Wave Motion

ISAT 241

ANALYTICAL METHODS III

Fall 2004

D. J. Lawrence

Wave Motion -- Examples Water waves Earthquake waves Mechanical waves in large structures

– (e.g., bridges and skyscrapers) Waves on stretched strings Sound waves Electromagnetic waves

– (e.g., radio, TV, and light)

Wave Motion

Wave -- the motion or “propagation” of a disturbance.

Mechanical waves require a body or “medium” that can be disturbed.

Electromagnetic waves do not require a medium. They can travel through vacuum.

All waves carry energy.

Wave Motion Waves travel or “propagate” with a specific speed that

depends on the properties of the “medium” through which they are traveling, for example:

velocity– Sound in air at 20°C 343 m/s = 767 mi/h– Sound in air at 0°C 331 m/s– Sound in water at 25°C 1493 m/s– Sound in aluminum 5100 m/s– Light in vacuum 3108 m/s = 186,000 mi/s– Light in diamond 1.2108 m/s

Wave Motion

Transverse Wave -- a wave in which the particles of the “disturbed medium” move perpendicular to the wave velocity, e.g.,

– mechanical wave on a rope or a string– electromagnetic waves (e.g., radio, TV, and

light; although these waves do not require a medium, the associated electric and magnetic fields vary in a direction perpendicular to the wave velocity)

Serway & Jewett, Principles of Physics, 3rd ed.

Figure 13.1

Wave Motion

Longitudinal Wave -- a wave in which the particles of the “disturbed medium” move in a direction parallel to the wave velocity, e.g.,– sound waves in air

Some waves are partly transverse and partly longitudinal, e.g.,– water waves

Serway & Jewett, Principles of Physics, 3rd ed.

Figure 13.3

Traveling Waves Consider a wave pulse on a string, moving from left to right

(along x-direction) with speed = v

x

y

x

yv vvt

A

Pulse at t = 0 Pulse at time t

• This is a transverse wave <=> the displacement of the string (the medium) is in the y-direction.

Traveling Waves

Pulse at t = 0 Pulse at time t

• A is called the amplitude of the wave = maximum displacement.

• At t = 0, peak of pulse is at x = 0. At a later time, t, peak of pulse is at x = vt.

x

y

x

yv vvt

A

Traveling Waves A mathematical function that describes a wave is

called a wave function. We can describe the wave pulse that we have been

considering by a function of the form

)(),( vtxftxyy

displacement(along y-axis)

position of pulse (along x-axis)

y is a function of the quantity (x - vt). y is a “function of two variables”.

Traveling Waves

A wave traveling to the right with speed v can be described by a wave function of the form:

)(),( vtxftxyy

)(),( vtxftxyy

A wave traveling to the left with speed v can be described by a wave function of the form:

Traveling Waves -- Example Consider the following wave function that describes a

pulse traveling along the x-axis:

where x and y are measured in centimeters and t in seconds.

1)3(

2),(

2

txtxy

Is this pulse moving to the left or to the right? What is the speed of the pulse? Is the wave transverse or longitudinal? What direction does a “particle” of the rope move and

what is its speed?

Serway & Jewett, Principles of Physics, 3rd ed.

Figure 13.5

Superposition and Interference of Waves

The Superposition Principle:– If two or more traveling waves are moving

through some medium, the resultant wave function at any point is the algebraic sum of the wave functions of the individual waves.

– The combination of separate waves in the same region of space to produce a resultant wave is called interference.

Serway & Jewett, Principles of Physics, 3rd ed.

Figure 14.1

Serway & Jewett, Principles of Physics, 3rd ed.

Figure 14.2

Reflection and Transmission of Waves

Whenever a traveling wave reaches a boundary (e.g., the end of the string, or a location where the medium changes in some way), part or all of the wave is reflected. Any part of the wave that is not reflected is said to be transmitted through the boundary.

– See the Figures in your text.

Serway & Jewett, Principles of Physics, 3rd ed.

Figure 13.12

Sinusoidal Waves A “sinusoidal wave” can be expressed

mathematically using a sine or cosine function (plus some phase angle).

y

x

“crests”

“troughs”

Serway & Jewett, Principles of Physics, 3rd ed.

Figure 13.7

Sinusoidal Waves

This sinusoidal wave has traveled a distance x1 in the time t1 (these are “snapshots” of the wave).

wave velocity = v = x1/ t1

t = 0 t = t1

y

xx1

Serway & Jewett, Principles of Physics, 3rd ed.

