University Physics: Waves and Electricity

Ch16. Transverse WavesLecture 3

Dr.-Ing. Erwin Sitompulhttp://zitompul.wordpress.com

2013

3/2Erwin Sitompul University Physics: Wave and Electricity

A sinusoidal wave of frequency 500 Hz has a speed of 350 m/s. (a) How far apart are two points that differ in phase by π/3 rad?(b) What is the phase difference between two displacements

at a certain point at times 1 ms apart?

Homework 2: Phase Differences

3/3Erwin Sitompul University Physics: Wave and Electricity

500 Hz,f 350 m sv

(a)v

f

350

500 0.7 m

2

x

3(0.7)

2 2x

11.7 cm

(b)1

Tf

1

500 0.002 s 2 ms

0.117

2

t

T

1 ms

2 22 ms

t

T rad

Solution of Homework 2: Phase Differences

3/4Erwin Sitompul University Physics: Wave and Electricity

(a) What is u, the transverse velocity of the element of the string, at x = 22.5 cm and t = 18.9 s?

A wave traveling along a string is described byy(x,t) = 0.00327sin(72.1x–2.72t),

in which the numerical constants are in SI units.

22.5 cm, 18.9 sx t

( , ) (3.27 mm)sin(72.1 2.72 )y x t x t ( , )

( , )y x t

u x tt

( 8.894 mm s)cos(72.1 2.72 )x t

(0.225 m,18.9 s) ( 8.894 mm s)cos( 35.1855 rad)u

7.197 mm s

( 2.72 rad s)(3.27 mm)cos(72.1 2.72 )x t

Example 1

3/5Erwin Sitompul University Physics: Wave and Electricity

(b) What is the transverse acceleration ay of the same element of the spring at that time?

A wave traveling along a string is described byy(x,t) = 0.00327sin(72.1x–2.72t),

in which the numerical constants are in SI units.

22.5 cm, 18.9 sx t

( , ) ( 8.894 mm s)cos(72.1 2.72 )u x t x t

2(0.225 m,18.9 s) (24.192mm s )sin( 35.1855 rad)ya 214.21mm s

( , )( , )y

u x ta x t

t

( 2.72 rad s)( 8.894 mm s) sin(72.1 2.72 )x t

2(24.192mm s )sin(72.1 2.72 )x t

Example 1

3/6Erwin Sitompul University Physics: Wave and Electricity

It often happens that two or more waves pass simultaneously through the same region (sound waves in a concert, electromagnetic waves received by the antennas).

Suppose that two waves travel simultaneously along the same stretched string, the displacement of the string when the waves overlap is then the algebraic sum.

The Principle of Superposition for Waves

3/7Erwin Sitompul University Physics: Wave and Electricity

Let y1(x,t) and y2(x,t) be two waves travel simultaneously along the same stretched string, then the displacement of the string is given by:

1 2( , ) ( , ) ( , )y x t y x t y x t

Overlapping waves algebraically add to produce a resultant wave (or net wave).

Overlapping waves do not in any way alter the travel of each other.

The Principle of Superposition for Waves

3/8Erwin Sitompul University Physics: Wave and Electricity

Suppose there are two sinusoidal waves of the same wavelength and the same amplitude, and they are moving in the same direction, along a stretched string.

The resultant wave depends on the extent to which one wave is shifted from the other.

We call this phenomenon of combining waves as interference.

1( , ) sin( )my x t y kx t

2 ( , ) sin( )my x t y kx t

Interference of Waves

3/9Erwin Sitompul University Physics: Wave and Electricity

Interference of Waves

The resultant wave as the superposition of y1(x,t) and y2(x,t) of the two interfering waves is:

1 2( , ) ( , ) ( , )y x t y x t y x t

sin( ) sin( )m my kx t y kx t

1 12 2sin sin 2sin ( ) cos ( )

1 12 22 sin( )cos( )my kx t

1 12 2( , ) 2 cos( ) sin( )my x t y kx t

The resultant sinusoidal wave – which is the result of an interference – travels in the same direction as the two original waves.

