Standing waves

Stationary waves.

From moving to Stationary •Waves we have considered so far have been PROGRESSIVE WAVES. •PROGRESSIVE WAVES : start from source & travel outwards •Transfer energy from one place to another.

From moving to Stationary •Next is STATIONARY WAVES or STANDING WAVES.

Nodes and antinodes •Nodes : points that do not move at all •Antinodes : points where the spring oscillates with a maximum amplitude Sections of spring in adjacent loops are always moving in anti-phase ; they are half a cycle out of phase with one another..

Formation of stationary waves •String stretched with 2 fixed ends •Pulling and releasing produced stationary wave •Node at each end •Antinode in the middle.

Formation of stationary waves •Releasing string produces 2 stationary waves travelling at OPPOSITE directions •These get reflected at the fixed ends •These then superimpose to produce stationary waves.

formation of stationary waves •Copy diagram from blackboard.

Formation of stationary waves •t = 0, left and right waves are in phase. => CI hence amplitude is twice of each wave •t = T/4, each travels λ/4 to the left and right. Hence in antiphase (phase difference 180˚) => DI giving zero displacement •t =T/2, back in phase again => CI •t = 3T/4, antiphase => DI => Zero displacement •t=T, CI similar as when t=0.

formation of stationary waves •Cycle repeats showing nodes and antinodes •Separation between N and AN tells us abt the progressive waves that produces the stationary waves •The separation is related to λ.

Formation of stationary waves •Impt CONCLUSIONS: •separation between 2 adjacent N (or AN) = λ/2 •separation between adjacent N and AN = λ/4 •λ of ANY progressive waves can be determined from separation of N or AN of standing wave (This is = λ/2).

Formation of standing waves •We can use this to determine either v or f •Using the wave eqn: v = f λ •NB: Stationary waves DO NOT TRAVEL hence has NO SPEED. •It also DOES NOT transfer ENERGY between 2 points like the progressive waves.

Summary Progressive waves Stationary waves wavelength λ λ frequency f f speed v zero.

Test yourself 1 •A stationary wave is set up on a vibrating spring. Adjacent nodes are separated by 25cm.Determine •a)the wavelength of the stationary wave •b) the distance from a node to an adjacent antinode.

Solutions •a) 50cm •b) 12.5cm.

Observing stationary waves •string attached to vibration generator driven by the signal generator. •weight to maintain tension in string •string vibrate with small amplitude •adjusting frequency - produces stationary waves with bigger amplitudes.

observing stationary waves •pulley and oscillator end are nodes. •Flashing stroboscope - to reveal motion of string that appear blurred to the eye •Frequency of stroboscope set to match the freq of the vibrations •see string in slow motion •This is known as Melde’s Experiment.

Extension of melde’s experiment •investigate the effect of changing •i) length of the string •ii) tension in the string •iii) thickness of the string.

Test yourself •For n=2, the vibrating section of the string is 60cm. •Determine the wavelength of the wave and the separation between two neighbouring antinodes. •The freq of vibration is increased until a stationary wave with three antinodes appears on the string •Sketch the stationary wave •What is the wavelength of the stationary wave?.

Solutions.

microwaves •Direct the microwave transmitter at a metal plate •Then it gets reflected back towards the source •By moving the probe receiver around will allow use to observe the positions of low and high intensities. •These represent the nodes and antinodes respectively.

Microwaves •The wavelength of the microwave can be determined from the distance between the nodes using the formula c=fλ.

An air column closed at one end.

an air column closed at one end •Glass tube open at both ends is clamped so that one end dips into a cylinder of water •adjusting height of clamp, we are changing the length of the column of air in the tube •hold a vibrating fork at one end •the air column is forced to vibrate and the note becomes louder.

An air column open at one end •This phenomenon is called resonance •The experiment described here is the resonance tube •The length of air column must be just right •Air at the bottom is unable to vibrate => node.

An air column closed at one end •The air at the open end is able to vibrate => an antinode. •length of air column must be 1/4 λ •alternatively can be set to 3λ/4.

NB •The representation of standing sound wave is misleading. •Sound wave : longitudinal wave •but we draw it like a transverse wave •Copy figure on board..

open ended columns •Air in open ended tubes vibrate similar to that in a closed column. •Take an open ended tube and blow gently across the top •Should hear a note whose pitch depends on the length of the tube •Now cover the bottom and repeat.