Figure 13.8

Sinusoidal Waves A = amplitude of the wave = maximum

value of the displacement (~ m) = wavelength of the wave (Greek

lambda)

A

-A

y

x

graph for t = constant

Sinusoidal Waves T = period of the wave (~ s)

= time it takes the wave to repeat

= time it takes the wave to travel a distance

of one wavelength v T

y

A

-A

t

T

graph for x = constant

How can we describe this wave mathematically?

Sinusoidal Waves Consider this sinusoidal wave, moving to the right

y

x

y

t

T

Sinusoidal Waves For t = 0,

For x = 0,

y

x

y A x

sin2

y

T

t

y Av

t

sin2

Sinusoidal Waves This sinusoidal wave is described by the expression

y A x vt

y f x vt

recall that vT vT

so

y Ax t

T

sin

,

sin

2

2

22

Tf angular frequency" "

2k wave number" "

so we can write

Sinusoidal Waves We can define two “new” quantities

)tkxsin(Ay

Sinusoidal Waves f = frequency of the wave

= number of times that a crest passes a fixed point each second

f ~ s-1 = hertz = Hz

v T vT

v f

fT

1

speed of a sinusoidal wave

m/s s-1 = Hz m

Sinusoidal Waves

)s/m(k

fv

so

)m/rad("numberwave"2

k

)s/rad("frequencyangular"f2

Interference of Sinusoidal Waves

The superposition principle can be applied to two or more sinusoidal waves traveling simultaneously through the same medium.

The term interference is also used to describe the result of combining two or more waves.

Serway & Jewett, Principles of Physics, 3rd ed.

Figure 14.3

Standing Waves / Modes of Vibration

At certain frequencies, a stationary pattern of vibration or oscillation is produced when two or more sinusoidal waves interfere (combine) in the medium. This stationary pattern is called a standing wave.

Such a pattern is also called a “mode of vibration” or a “normal mode”.

These modes of vibration can be found in musical instruments, bridges, buildings, molecules ...

Serway & Jewett, Principles of Physics, 3rd ed.

Figure 14.8

Standing Waves in a String

If a stretched string is clamped at both ends, waves traveling in both directions can be reflected from the ends.

The incident and reflected waves combine according to the superposition principle.

Consider these two waves: y1 = A sin (kx - t) y2 = A sin (kx + t)

Adding these two functions gives y = y1 + y2 = (2A sin kx) cos (t)

This is the wave function of a standing wave.

Serway & Jewett, Principles of Physics, 3rd ed.

Figure 14.7

Standing Waves in a String

y = y1 + y2 = (2A sin kx) cos (t)

Standing wave has angular frequency = – Every particle of string vibrates (in SHM) with

the same frequency, f = .

Amplitude = 2A sin kx– Amplitude of motion of a particle of string

depends on x.

Standing Waves in a String

Consider a string of length L that is clamped at both ends.

The ends of the string cannot move. Points along the string that do not move are

called nodes. If the string is displaced at its midpoint and

released, a vibration is produced in which the center of the string undergoes the greatest movement.

The center of the string is called an antinode.

Standing Waves in a String For this pattern of vibration, called a normal

mode, the length of the string equals /2 , i.e., L = /2 >>>>> =2L

We can produce vibrations (normal modes) in which there is more than one antinode.

In general, the wavelengths of the various normal modes can be written

n

L

nn

21 2 3( , , , )

Standing Waves in a String The frequencies of the normal modes can be written

fv nv

Lnn

n

21 2 3( , , ,...)

T

v

,...)3,2,1(2

nT

L

nf n

If T is the tension in the string and is its mass per unit length, then the speed of the wave is given by

So we can write

Standing Waves in a String

We get the lowest frequency when n = 1. This is called the fundamental frequency (or the first harmonic) and it is

T

Lf

2

11

The remaining modes are integral multiples of the fundamental frequency and are called higher harmonics, e.g., 2nd harmonic, 3rd harmonic, etc.

Serway & Jewett, Principles of Physics, 3rd ed.

Figure 14.8

Modes of Vibration / Natural Frequencies / Resonance

These normal modes of vibration occur in stringed musical instruments.

There are natural frequencies, normal modes, or resonant frequencies associated with many objects and phenomena, in addition to strings.

Natural frequencies can be found in air columns (as in organ pipes and wind instruments), tuning forks, bridges, buildings, automobile suspensions, molecules, antennas, radio and TV tuners, playground swings ...

Modes of Vibration / Natural Frequencies / Resonance

If a periodic force is applied to such a system, the amplitude of the resulting motion will be large if the frequency of the applied force is equal or nearly equal to one of the natural frequencies of the system.

The natural frequencies of oscillation of a system are often referred to as resonant frequencies.