3/10Erwin Sitompul University Physics: Wave and Electricity

Interference of Waves

Fully constructiveinterference

Fully destructiveinterference

Intermediateinterference

3/11Erwin Sitompul University Physics: Wave and Electricity

Phase Difference and Resulting Interference Types

3/12Erwin Sitompul University Physics: Wave and Electricity

Checkpoint

Here are four possible phase differences between two identical waves, expressed in wavelengths: 0.2, 0.45, 0.6, and 0.8.Rank them according to the amplitude of the resultant wave, greatest first.

Rank: 0.2 and 0.8 tie, 0.6, 0.45

1 2 radians 360

0.2 0.4 radians 72 0.45 0.9 radians 162 0.6 1.2 radians 216 0.8 1.6 radians 288

122 cos( )m mAmplitude y y

12cos( 72 ) 0.809 12cos( 162 ) 0.156 12cos( 216 ) 0.309 12cos( 288 ) 0.809

3/13Erwin Sitompul University Physics: Wave and Electricity

Example 2

Two identical sinusoidal waves, moving in the same direction along a stretched string, interfere with each other. The amplitude ym of each wave is 9.8 mm, and the phase difference Φ between them is 100°. (a) What is the amplitude ym’ of the resultant wave due to the

interference, and what is the type of this interference?

122 cos( )m my y 1

22(9.8 mm)cos( 100 ) 12.599 mm

The interference is intermediate, which can be deducted in two ways:

1. The phase difference is between 0 and π radians.2. The amplitude ym’ is between 0 and 2ym.

3/14Erwin Sitompul University Physics: Wave and Electricity

Example 2

(b) What phase difference, in radians and wavelengths, will give the resultant wave an amplitude of 4.9 mm?

Two identical sinusoidal waves, moving in the same direction along a stretched string, interfere with each other. The amplitude ym of each wave is 9.8 mm, and the phase difference Φ between them is 100°.

122 cos( )m my y

124.9 mm 2(9.8 mm)cos( )

12

4.9 mmcos( )

2(9.8 mm)

12cos( ) 0.25

12 1.3181 or 1.8235

2.6362 or 3.6470

2.636 rad

2

x

2.636

2x

0.420 wavelength

0.420

3/15Erwin Sitompul University Physics: Wave and Electricity

Standing Waves

The following figures shows the superposition of two waves of the same wavelength and amplitude, traveling in opposite direction.

There are places along the string, called nodes, where the string never moves. Halfway between adjacent nodes, we can see the antinodes, where the amplitude of the resultant wave is a maximum.

The resultant wave is called standing waves because the wave pattern do not move left or right.

•Where?

•Where?

3/16Erwin Sitompul University Physics: Wave and Electricity

Standing Waves

3/17Erwin Sitompul University Physics: Wave and Electricity

Standing Waves

To analyse a standing wave, we represent the two combining waves with the equations:

2 ( , ) sin( )my x t y kx t 1( , ) sin( )my x t y kx t

The principle of superposition gives:

1 2( , ) ( , ) ( , )y x t y x t y x t

sin( ) sin( )m my kx t y kx t

1 12 2sin sin 2sin ( ) cos ( )

( , ) 2 sin cosmy x t y kx t

3/18Erwin Sitompul University Physics: Wave and Electricity

Standing Waves

For a standing wave, the amplitude 2ymsinkx varies with position.

For a traveling wave, the amplitude ym is the same for all position.

N N N N

AN AN AN

x0

3/19Erwin Sitompul University Physics: Wave and Electricity

Standing Waves

In the standing wave, the amplitude is zero for values of kx that give sinkx = 0.

( , ) 2 sin cosmy x t y kx t

, for 0,1,2,kx n n

, for 0,1, 2,2

x n n

•Nodes

In the standing wave, the amplitude is zero for values of kx that give sinkx = ±1

1 3 5

2 2 2, , ,kx

1, for 0,1, 2,

2 2x n n

•Antinodes1

2( ) , for 0,1, 2,kx n n

3/20Erwin Sitompul University Physics: Wave and Electricity

Checkpoint

Two waves with the same amplitude and wavelength interfere in three different situations to produce resultant waves with the following equations:(a) y’(x,t) = 4sin(5x–4t)(b) y’(x,t) = 4sin(5x)cos(4t) (c) y’(x,t) = 4sin(5x+4t)

In which situation are the two combining waves traveling (i) toward positive x, (ii) toward negative x, and (ii) in opposite directions?

• Toward positive x: (a), the sign before t is negative• Toward negative x: (c), the sign before t is positive• In opposite directions: (b), resulting standing wave

3/21Erwin Sitompul University Physics: Wave and Electricity

Standing Waves and Resonance

Consider a string, such as a guitar string, that is stretched between two clamps.

If we send a continuous sinusoidal wave of a certain frequency along the string, the reflection and interference will produce a standing wave pattern with nodes and antinodes like those in the figure.

Such a standing wave is said to be produced at resonance. The string is said to resonate at a certain resonant frequencies.

3/22Erwin Sitompul University Physics: Wave and Electricity

Standing Waves and Resonance

For a string stretched between two clamps, we note that a node must exist at each of its end, because each end is fixed and cannot oscillate.

The simplest patterns that meets this requirement is a single-loop standing wave, with two nodes and one antinode.

A second simple pattern is the two loop pattern. This pattern has three nodes and two antinodes.

A third pattern has four nodes, three antinodes, and three loops

3/23Erwin Sitompul University Physics: Wave and Electricity

Standing Waves and Resonance

Thus, a standing wave can be set up on a string of length L by a wave with a wavelength equal to one of the values:

2, for 1, 2,3,L

nn

The resonant frequencies that correspond to these wavelengths are:

, for 1, 2,3,2

v vf n n

L

The last equation tells us that the resonant frequencies are integer multiples of the lowest resonant frequency, f = v/2L, for n = 1.

The oscillation mode with the lowest frequency is called the fundamental mode or the first harmonic.

3/24Erwin Sitompul University Physics: Wave and Electricity

Standing Waves and Resonance

The second harmonic is the oscillation mode with n = 2, the third harmonic is that with n = 3, and so on.

The collection of all possible oscillation modes is called the harmonic series.

n is called the harmonic number.

3/25Erwin Sitompul University Physics: Wave and Electricity

Checkpoint

In the following series of resonant frequencies, one frequency (lower than 400 Hz) is missing: 150, 225, 300, 375 Hz. (a) What is the missing frequency? (b) What is the frequency of the seventh harmonic?

, for 1, 2,3,2

v vf n n

L

1 ( 1)2 2h h

v vf f h h

L L 2

v

L

•The most possible value for v/2L from the above series is 75 Hz?•The missing frequency below 400 Hz is thus 75 Hz.•The seventh harmonic has the frequency of f5 + 2 v/2L = 375 + 2·75 Hz = 520 Hz.

3/26Erwin Sitompul University Physics: Wave and Electricity

Homework 3: Standing Waves

Two identical waves (except for direction of travel) oscillate through a spring and yield a superposition according to the equation

(a) What are the amplitude and speed of the two waves?(b) What is the distance between nodes?(c) What is the transverse speed of a particle of the string at the position x = 1.5 cm when t = 9/8 s?

1 1

3(0.50 cm)sin mm cos (40 min )y x t

3/27Erwin Sitompul University Physics: Wave and Electricity

Homework 3A: Standing Waves

2. Two identical waves (except for direction of travel) oscillate through a spring and yield a superposition according to the equation

(a) What are the amplitude and speed of the two waves?(b) What is the distance between nodes?(c) What is the transverse speed of a particle of the string at the position x = 2.70 m when t = 0.25 min?

11 1

8(0.8 m)sin 3 cm cos ( s )y x t

1. Two waves propagate in one direction on a stretched rope. The frequency of the waves is 120 Hz. Both have the same amplitude of 4 cm and wavelength of 0.04 m. (a) Determine the amplitude of the resultant wave if the two original waves differ in phase by π/3? (b) What is the phase difference between the two waves if the amplitude of the resultant wave is 0.05 cm?