1.2 Introduction
1.2 Constructive and Destructive Iinterference
1.2 Phase
1.2 Standing Waves
1.2 Summary
Unit 1.1 Multiple Choice Questions - Waves
Unit 1.1 Multiple Choice Answers - Waves
 

Unit 2 Superposition and Interference

guitar

Fig. SW 1

Standing Waves

Strings which are fixed at both ends.

I always wanted to play the guitar... I learnt music and we had a guitar at home so I attempted to play it. I wasn't very good or so my siblings say! Becoming discouraged I attempted plyed with the guitar and was facinated by the behaviour of the strings when they were plucked...

The pitch changed

— when the strigs were tightened!

— from string to string (different thickness)

— when the finger pressed the string and it was shortened.

Recall that if a wave is propagated on a string which is firmly attahed at one end, the reflected wave is out of phase with the incident wave. by Newton;s Third law the force for generating the reflected wave comes from the wall in response to the force from the incident wave on the wall. Being opposite it move the reflected wave in the opposite direction (Newton's first and second laws).

This standing oscillation is really the effect of two traveling waves - an interference pattern caused by an oncomming wave travelling forward and the reflected wave which travels backward along the string. The pattern called a standing or stationary wave.

 

standing wave 1

Fig. SW 2

standing waves2

Fig. SW 3

unit 2 standing waves3

Fig. SW 4

Consider my guitar string which is fixed at both ends...

If I pluck it to cause a wave, this simplest form will be like the diagram on the right.

As the string vibrates and the incident and reflected wave interact we get no movement at either end. These are called nodes where there has been destructive interfernce. At the centre where the string vibrates with maximum amplitude we have antinodes.

These remain fixed when I plucked a particular string if generated one and only one frequency. Actually this is the lowest frequency I could generate with that particular string with all parameters fixed.

Standing waves can occur at more than one frequency. These are called harmonics. The vibration frequencies at which standing waves can exist depend on two factors:

— the length of the string.

— the speed of wave propagation along the string.

Two examples are shown on the left, but more can be generated provided the ends of the standing waves are nodes.

 

 

So what is the relationship between frequency and the length of the string?

Observe that the string length (l) in fig.

— SW1 = ½ wavelength

— SW2 = 1 wavelength

— SW1 = 1½ wavelength

From this we develop:

λn = 2l/n

⇒ λ ∝ 1/n since 2l is a constant for that system.

From this we develop:

If λ1 is the longest wavelength possible, the other possible wavelengths will be ½λ1, ⅓λ1, ¼λ1... (1/n

Recall that frequency is inversely proportional to wavelength

⇒ f ∝ 1/λ

⇒ f ∝ n

The lowest frequency is called the fundamental frequency and the higher frequencies are called overtones.

Exact multiples of a frequency are called harmonics, but note that while in the ideal string the ovetones are harmonics, the overtones of real musical instruments may not be harmonics. This leads to the distinct sounds of each istrument.

 standing waves4

Fig SW5

Standing waves in Tubes

Air columns can vibrate in tubes in air instruments. However, there are significant differences between closed and open tubes. The open end of a tube will have an antinode and a closed ens a node.

Fig. SW5 shows a tube with both ends open and another with one end closed.

In the open tube the length of the tube = ½ fundamental wavelength, but the tube open at one length of the tube = ¼ fundamental wavelength

 

standing waves5

Fig SW6

 

Exercise 2.4.1

Develop the relationship between harmonic frequencies for open ended tubes with vibrating columns and their length. You may use the Fig SW5.

 standing waves6

Fig SW7

 

Exercise 2.4.1

Develop the relationship between harmonic frequencies for tubes closed at one end with vibrating columns and their length. You may use the Fig SW6.

