Waves

Progressive waves.

Progressive waves are wave where energy is transferred by moving particles. As the wave passes through the material the particles vibrate about their equilibrium position, once the wave has passed the particles return to their original position.

The following progressive waves are all mechanical waves, they transfer energy through a material known as a medium.

Wave Medium Cause

Water waves

Wave in strings E.g. instruments

Seismic Waves

Sound waves

Types of wave.All waves can be grouped into 2 categories, transverse and longitudinal waves.

Longitudinal waves.Seismic p-waves and sound waves are example of longitudinal waves, the particles in longitudinal waves vibrate parallel to the direction the wave is travelling in.Sound waves travel in a series of compressions and rarefactions. In a compression the air pressure is relatively high as is the density, because the molecules are closer together, in a rarefaction the opposite is true.

Transverse waves.

In transverse waves, e.g. water, EM waves.

The movement of the particles in transverse is at right angles to the direction the waves travel.

(EM waves are actually linked electric and magnetic fields rather than particles)

All EM waves travel at 3x108 m/s in a vacuum.

Displacement-distance graphs.

These graphs allow waves to be frozen in time, they show how the displacement of particles depends on distance at a certain time.

distance

displacement

Displacement-time graphs

Rather than use the displacement distance graphs, we commonly concentrate on one point in the cycle and sketch how the displacement changes with time, although the traces look extremely similar it is important to remember that similar points are separated by a time period and not distance

time

displacement

The time taken for 1 wave to pass is known and the period (T), the period is related to the frequency by the formula f=1/T.

Phase difference.

The phase difference between two points on a wave or between 2 waves is expressed as an angle.

One whole cycle represents an angle of 360, two points with a phase difference of 360 are in phase with each other.

If two points have a phase difference of 180 they are out of phase.

We can also express phase difference using radians where there are 2 rad in every complete wave cycle.

Path difference

If we express the difference between points of a wave or between waves in terms of wavelength then we sometimes talk about path difference. Where is the wavelength.

A points on a wave with phase difference of 90 will be a least ¼ apart.

Path difference Phase difference (deg) Phase difference (rad)

1, 2, 3 …. 360, 720, 1080….. 2,4, 6 …

/2, 3/2, 5/2 …. 180, 540, 900….. ,3, 5 …

/4, 5/4, 9/4 …. 90, 450, 810….. 2,4, 6 …

/3, 4/3, 7/3 …. 120, 480, 840….. 2/3,8/3, 14/3 …

For 2 coherent waves of equal wavelength

Speed of a wave.

The speed of a wave, c, is related to both its wavelength, , and its frequency.

The formula for the speed of a wave is:

c = f

We can use this formula to calculate the speed, frequency or wavelength of any wave so long as the other two are known.

Wave Frequency Wavelength Speed Medium

Sound 2000 Hz 340 m/s In air

Yellow light 3.6 E14 Hz 0.61 um water

X-rays 0.1nm In air

Microwave 6GHz In vacuum

Water ripples 8Hz 400 m/s

sonar 30 kHz 45mm In water

Radio from space

0.8m

Ultra sound 3.5 MHz 1600 m/s In body tissue

Superposition of waves.

There are a number of things that can happen when waves are superposed on each other.

These 2 waves constructively interfere to form a larger wave. Waves can also destructively interfere, this will make the wave smaller.

=

=

Constructive.

(Amplitude increases)

=

=

Destructive.

(Amplitude decreases)

+

+

+

+

Ripple tank.

• http://www.ngsir.netfirms.com/englishhtm/Interference.htm

• http://phet.colorado.edu/simulations/sims.php?sim=Wave_Interference

The animation shows two sinusoidal waves travelling in the same direction. The phase difference between the two waves varies increases with time so that the effects of both constructive and destructive interference may be seen.

Standing waveIf two sinusoidal waves having the same frequency (wavelength) and the same amplitude are travelling in opposite directions.When the two waves are 180° out-of-phase with each other they cancel, and when they are in-phase with each other they add together.

BeatsTwo waves with slightly different frequencies are travelling to the right. The resulting wave travels in the same direction and with the same speed as the two component waves. The "beat" wave oscillates with the average frequency, and its amplitude envelope varies according to the difference frequency.

Standing waves in pipes and strings

Pipes and strings produce notes when they vibrate at certain frequencies. These vibrations set up standing waves in the pipes or the strings.

There are limits to the waves that are able to resonate and these limits are related to the length of the string or pipe.

Standing waves in strings.

The first standing wave to form on a string has a wavelength of /2, as the sting is a fixed length and the speed of sound is constant then there is only one frequency that can cause this to happen. This is the fundamental frequency of the string.

There are other higher frequencies that will cause the string to vibrate in a standing wave, these are called the overtone frequencies.

Fundamental frequency. l = /2

1st overtone. l =

2nd overtone. l = 3/2 (1.5)

3rd overtone. l = 2

4th overtone. l = 5/2 (2.5)

5th overtone. l = 3

6th overtone. l = 7/2 (3.5)

node nodeAnti-node

n n na a

The first point at which the string will resonate is when the length of the string is equal to half of the wave length determined by the fundamental frequency.

StringUnobserved 2nd half of the wave

ExperimentSig Gen

Vibrator

String of known length l

50 Hz

Use the sig gen to set up the standing wave with the natural frequency of the string. Use the wave formula to predict the frequency of the next 4 standing waves.

Test your hypothesis

f1 f2 f3 f4 f5

1 2 3 4 5

c1 c2 c3 c4 c5

In a pipe with one enclosed end the natural (fundamental) frequency sets up a standing wave with a wavelength of /4 the overtones then go up in steps on /2

For a pipe with an enclosed end to resonate the standing must have a node at the closed end and an anti-node at the open end.

The first situation when this arises is when the wave created by the fundamental frequency has a wave length 4 times greater than the length of the pipe.

L= /4 or = 4L

Each further fundamental needs an extra half wave in the pipe to satisfy the conditions so the next fundamental will have a wave length of = 4/3 L or a length of L= 3/4

In a pipe with open ends the natural (fundamental) frequency sets up a standing wave with a wavelength of /2 the overtones then go up in steps on /2

ExperimentA speaker playing a constant tone is placed above a tube containing a fluid, the size of the air column above the fluid can be altered by draining the fluid.

Allow the fluid to slowly drain until the tube starts to resonate, this is the fundamental harmonic.

In this experiment you would expect the first resonance to occur when l = /4

Find the next harmonic, the difference between the two water levels = /2

Speaker playing constant tone

From the length of the tube and the frequency of the source you should be able to calculate the speed of sound in the air column.

Speaker playing constant tone

l