Waves Physics 23

Armen KocharianLecture 2: Sep 12. 2006

Chapter 16

Wave Motion

What is a Wave

There are two main types of wavesMechanical waves

Some physical medium is being disturbedThe wave is the propagation of a disturbance through a medium

Electromagnetic wavesNo medium requiredExamples are light, radio waves, x-rays

General Features of Waves

In wave motion, energy is transferred over a distanceMatter is not transferred over a distanceAll waves carry energy

The amount of energy and the mechanism responsible for the transport of the energy differ

Mechanical Wave Requirements

Some source of disturbanceA medium that can be disturbedSome physical mechanism through which elements of the medium can influence each other

Combined long. and transv. wave on liquid surface

ExamplesTransverse wave on a string

Longitudinal wave in a fluid (sound)

Complex Waves

Some waves exhibit a combination of transverse and longitudinal wavesSurface water waves are an example

The Motion of a Water Wave

Longitudinal vs. Transverse

Longitudinal: the disturbance is parallel to the direction of propagation.Transverse: the disturbance is perpendicularto the direction of propagation.

Pulse on a RopeThe wave is generated by a flick on one end of the ropeThe rope is under tensionA single bump is formed and travels along the rope

The bump is called a pulse

Pulse on a RopeThe rope is the medium through which the pulse travelsThe pulse has a definite heightThe pulse has a definite speed of propagation along the mediumA continuous flicking of the rope would produce a periodic disturbance which would form a wave

Transverse WaveA traveling wave or pulse that causes the elements of the disturbed medium to move perpendicular to the direction of propagation is called a transverse waveThe particle motion is shown by the blue arrowThe direction of propagation is shown by the red arrow

Longitudinal Wave

A traveling wave or pulse that causes the elements of the disturbed medium to move parallel to the direction of propagation is called a longitudinal waveThe displacement of the coils is parallel to the propagation

A Wave on a Slinky

Example: Earthquake Waves P waves

“P” stands for primaryFastest, at 7 – 8 km / sLongitudinal

S waves“S” stands for secondarySlower, at 4 – 5 km/sTransverse

A seismograph records the waves and allows determination of information about the earthquake’s place of origin

Traveling PulseThe shape of the pulse at t = 0 is shownThe shape can be represented by y (x,0) = f (x)

This describes the transverse position yof the element of the string located at each value of x at t = 0

Traveling Pulse, 2The speed of the pulse is vAt some time, t, the pulse has traveled a distance vtThe shape of the pulse does not changeIts position is now y = f (x – vt)

Traveling Pulse, 3For a pulse traveling to the right

y (x, t) = f (x – vt)For a pulse traveling to the left

y (x, t) = f (x + vt)The function y is also called the wave function: y (x, t)The wave function represents the ycoordinate of any element located at position x at any time t

The y coordinate is the transverse position

Traveling Pulse, final

If t is fixed then the wave function is called the waveform

It defines a curve representing the actual geometric shape of the pulse at that time

Wave Velocity (Speed)

The "disturbance" propagates in space

t = t0 t = to + Δt

Δx

Do not confuse the velocity of the wave with the velocity of the particles in the medium

Wave pulses vs. periodic waves

v

v

In a periodic wave the disturbance repeats itself and the motion of the particles of the medium is periodic

Sinusoidal wavesSpecial case of periodic wave where the motion of each particle in the medium is simple harmonic with the same amplitudeand the same frequency.Resulting wave is a symmetrical sequence of crests and troughs:

Important because any periodic wave can be represented as the sum of sinusoidal waves (Fourier decomposition)

Sinusoidal waves (cont.)

Wavelength (λ):Distance over which displacement repeats

Period (T):Time over which displacement repeats

Frequency (f):f=1/T

Velocity (v):v = λ/T = λf

The Motion of a Wave on a String

The Speed of a Wave

Mathematical DescriptionWaves on a string for concreteness

Wave Function: a function that describes the position of any particle at any time:y(x,t)Take y=0 as equilibrium position, i.e., unperturbed, stretched string

x

y

Mathematical description (sinusoidal wave)Take particle at x=0. Its motion is described by y(x=0,t).It undergoes Simple Harmonic Motion (SHM)

y(x=0,t) = A cos(ωt+δ) = A cosωt (set δ=0)

What is then y(x,t), i.e., the displacement at any other position along the string?The disturbance travels with velocity v.The motion at any x at a given time t is the same as it was at x=0 at a time t’=t-x/v

Nothing fundamental here: just different ways of writing the same thing.........

Note: this is for waves moving in positive x-direction. For a wave moving in negativex-direction, kx-ωt becomes kx+ωt

Sinusoidal WavesThe wave represented by the curve shown is a sinusoidal waveIt is the same curve as sin θ plotted against θThis is the simplest example of a periodic continuous wave

It can be used to build more complex waves

Sinusoidal Waves, contThe wave moves toward the right

In the previous example, the brown wave represents the initial positionAs the wave moves toward the right, it will eventually be at the position of the blue curve

Each element moves up and down in simple harmonic motionDistinguish between the motion of the wave and the motion of the particles of the medium

Terminology: Amplitude and Wavelength

The crest of the wave is the location of the maximum displacement of the element from its normal position

This distance is called the amplitude, A

The wavelength, λ, is the distance from one crest to the next

Terminology: Wavelength and Period

More generally, the wavelength is the minimum distance between any two identical points on adjacent wavesThe period, T , is the time interval required for two identical points of adjacent waves to pass by a point

The period of the wave is the same as the period of the simple harmonic oscillation of one element of the medium

Terminology: Frequency

The frequency, ƒ, is the number of crests (or any point on the wave) that pass a given point in a unit time interval

The time interval is most commonly the secondThe frequency of the wave is the same as the frequency of the simple harmonic motion of one element of the medium

Terminology: Frequency, cont

The frequency and the period are related

When the time interval is the second, the units of frequency are s-1 = Hz

Hz is a hertz

1ƒT

=

Terminology, ExampleThe wavelength, λ, is 40.0 cmThe amplitude, A, is 15.0 cmThe wave function can be written in the form y = A cos(kx – ωt)

Speed of WavesWaves travel with a specific speed

The speed depends on the properties of the medium being disturbed

The wave function is given by

This is for a wave moving to the rightFor a wave moving to the left, replace x – vt with x + vt

( )2( , ) siny x t A x vtπλ

⎡ ⎤= −⎢ ⎥⎣ ⎦

Wave Function, Another Form

Since speed is distance divided by time, v = λ / TThe wave function can then be expressed as

This form shows the periodic nature of y

( , ) sin 2 x ty x t AT

πλ

⎡ ⎤⎛ ⎞= −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Wave Equations

We can also define the angular wave number (or just wave number), k

The angular frequency can also be defined

2k πλ

=

2Tπω =

Wave Equations, cont

The wave function can be expressed as y = A sin (k x – ωt)The speed of the wave becomes v = λƒIf x ≠ 0 at t = 0, the wave function can be generalized to

y = A sin (k x – ωt + φ)where φ is called the phase constant

Sinusoidal Wave on a StringTo create a series of pulses, the string can be attached to an oscillating bladeThe wave consists of a series of identical waveformsThe relationships between speed, velocity, and period hold

Sinusoidal Wave on a String, 2

Each element of the string oscillates vertically with simple harmonic motion

For example, point P

Every element of the string can be treated as a simple harmonic oscillator vibrating with a frequency equal to the frequency of the oscillation of the blade

Sinusoidal Wave on a String, 3The transverse speed of the element is

or vy = -ωA cos(kx – ωt)This is different than the speed of the wave itself

constanty

x

dyvdt =

⎤= ⎥⎦

Sinusoidal Wave on a String, 4The transverse acceleration of the element is

or ay = -ω2A sin(kx – ωt)

constant

yy

x

dva

dt =

⎤= ⎥

⎦

Sinusoidal Wave on a String, 5The maximum values of the transverse speed and transverse acceleration are

vy, max = ωAay, max = ω2A

The transverse speed and acceleration do not reach their maximum values simultaneously

v is a maximum at y = 0a is a maximum at y = ±A

Speed of a Wave on a StringThe speed of the wave depends on the physical characteristics of the string and the tension to which the string is subjected

This assumes that the tension is not affected by the pulseThis does not assume any particular shape for the pulse

tensionmass/length

Tvμ

= =

Reflection of a Wave, Fixed EndWhen the pulse reaches the support, the pulse moves back along the string in the opposite directionThis is the reflection of the pulseThe pulse is inverted

A Reflected Wave Pulse: Fixed End

Reflection of a Wave, Free End

With a free end, the string is free to move verticallyThe pulse is reflectedThe pulse is not inverted

A Reflected Wave Pulse: Free End

Transmission of a WaveWhen the boundary is intermediate between the last two extremes

Part of the energy in the incident pulse is reflected and part undergoes transmission

