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Linear Wave Equation The maximum values of the transverse

speed and transverse acceleration are vy, max = A ay, max = 2A

The transverse speed and acceleration do not reach their maximum values simultaneously v is a maximum at y = 0 a is a maximum at y = A

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The Linear Wave Equation, cont. The wave functions y (x, t) represent

solutions of an equation called the linear wave equation

This equation gives a complete description of the wave motion

From it you can determine the wave speed The linear wave equation is basic to many

forms of wave motion

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Linear Wave Equation, General

The equation can be written as

This applies in general to various types of traveling waves y represents various positions

For a string, it is the vertical displacement of the elements of the string

For a sound wave, it is the longitudinal position of the elements from the equilibrium position

For em waves, it is the electric or magnetic field components

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Linear Wave Equation, General cont The linear wave equation is satisfied by

any wave function having the form y (x, t) = f (x vt)

Nonlinear waves are more difficult to analyze A nonlinear wave is one in which the

amplitude is not small compared to the wavelength

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13.4 Linear Wave Equation Applied to a Wave on a String

The string is under tension T

Consider one small string element of length s

The net force acting in the y direction is

This uses the small-angle approximation

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Linear Wave Equation and Waves on a String, cont s is the mass of the element Applying the sinusoidal wave function to

the linear wave equation and following the derivatives, we find that

This is the speed of a wave on a string It applies to any shape pulse

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13.5 Reflection of a Wave, Fixed End

When the pulse reaches the support, the pulse moves back along the string in the opposite direction

This is the reflection of the pulse

The pulse is inverted when it is reflected from a fixed boundary

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Reflection of a Wave, Free End With a free end, the

string is free to move vertically

The pulse is reflected

The pulse is not inverted when reflected from a free end

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Transmission of a Wave When 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

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Transmission of a Wave, 2 Assume a light string is attached to a

heavier string The pulse travels through the light string

and reaches the boundary The part of the pulse that is reflected is

inverted The reflected pulse has a smaller

amplitude

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Transmission of a Wave, 3 Assume a heavier

string is attached to a light string

Part of the pulse is reflected and part is transmitted

The reflected part is not inverted

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Transmission of a Wave, 4 Conservation of energy governs the

pulse When 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

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13.6 Energy in Waves in a String Waves transport energy when they

propagate through a medium We 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

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Demonstration for energy transfer by wave propagation

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A sinusoidal wave on a string

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Energy, cont. Each element can be considered to have a mass of

m Its kinetic energy is K = 1/2 (m) vy

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The mass m is also equal to x and K = 1/2 (x) vy

2

As the length of the element of the string shrinks to zero, the equation becomes a differential equation: dK =1/2 (x) vy

2 = 1/2 2A2cos2(kx – t) dx

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Energy, final Integrating over all the elements, the total

kinetic energy in one wavelength is K = 1/42A 2

The total potential energy in one wavelength is U = 1/42A 2

This gives a total energy of E = K + U = 1/22A 2

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Power Associated with a Wave The 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 Square of the frequency Square of the amplitude Speed of the wave

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13.7 Introduction to Sound Waves Sound waves are longitudinal waves They travel through any material

medium The speed of the wave depends on the

properties of the medium The mathematical description of

sinusoidal sound waves is very similar to sinusoidal waves on a string

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Speed of Sound Waves Use a compressible gas as

an example with a setup as shown at right

Before the piston is moved, the gas has uniform density

When the piston is suddenly moved to the right, the gas just in front of it is compressed Darker region in the diagram

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Speed of Sound Waves, cont When the piston comes to

rest, the compression region of the gas continues to move

This corresponds to a longitudinal pulse traveling through the tube with speed v

The speed of the piston is not the same as the speed of the wave

The light areas are rarefactions

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Description of a Sound Wave The distance between two successive

compressions (or two successive rarefactions) is the wavelength,

As these regions travel along the tube, each element oscillates back and forth in simple harmonic motion

Their oscillation is parallel to the direction of the wave

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Displacement Wave Equation The displacement of a small element is

s(x,t) = smax sin (kx – t) smax is the maximum position relative to

equilibrium This is the equation of a displacement

wave k is the wave number is the angular frequency of the piston

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Pressure Wave Equation The variation P in the pressure of the gas as

measured from its equilibrium value is also sinusoidal

P = Pmax cos (kx – t) The pressure amplitude, Pmax is the maximum

change in pressure from the equilibrium value The pressure amplitude is proportional to the

displacement amplitude Pmax = v smax

V is the speed of the wave.

