since its horizontal axis is in cm (not s as in Graph 1).

EXAMPLE:


(c) Use the graphs to determine the period.

Period is measured in seconds and is the time for one complete wave. T = 0.3 s.



Graph 1 must be used since its horizontal axis is in s (not cm as in Graph 2).

EXAMPLE:


(d) Use the graphs to find the frequency.

This can be calculated from the period.f = 1/T = 1/0.3 = 3 Hz. [3.333 Hz]



EXAMPLE:


(e) Use the graphs to find the wave speed.

This can be calculated from and T.v = /T = 0.024 / 0.3 = 0.08 m s-1.



PRACTICE: Graph 1 shows the variation with time t of the displacement y of a traveling wave. Graph 2 shows the variation with distance x along the same wave of its displacement.

(a) Use the graphs to determine the amplitude and wavelength of the wave motion.

Amplitude (maximum displacement) is 0.002 m.Wavelength is 0.3 cm = .003 m.



Graph 2 must be used for since its horizontal axis is in cm.


(b) Use the graphs to determine the period and the frequency.

Period (cycle time) is 0.25 ms = 0.00025 s.Frequency is calculated from period.f = 1/T = 1/0.00025 = 4000 Hz.



Graph 1 must be used for T since its horizontal axis is in ms.


(c) Use the graphs to determine the wave speed.

Wave speed is a calculation.

v = /T = 0.003/0.00025 = 10 m s-1 [12 m s-1].



EXAMPLE:

Graph 1 shows the variation with time t of the displacement x of a single particle in the medium carrying a longitudinal wave moving in the +x direction.

(a) Use the graph to determine the period and the angular frequency of the particle’s SHM.

The period is the time for one cycle. T = 0.2 s. = 2/T = 2/0.2 = 30 rad s-1. [31.4]



Graph 1

EXAMPLE:


(b) Use the graph to determine the maximum velocity of the particle.

The particle’s amplitude is x0 = 2 cm = .02 m.

v0 = vmax = x0 = 31.4(0.02) = 0.6 m s-1.



Graph 1

EXAMPLE:


(c) Use the graph to determine the maximum acceleration of the particle.

amax = 2x0 = 31.42(0.02) = 20 m s-2.



Graph 1

EXAMPLE:


(d) Use the graph to determine the velocity of the particle at t = 0.125 s.

v is the slope of the tangent in x vs. t:v = ∆x/∆t = 0.04/0.9 = .04 m s-1 (+x-dir).



Graph 1 rise = .04 m

Always maximize your slope

triangles for accuracy.

run = 0.9 s

EXAMPLE:

Graph 2 shows the variation of the displacement x with distance d from the beginning of the wave at a particular instant in time.

(e) Use the graph to determine the wavelength and wave velocity of the longitudinal wave motion.

Wavelength is the length of a complete cycle. = 16 cm = 0.16 m.v = /T = 0.16/0.2 = 0.8 m s-1.



Graph 2

EXAMPLE:


(f) The above diagram shows the equilibrium position of 6 particles in the medium. Using ’s, indicate the actual positions of each of the six particles at the instant shown in Graph 2.



Graph 2

EXAMPLE:


(g) In the diagram above label the center of a compression with a C and the center of a rarefaction with an R.



Graph 2

C R

Derive and apply the relationship between wave speed, wavelength and frequency.

From the above relations we can very quickly derive the last formula for this section:

v = /T v = (1/T) v = f.


v = /T relation between v, and T

f = 1/T relation between f and T

v = f relation between v, and f

EXAMPLE:

A traveling wave has a wavelength of 2.0 cm and a speed of 75 m s-1. What is its frequency?

Since v = f we have 75 = .02f or f = 3800 Hz.

State that all electromagnetic waves travel with the same speed in free space.

All of us are familiar with light. But visible light is just a tiny fraction of the complete electromagnetic spectrum.


104 106 108 1010 1012 1014 1016 1018

Frequency f / Hz

Radio, TV Cell Phones Infrared Light X-Rays

The Electromagnetic Spectrum Microwaves Ultraviolet Light Gamma Rays

700 600 500 400Wavelength / nm

State that all electromagnetic waves travel with the same speed in free space.

In free space (vacuum), all electromagnetic waves travel with the same speed v = 3.00108 m s-1.

We use the special symbol c for the speed of light. Thus


c = f relation between c, and f

where c = 3.00108 m s-1

PRACTICE: The wavelength of a particular hue of blue light is 475 nm. What is its frequency?

1 nm is 110-9 m so that = 47510-9 m.

c = f so that 3108 = (47510-9)f.

f = 6.321014 Hz.

700 600 500 400Wavelength / nm

4.4.1describe a wave pulse and a continuous progressive (traveling) wave. 4.4.2state that...

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