Some energy passes through the boundary

Transmission of a Wave, 2Assume a light string is attached to a heavier stringThe pulse travels through the light string and reaches the boundaryThe part of the pulse that is reflected is invertedThe reflected pulse has a smaller amplitude

Transmission of a Wave, 3Assume a heavier string is attached to a light stringPart of the pulse is reflected and part is transmittedThe reflected part is not inverted

Transmission of a Wave, 4

Conservation of energy governs the pulseWhen a pulse is broken up into reflected and transmitted parts at a boundary, the sum of the energies of the two pulses must equal the energy of the original pulse

The Linear Wave EquationThe wave functions y (x, t) represent solutions of an equation called the linear wave equationThis equation gives a complete description of the wave motionFrom it you can determine the wave speedThe linear wave equation is basic to many forms of wave motion

The wave equationFamiliar SHM properties:

But also:

So:

Wave Equation:

One of the most important equations in physics.Whenever we encounter it we can say that adisturbance ("y") can propagate along the x-axis with velocity v.

Any function y(x,t)=y(x±vt) is a solution of the wave equation.

or

etc..........All these functions represent disturbances propagating to the right or the left with speed v

For example:

Proof that y(x±vt) is solution of wave equation

So left-hand side of wave equation:

Let z = x ± vt so that y(x,t)=y(x±vt)=y(z)

Right-hand side of wave equation:

Left-hand side was

Wave Speed on a String: First Method

y yF t mv=

y y yy

F v t vF F

F vt v= → =

yy

v FF t vtv vv

μμ

= ⇒ =

vt

v

vy

A

F

Fvyt

vy vy vy

F

F

Transverse momentum= ( )y ymv vt vμ=

P

CB

D

ABC similar DCP

Wave Speed on a String: Second Method

The string is under tension TConsider one small string element of length ΔxThe net force acting in the y direction is

This uses the small-angle approximation

(tan tan )y B AF T θ θΣ ≈ −

Linear Wave Equation Applied to Wave on a String

Applying Newton’s Second Law gives

In the limit as Δx→0, this becomes

This is the linear wave equation as it applies to waves on a string

( ) ( )2

2B A

y x y xyT t xμ ∂ ∂ − ∂ ∂∂

=∂ Δ

2 2

2 2y y

T t xμ ∂ ∂

=∂ ∂

Linear Wave Equation and Waves on a String, cont

The sinusoidal wave function represents a solution of the linear wave equationApplying the sinusoidal wave function to the linear wave equation and following the derivatives, we find that

This verifies the speed of a wave on a string

Tvμ

=

How does the wave speed change?

A Wave on a Rope

A Wave on a String

Sound Produced by a Speaker

Water Waves from a Disturbance

Linear Wave Equation, General

The equation can be written as

This applies in general to various types of traveling waves

y represents various positionsFor a string, it is the vertical displacement of the elements ofthe stringFor a sound wave, it is the longitudinal position of the elements from the equilibrium positionFor em waves, it is the electric or magnetic field components

2 2

2 2 21y y

x v t∂ ∂

=∂ ∂

Linear Wave Equation, General cont

The linear wave equation is satisfied by any wave function having the form y = f (x ± vt)The linear wave equation is also a direct consequence of Newton’s Second Law applied to any element of a string carrying a traveling wave

Energy in Waves in a String

Waves transport energy when they propagate through a mediumWe can model each element of a string as a simple harmonic oscillator

The oscillation will be in the y-direction

Every element has the same total energy

Energy, cont.

Each element can be considered to have a mass of ΔmIts kinetic energy is ΔK = ½ (Δm) vy

2

The mass Δm is also equal to μΔxAs the length of the element of the string shrinks to zero, and ΔK = ½ (μΔx) vy

2

Energy, final

Integrating over all the elements, the total kinetic energy in one wavelength is Kλ = ¼μω2A 2λThe total potential energy in one wavelength is Uλ = ¼μω2A 2λThis gives a total energy of

Eλ = Kλ + Uλ = ½μω2A 2λ

Power Associated with a WaveThe power is the rate at which the energy is being transferred:

The power transfer by a sinusoidal wave on a string is proportional to the

Frequency squaredSquare of the amplitudeWave speed

2 2

2 2

1122

AE A vt T

μω λμωΔ

℘ = = =Δ

How to control the frequency of the normal modes

Longer strings lower frequencies.Cello vs violin

Higher tension (F) higher frequencies.More massive strings lower frequencies.