Superposition and standing wavesSuperposition and interference of sinusoidal wavesPhasor Addition of WavesStanding waves in a stringResonanceStanding waves in air columnStanding waves in rod and plates

),(),(),(),( 321 txytxytxytxy

Principle of superpositionThe superposition principle says that when two waves pass through the same point, the displacement is the arithmetic sum of the individual displacements.

1 01 1

2 02 2

1 2

sin( )

sin( )

Resultant disturbance

E E t

E E t

E E E

i) Algebraic method of Addition

Addition of wave of same frequency

0 sin( )E E t where,

2 2 20 01 02 01 02 2 1

1 01 1 02 2

01 1 02 2

2 cos( )

sin sintan

cos cos

E E E E E

E E

E E

(Cosine rule )

For

20201

10201

EE

EE

N

iii

N

iii

N

ij

N

ijiji

N

ii

N

iii

E

E

EEEE

tEtEE

10

10

100

1

20

20

01

0

cos

sintan

)cos(2

)cos()cos(

Superposition of many waves:

ii) Phasor Addition of Waves

• Let us consider sinusoidal waves whose electric field component are given by

and

39.174928.610 22 RE

01 28.13928.610

4tan

028.13sin39.17 ty

030

RE

10

928.630cos8 0

430sin8 0 8

02

1

30sin0.8

sin10

ty

ty

02

1

30sin0.8

sin10

ty

ty

tEy R sin

ResultantAddition of two phasors

1 0

02 0

03 0

sin

sin 60

sin 30

E E t

E E t

E E t

Addition of three phasors

0 0 0 2 0 0 20 (cos0 cos 60 cos30 ) (sin 60 sin 30 )E E

00

00

30cos60cos1

30sin60sintan

Eg.

Eg.

Argand DiagramRepresentation of a complex number in terms of real and imaginary components

ImComplex plane

A Sin wt

A Cos wtRe

y = A cos(wt + f) = A cosf + iA sinf

The composite wave is

(i) harmonic

(ii) of same frequency as that of constituents

(iii) of different amplitude and phase from that of constituents

standing wave on a string, fixed end at x = 0

Standing Waves in a string

Standing waves occur when both ends of a string are fixed. In that case, only waves which are motionless at the ends of the string can persist. There are nodes, where the amplitude is always zero, and antinodes, where the amplitude varies from zero to the maximum value.)cos)s(),( ωtinkxAtxy sw

The distance between adjacent antinodes is equal to l/2.The distance between adjacent nodes is equal to l/2.The distance between a node and an adjacent antinode is l/4.

Mathematical representation of a standing wave:

Consider two traveling waves moving in opposite directions on a string

)(s),(1 ωtkxinAtxy )(s),(2 ωtkxinAtxy

)(s)(s21 ωtkxinAωtkxinAyyy

and

212121 2

1cos

2

1sin2sinsin

ωtinkxAtxy coss2),(1

From

Therefore

For a standing wave y(x, t) = 0 at x = 0 and x = L .Satisfied when sinkL = 0 giving kL = p, 2p, 3p,..np

Since k = 2p/l,

Eg. Two waves traveling in opposite directions on a string fixed at x = 0 produce a standing wave. The waves are described by the functions

m )0.40.2(s20.01 txiny

m )0.40.2(s20.02 txiny

a) Determine the function for the standing waveb) What is the maximum amplitude at x = 0.45 m?c) What is the maximum amplitude and where does it occur?

Ex. Add the two waves using a phasor diagram.

Standing Waves in string - Resonance A standing waves can be set up in the string by a continuous superposition of waves incident on and reflected from the ends. Notice that there is a boundary condition for the waves on the string. Because the ends of the string are fixed, they must necessarily have zero displacement and are therefore nodes by definition.

Example, Standing Waves and Harmonics:

Figure 16-22 shows a pattern of resonant oscillation of a string of mass m =2.500 g and length L =0.800 m and that is under tension t =325.0 N. What is the wavelength l of the transverse waves producing the standing-wave pattern, and what is the harmonic number n? What is the frequency f of the transverse waves and of the oscillations of the moving string elements? What is the maximum magnitude of the transverse velocity um of the element oscillating at coordinate x =0.180 m ? At what point during the element’s oscillation is the transverse velocity maximum?

Calculations:

By counting the number of loops (or half-wavelengths) inFig. 16-22, we see that the harmonic number is n=4.

Also,

For the transverse velocity,

We need:

But ym =2.00 mm, k =2p/l =2p/(0.400 m), and w= 2pf =2 p (806.2 Hz).

Then the maximum speed of the element at x =0.180 m is

Standing Waves in Air Columns

In a pipe closed at one end, the closed end is a displacement node because the rigid barrier at this end does not allow longitudinal motion of the air. Because the pressure wave is 90° out of phase with the displacement wave , the closed end of an air column corresponds to a pressure antinode (that is, a point of maximum pressure variation).The open end of an air column is approximately a displacement antinode and a pressure node. We can understand why no pressure variation occurs at an open end by noting that the end of the air column is open to the atmosphere; therefore, the pressure at this end must remain constant at atmospheric pressure.

A. Pipe open at both ends B. Pipe open at one end only

Example, Double Open and Single Open Pipes:

Normal-mode longitudinal vibrations of a rod of length L (a) clamped at the middle to produce the first normal mode and (b) clamped at a distance L/4 from one end to produce the second normal mode. Notice that the red-brown curves are graphical representations of oscillations parallel to the rod (longitudinal waves).

Standing waves in rod

Two-dimensional oscillations can be set up in a flexible membrane stretched over a circular hoop such as that in a drumhead. As the membrane is struck at some point, waves that arrive at the fixed boundary are reflected many times. The resulting sound is not harmonic because the standing waves have frequencies that are not related by integer multiples. Without this relationship, the sound may be more correctly described as noise rather than as music. The production of noise is in contrast to the situation in wind and stringed instruments, which produce sounds that we describe as musical.Whereas nodes are points in one-dimensional standing waves on strings and in air columns, a two-dimensional oscillator has curves along which there is no displacement of the elements of the medium. The lowest normal mode, which has a frequency f1, contains only one nodal curve; this curve runs around the outer edge of the membrane. The other possible normal modes show additional nodal curves that are circles and straight lines across the diameter of the membrane.

Standing waves in plates

Some possible normal modes of oscillation for a two-dimensional circular membrane: