11physics_lecture_26 apr 2014

MS101: Physics

Chapter 11: Waves

Dr. Ahmed Amin Hussein01007903935

ahussein32125@gmail.com2013-2014

April 19, 2023 Prepared By: Dr. Ahmed Amin 1

Chapter 11: Waves

Energy Transport by Waves

Longitudinal and Transverse Waves

Transverse Waves on Strings

Periodic Waves

Mathematical and Graphical Descriptions of Waves

Reflection and Refraction of Waves

Interference and Diffraction

Standing Waves on a String

What can waves do?

A wave can do many things. Travel e.g. pulse on a string,

telecommunications signal down optical fiber. Carry energy and momentum from one point to

another. Bounce off surfaces - reflection. Go across boundaries - refraction. Go round corners - diffraction. Interact and superimpose - interference. Change shape - dispersion. Loose energy - dissipation.

Types of waves

There are several different types of wave that we must consider.

Mechanical Waves:- These need a medium to propagate in - sound waves.

• A Mechanical Wave travels with a material called a medium• As wave travels through medium particles in the medium

undergo displacement• The speed of travel depends upon the mechanical properties of

the medium

Non-mechanical waves:-These waves do not need a medium in which to propagate - light waves.

Matter waves:- Particles such as protons and electrons can be treated as waves. This forms the basis of quantum mechanics. We will not be discussing this type of wave in this course.

Waves• Every sound we hear, every photon of light that hits our eyes, the movement of

grass blown by the wind and the regular beat of the tides are all examples of waves.

• Two types of waves

– Mechanical (Sound waves , water waves)• Use matter to transfer energy through a medium (solid,

liquid, or gas - ropes, water, air)

– Electromagnetic• Do not need matter to transfer energy (light, radio,

radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays)

§11.1 Waves and Energy Transport

A wave is a disturbance that travels outward from its source.

Waves carry energy. The energy is transported outward from the source; matter is not.

Water waves are able to transfer energy without transferring matter. How this is done is by the wave's energy traveling through the water and leaving the water molecules in place.

• a wave is a disturbance that travels through a medium from one location to another.

• a wave is the motion of a disturbance

April 19, 2023 7Prepared By: Dr. Ahmed Amin

When a stone is dropped into a pond, the water is disturbed from its equilibrium positions as the wave passes; it returns to its equilibrium position after the wave has passed.

The water moves up and down as the disturbance moves outward.

Intensity is a measure of the amount of energy/sec that passes through a square meter of area perpendicular to the wave’s direction of travel.

22 r4r4

I Intensity has units of watts/m2 .

This is an inverse square law. The intensity drops as the inverse square of the distance from the source. (Light sources appear dimmer the farther away from them you are.)

Checkpoint 11.1 page 395 408

Checkpoint 11.1A siren in a fire tower 20 m high generates a sound wave with intensity 0.090 W/m2

at a point on the ground below the tower. What is the intensity of the sound wave 2.0 km from the tower? Assume the siren is an isotropic.

For an isotropic source, I α 1/r2

At a distance 102 times as far from the tower, the intensity is 10-4 x 0.090 W/m2 = 9.0 μ W/m2.

Example: At the location of the Earth’s upper atmosphere, the intensity of the Sun’s light is 1400 W/m2. What is the intensity of the Sun’s light at the orbit of the planet Mercury?

sune 4 r

sunm 4 r

Divide one equation by the other:

W/m920057.6

57.6m 1085.5

m 1050.1

§11.2 Transverse and Longitudinal Waves

A transverse wave is where the motions of the particles are transverse (perpendicular) to the direction of wave travel.

Transverse waves may occur on a string, on the surface of a liquid and throughout a solid.

A ripple on a pond and a wave on a string are easily visualized transverse waves.

A longitudinal wave is where the motions of the particles are along the same direction as the wave propagation.

A wave in a "slinky" is a good visualization.

Sound waves in air are longitudinal waves.

Rarefaction, a region of low density

Compression, a region of high density

Displacement

Velocity propagation

Both types of waves can move through solids. Only longitudinal waves can move through a fluid. A transverse wave can move along the surface of a fluid.

§11.3 Transverse Waves on a String

Attach a wave driver here

Attach a mass to a string to provide tension. The string is then shaken at one end with a frequency f.

A wave traveling on this string will have a speed ofF

where F is the force applied to the string (tension) and is the mass/unit length of the string (linear mass density).

A long piece of piano wire of radius 0.4 mm is made of steel of density 7.8 X 103 kg/m3 The wire is under a tension of 1.0 X103 N. What is the speed of transverse waves on this wire? What is the wavelength of a wave on this wire if its frequency is 262 Hz?

SOLUTION: Consider a 1-m-long piece of this wire. The volume of this piece is p X (0.4 X 10-3 m)2 X 1 m = 5.0 X 10-7 m3 and the mass is 5.0 X 10-7 m3 X 7.8 X 103 kg/m3 = 3.9 X 10-3 kg. Hence, the mass per unit length of the wire is 3.9 X 10-3 kg/m. From Eq. (19), the wave speed is then

Consequently, the wavelength is

Example 1

A string is tied to a pole at one end and 100 g mass at the other, and wound over a pulley. The string’s mass is 100 g, and it is 2.5 m long. If the string is plucked, at what speed do the waves travel along the string? How could you make the waves travel faster? Assume the acceleration due to gravity is 10 m/s2.

The tension in the string is the force of gravity pulling down on the weight, T = mg = (0.1 kg)(10 m/s2) = 1 N. The equation for calculating the speed of a wave on a string is:

This equation suggests two ways to increase the speed of the waves: increase the tension by hanging a heavier mass from the end of the string, or replace the string with one that is less dense.

Example 2

Since the formula for the speed of a wave on a string is expressed in terms of the mass density of the string, we’ll need to calculate the mass density before we can calculate the wave speed.

Example (text problem 11.8): When the tension in a cord is 75.0 N, the wave speed is 140 m/s. What is the linear mass density of the cord?

v The speed of a wave on a string is

kg/m 108.3

m/s 140

N 0.75 322

Solving for the linear mass density:

§11.4 Periodic WavesA periodic wave repeats the same pattern over and over.

For periodic waves: v = fv is the wave’s speed

f is the wave’s frequency

is the wave’s wavelength

All waves can be made by adding up sine waves. The sine wave has a pattern that repeats. The length of this repeating piece of the sine wave is called the wavelength. The wavelength can be found by measuring the length or distance between one peak of a sine wave and the next peak.

The period T is measured by the amount of time it takes for a point on the wave to go through one complete cycle of oscillations. The frequency is then f = 1/T.

Parts of a Wave

• Crest: The highest point on the wave• Trough: The lowest point on the

The maximum displacement from equilibrium is amplitude (A) of a wave.

One way to determine the wavelength is by measuring the distance between two consecutive crests.

Example (text problem 11.13): What is the wavelength of a wave whose speed and period are 75.0 m/s and 5.00 ms, respectively?

m 3750s 10005m/s 075 3 ...vT

Solving for the wavelength:

§11.5 Mathematical Description of a Wave

To describe a wave, we must know the position of the particles in the medium. This requires a function of the form y(x,t).

1.2 Mathematical Description of a wave

Need to make several assumptions before we can describe a wave.

1. Wave depends on both position, x, and time, t.

2. We have a random disturbance y(x,t) = f(x,t).

3. Wave travels in straight line in x direction.

4. Wave travels at a constant speed v.

5. Wave does not change shape - Non-dispersive.

6. Wave does not loose energy - Non-dissipative.

7. Need to define a frame of reference to understand pulse propagation.

1.2 Mathematical Description of a wave

A frame of reference allows us to quantify processes. The speed of a car is measured with respect to the road. The road is the frame of reference.

For a pulse there are two possible frames of reference:

1) Laboratory Frame of Reference:-

Here we define at set of axes x, y, z and as time changes the pulse moves away from the origin.

2) Pulse Frame of Reference:-

Here we define a set of axes x’, y’, z’ that move with the pulse and at the same speed as the pulse. The position of pulse is stationary in this frame and so pulse is time invariant, i.e. independent of time.

Laboratory Frame

0 50 100 150 200 250 300

Position

Y(x,t=0s)

Pulse Frame

0 50 100 150

Position

Y’(x’)

At time t = 0 s both frames of reference coincide. Consider two points, x in laboratory frame and x’ in pulse frame.

Laboratory Frame

0 50 100 150 200 250 300

Position

Y(Dx,t=t)

After time t, pulse in laboratory frame moves and point is now Dx from origin.

Pulse Frame

0 50 100 150

Position

Y’(x’)

In pulse frame the point x’ is unchanged but

axes have moved a distance vt.

1.2 Mathematical description of a wave

Is there anyway we can relate the two frames of reference?

Let us look at pulse in the pulse frame of reference:

• Pulse is described by an arbitrary function.

y’(x’) = f(x’)

• The pulse has the same profile irrespective of the frame of

reference. So in the laboratory frame of reference

y(x,t) = f(x’)

• It is easy to show that

Dx = x’+vt

x’ = Dx-vt

Laboratory Frame

0 50 100 150 200 250 300

Position

Y(x,t=0s)

Pulse Frame

0 50 100 150

Position

Y’(x’)

Laboratory Frame

0 50 100 150 200 250 300

Position

Y(Dx,t=t)

Pulse Frame

0 50 100 150

Position

Y’(x’)

kxtAtxy cos),(

+ is used for a wave traveling in the x direction, and is used for a wave traveling in the +x direction.

is called the wave number.

Note: it would also be valid to use the sine function in the above description.

kxt is called the phase (radians).

The above picture is a snapshot (time is frozen). Two points on the wave are “in phase” if:

12 2(n = 1, 2, 3,…)

Example (text problem 11.21): A wave on a string has an equation:

xttxy rad/m 00.6 rad/sec 600sinmm 00.4),(

(a) What is the amplitude of the wave?