Resonance

Natural frequencies or resonant frequencies are the frequencies at which standing waves are produced. So far we have been examining the different resonant frequencies in standing waves on a cord and air columns in open and closed end tubes. An oscillating spring or pendulum will oscillate at its resonance frequency when set in motion. But while a spring or pendulum has only one resonant frequency most objects like the cord and air columns have an infinite number of resonant frequencies, each of which is related to the lowest resonant frequency.

swing

 

 

 

 

 

Fig SW8

 

Forced Oscillations & Resonance

Consider a child who is on a swing or a hammock. The swing or hammock may be set into motion by an push and it oscillates at its natural frequency pendulum. Friction and air resistance decrease the amplitude until it dies out. To maintain the swing in motion the we apply a period external force "in sync" with the natural motion. So we stand behind the swing and quickly push it after stops on its backwward swing.

 

resonance2

Fig SW9

Fig SW10

Fig SW10

 

 

What happens if you try to push it before it stops on the backward swing?

How difficult is it to push after it stops on the backward swing?

Applying the forced oscillations we can make the amplitude very large.

But what is the relationship between a forced frequency and the natural or resonant frequency fo?

The graph on the left shows that as the forced frequency approaches that of the natural frequency the amplitude increases significantly.

This is why you have to apply the force at a certain frequency to maintain large aplitudes on a swing or hammocks.

Some observations about natural frequency.

— Bridges have a natural frequency. Winds and traffic movement over them can cause them to oscillate. If the forced oscillations have the same frequency as the bridges natural frequency the bridge may oscillate with dangerous frequency. Fig SW10 is a suspension bridge on the Marian River in Blanchessuise. I had to hold on to the support as a car (out of the picture) drove onto the bridge towards me and the bridge oscillated with an alarming amplitude! More spectular is the case of the Tacoma Narrows Bridge which opened "in in 1940, the Tacoma Narrows Bridge was the third-longest suspension bridge in the world... On November 7, with a steady wind blowing at 42 mph, the roadway began to twist back and forth in an increasingly violent fashion... the center span broke off... most agree the collapse was related to resonance..." [ http://www.history.com/this-day-in-history/tacoma-bridge-collapses]Click here to see an 'Exhibit: History of the Tacoma Narrows Bridge' University Libraries University of Washington.

— crystal glass can be shattered by using sound waves to force oscillations beyond the elastic limit of the glass.

— infrastuctre can collapse in earthquakes. People are amazed that one building collapses next to a weaker structure which is left standing with little damage. The simple answer is that the earthquake waves caused resonance in the buliding to create an amplitude beyond the elastic limit of the materials. Earthquakes are always felt stronger in the DeigoMartin area. This is because that area sits on a bed of sediment whose resonant frequency is in the range of the earthquakes waves experienced in this region.

— Musical instruments are designed to have strings or air columns or rods, etc. resonate at particular frequencies so that we can play the notes.

— Vibrating parts in a vehicle may often be reduced by adding or removing some material. A friend of mine had a jeep with a fibreglass roof. It vibrated incessantly when driving at normal speeds. Sticking a think rubber sheet to the underside of the roof changed the resonant frequency and hence stopped the vibration!

Example:

A guitar string of mass 11.00g is 0.60 m long and is tuned to the note C2 (65Hz.)

(a) What is the tension on the string?

(b) What are the frequencies of the first four harmonics?

Answer:

The wavelength (λo) of the fundamental frequency (fo) = 2 x length of string.

λo = 1.20 m

v = fλ

v = 1.20 m X 65 s-1

v = 78 ms-1

Recall from Eq1 in Unit 1.1 Speed of Waves formulawhere μ is mass per unit length.

FT = μv2

FT = (11X10-3 kg) ÷ (0.60 m) X (78 ms-1)2

FT = 112 N


(b) f = n fo where n is the number of the harmonic and fo the first harmonic.

f2= 224 Hz

f3= 336 Hz

f4= 448 Hz

f5= 560 Hz

 


So what happens when incoherent waves meet?

Exercise 2.1.1

Concept by Kishore Lal. Programmed by Kishore Lal... Copyright © 2015 Kishore Lal. All rights reserved.