Open ended columns •pitch produced is now an octave higher that previous note •means freq is approximately twice the original freq •how is this possible? •in this situation there must be an antinode at each end. •hence the node at the midpoint.

open ended air columns •tube of length, l •in closed tube: standing wave is formed at 1/4 λ •therefore wavelength = 4l •in open tube : 1/2 λ •therefore wavelength = 2l •closing a tube halves the wavelength hence the freq doubles.

26.3.01 Standing waves in air columns X=2L Standing wave Wavelength column closed at one end colunm open at both ends Harmonic 1st harmonic (fundamental) 2nd hannonic 3rd hannonic 1st harmonic (fundamental) 2nd hannonic 3rd Fre quency f =v/4L (natural) f = v/2L (natural) 2f n n a a "GORE 0-0 Srarxling in pipe- end a lid i b} pipe are ',viü Siandinæ u.x.e The of tnnxirnurn and an open Only the ruruhn-wntal shoon. a a a X =4ßL a a n a n = node, a = antinode.

musical instruments •production of diff notes dep on creation of stationary waves •for stringed instruments, nodes are at the two fixed ends •string is plucked halfway hence antinode is at the midpoint. •This is known as the fundamental mode of vibration.

musical instruments •Fundamental freq is the minimum freq of a standing wave for a given system or arrangement..

musical instruments •by changing the length of the air column, the note can be changed. •holes can be uncovered so that air can vibrate freely. •this gives diff pattern for nodes and antinodes •in practice, sounds are produced made up different standing waves having diff nodes and antinodes.

musical instruments •eg guitar may vibrate with 2 antinodes along its length •give a note having twice the freq •this is described as the harmonic of fundamental •the skill of musician is in stimulating the string or air column to produced desired mixture of freq..

(a) Displacement of air TUBE CLOSED AT ONE END (b) Pressure variation in the air First harmonic = fundarnental Third harmonic Fifth harmonic Copyright 0 2005 Pearson Prentice Han. Inc. A First Harmonic or Fundamental - f Second Harmonic or First Overtone Third Harmonic or Second Overtone - f Fourth Harmonic or Third Overtone - f.

Determining wavelength and speed of sound •recall separation of nodes and antinodes of stationary waves are separated by 1/2 λ •we can use eqn v=f λ •one approach is the Kundt’s dust tube •copy diagram on the board.

Kundt’s dust tube •Loudspeaker send sound waves along the inside of the tube •sound is reflected at the closed end •when stationary wave is established- dust at antinodes vibrate violently. •accumulates at the nodes where movement of air is ZERO.

kundt’s dust tube •hence we can observe the positions of nodes and antinodes clearly •alternative method is using same arrangement as the microwave experiment in the previous slides. •copy diagram on the board •loudspeaker- sound waves •reflected by vertical board.

•microphone detects sound wave •output displayed on the oscilloscope •turn off the time base of the oscilloscope so that the spot no longer moves across the screen •spot moves up and down •height measures intensity of sound •moving the mic help to detect N and A. •max accuracy DO NOT measure separation of adjacent nodes •but measure distance across several nodes.

•resonance tube expt can also determine the λ & v •however, need to take account of systematic error in the expt..

Eliminating errors •look at how the standing waves in the tube are shown •A is at the open end •slightly extending beyond the tube •reason: expt shows that the air slightly beyond the tube vibrates as part of the stationary wave.

Eliminating errors •A is at a distance c beyond the end of the tube •c is called the end correction •we do not know c •it cannot be measured directly •however we can write •for shorter tube = λ/4 = l1 + c •for longer tube = 3 λ/4 = l2 + c.

eliminating errors •show on the board •We can make 2 measurements for l1 and l2 and obtain accurate value of λ •end correction c is the example of systematic error.

summary •stationary waves are formed when two identical waves travelling at opposite directions meet and superimpose. •this usually happens when one wave is a reflection of the other •a stationary wave has a characteristic pattern of nodes and antinodes •a.

summary •a node is a point where the amplitude is always zero •an antinode is the point of maximum amplitude •adjacent nodes or antinodes are separated by distance equal to half wavelength •use v=fλ to determine speed or freq. •wavelength can be found using nodes and antinodes of stationary wave pattern..