(b) What is the wavelength?

A = 4.00 mm

m 051rad/m 006

The wave number k is 6.00 rad/m.

kxtAtxy sin),(Compare this to

(d) What is the wave speed?

(e) What direction is the wave traveling.

(c) What is the period?

sec 10051rad/sec 600

22 2 .T

m/s 100rad/m 00.6

rad/sec 6002

Along the +x direction.

Example continued:

§11.6 Graphing Waves

The next two slides show three “snapshots” of a traveling wave y(x,t) = A cos (t kx) where A = 1.0 m, k = 1 rad/m, and = rad/sec.

Wave travels to the left

(x-direction)

Wave travels to the right

(+x-direction)

§11.7 The Principle of Superposition

For small amplitudes, waves will pass through each other and emerge unchanged.

Superposition Principle: When two or more waves overlap, the net disturbance at any point is the sum of the individual disturbances due to each wave.

Displacement

• The combination of two overlapping waves is called superpositon.

• Displacement in the same direction produce constructive interference.

• When two waves are added together the resultant wave is larger than the individual displacements and this is constructive interference.

Destructive Interference

• Displacements in opposite directions produce destructive interference.

• When positive and negative displacements are added, the resultant wave is the difference between the pulses, this is called

destructive interference.

CONSTRUCTIVE INTERFERENCE

DESTRUCTIVE INTERFERENCE

Two traveling wave pulses: left pulse travels right; right pulse travels left.

X§11.8 Reflection and Refraction

At an abrupt boundary between two media, a reflection will occur. A portion of the incident wave will be reflected backward from the boundary.

When you have a wave that travels from a “low density” medium to a “high density” medium, the reflected wave pulse will be inverted.

The frequency of the reflected wave remains the same.

When a wave is incident on the boundary between two different media, a portion of the wave is reflected, and a portion will be transmitted into the second medium.

The frequency of the transmitted wave also remains the same. However, both the wave’s speed and wavelength are changed such that:

The transmitted wave will also suffer a change in propagation direction (refraction).

Example (text problem 11.36): Light of wavelength 0.500 m in air enters the water in a swimming pool. The speed of light in water is 0.750 times the speed in air. What is the wavelength of the light in water?

m3750m 50007500

airair

waterwater

Since the frequency is unchanged in both media:

X§11.9 Interference and Diffraction

Two waves are considered coherent if they have the same frequency and maintain a fixed phase relationship.

Two waves are considered incoherent if the phase relationship between them varies randomly.

When waves are in phase, their superposition gives constructive interference.

When waves are one-half a cycle out of phase, their superposition gives destructive interference.

When two waves travel different distances to reach the same point, the phase difference is determined by:

difference phase21 dd

difference path21

Diffraction is the spreading of a wave around an obstacle in its path.

§11.10 Standing Waves

Pluck a stretched string such that y(x,t) = A sin(t + kx)

When the wave strikes the wall, there will be a reflected wave that travels back along the string.

The reflected wave will be 180° out of phase with the wave incident on the wall. Its form is y(x,t) = A sin (t kx).

Apply the superposition principle to the two waves on the string:

kxtkxtA

txytxytxy

sincos2

sinsin

),(),(),( 21

The previous expression is the mathematical form of a standing wave.

A node (N) is a point of zero oscillation. An antinode (A) is a point of maximum displacement. All points between nodes oscillate up and down.

The nodes occur where y(x,t) = 0.

0sincos2, kxtAtxy

The nodes are found from the locations where sin kx = 0, which happens when kx = 0, , 2,…. That is when kx = n where n = 0,1,2,…

The antinodes occur when sin kx = 1; that is where

2 1 0 and 2

If the string has a length L, and both ends are fixed, then y(x = 0, t) = 0 and y(x = L, t) = 0.

kLtLxy

00sin,0

The wavelength of a standing wave: where n = 1, 2, 3,…

These are the permitted wavelengths of standing waves on a string; no others are allowed.

The speed of the wave is:nn fv

The allowed frequencies are then:L

n =1, 2, 3,…

The n = 1 frequency is called the fundamental frequency.

All allowed frequencies (called harmonics) are integer multiples of f1.

Example (text problem 11.51): A Guitar’s E-string has a length 65 cm and is stretched to a tension of 82 N. It vibrates with a fundamental frequency of 329.63 Hz. Determine the mass per unit length of the string.

kg/m 1054

m 650*2Hz63329

v For a wave on a string:

Solving for the linear mass density:

Summary

• Intensity• Wave Properties (f, , v, amplitude)• Transverse vs. Longitudinal Waves• Mathematical Description of a Wave• Reflection, Refraction, Interference, and Diffraction• Superposition of Waves• Standing Waves on a String

THANK YOU

QUESTIONS ?

11physics_lecture_26 apr 2014

matter waves

light waves

radio waves

diffractionstanding

examples of waves

waves energy transport

medium nonmechanical

energy light

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