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Sound Waves as Displacement or Pressure Wave A sound wave may

be considered either a displacement wave or a pressure wave

The pressure wave is 90o out of phase with the displacement wave

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Speed of Sound Waves, General The speed of sound waves in air

depends only on the temperature of the air

v = 331 m/s + (0.6 m/s . oC) TC

TC is the temperature in Celsius The speed of sound in air at 0o C is 331

m/s

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Speed of Sound in Gases, Example Values

Note: temperatures given, speeds are in m/s

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Speed of Sound in Liquids, Example Values

Speeds are in m/s

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Speed of Sound in Solids, Example Values

Speeds are in m/s; values are for bulk solids

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13.8 The Doppler Effect The Doppler effect is the apparent change in

frequency (or wavelength) that occurs because the relative motion between the source of a wave and the observer

When the motion of the source or the observer moves toward the other, the frequency appears to increase When the motion of the source or the observer

moves away from the other, the frequency appears to decrease

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Doppler Effect, Observer Moving

The observer moves with a speed of vo

Assume a point source that remains stationary relative to the air

It is convenient to represent the waves with a series of circular arcs concentric to the source

These surfaces are called a wave front

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Doppler Effect, Observer Moving, cont The distance between adjacent wave

fronts is the wavelength The speed of the sound is v, the

frequency is ƒ, and the wavelength is When the observer moves toward the

source, the speed of the waves relative to the observer is vrel = v + vo The wavelength is unchanged

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Doppler Effect, Observer Moving, final The frequency heard by the observer, ƒ ’,

appears higher when the observer approaches the source

The frequency heard by the observer, ƒ ’, appears lower when the observer moves away from the source

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Doppler Effect, Source Moving Consider the source

being in motion while the observer is at rest

As the source moves toward the observer, the wavelength appears shorter

As the source moves away, the wavelength appears longer

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Doppler Effect, Source Moving, cont When the source is moving toward the

observer, the apparent frequency is higher

When the source is moving away from the observer, the apparent frequency is lower

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Doppler Effect, General Combining the motions of the observer and

the source

The signs depend on the direction of the velocity A positive value is used for motion of the observer

or the source toward the other A negative sign is used for motion of one away

from the other

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Doppler Effect, final Convenient rule for signs

The word toward is associated with an increase in the observed frequency

The words away from are associated with a decrease in the observed frequency

The Doppler effect is common to all waves

The Doppler effect does not depend on distance

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Shock Wave The speed of the

source can exceed the speed of the wave

The concentration of energy in front of the source results in a shock wave

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13.9 Speed of Sound Waves, General The speed of sound waves in a medium

depends on the compressibility and the density of the medium

The compressibility can sometimes be expressed in terms of the elastic modulus of the material

The speed of all mechanical waves follows a general form:

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Speed of Transverse Wave in a Bulk Solid The shear modulus of the material is S The density of the material is The speed of sound in that medium is

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Speed of Sound in Liquid or Gas The bulk modulus of the material is B The density of the material is The speed of sound in that medium is

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Speed of a Longitudinal Wave in a Bulk Solid The bulk modulus of the material is B The shear modulus of the material is S The density of the material is The speed of sound in that medium is

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Seismic Waves When an earthquake occurs, a sudden

release of energy takes place at its focus or hypocenter.

The epicenter is the point on the surface of the Earth radially above the focus

The released energy will propagate away from the focus by means of seismic waves

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Types of Seismic Waves P waves

P stands for primary They are longitudinal waves They arrive first at a seismograph

S waves S stands for secondary They are transverse waves They arrive next at the seismograph

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Seismograph Trace

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Cross-section of the Earth showing paths of waves produced by an earthquake

Exercises

6, 13, 18, 23, 27, 33, 42, 46, 59, 60, 61, 68