The Wave part of Phys 103

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Wave Propagation A wave is a disturbance in an elastic medium which travels, or propagates through the medium. The wave is intangible. The medium itself does not travel, but only oscillates back and forth. So there is not a net transport of matter from place to place.

However, a wave transports energy from place to place, through the medium.

Waves come in many forms, all with certain common properties. There are waves in a plucked string, seismic waves, sound waves, electromagnetic waves. These are different sorts of disturbances propagating in different sorts of media.

In this course, we will consider the common properties.

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Wave Motion

y



Af

  

x

 

T t

   y is the displacement from equilibrium at position x and at time t. f is an unspecified function.

i) wave speed is a property of the medium.

ii) shape of the wave pulse is unchanged as it travels iii) two or more wave pulses that exist at the same place & time in a medium add—superimpose.

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Harmonic wave – a wave of a particular shape that repeats itself. It’s periodic.

y



A

cos 2   

x

 

t T

  

A

cos 2 

f

 

x c



t

 

y

  

A

cos 

kx

 

t

 Each point in the medium (x) is displaced from equilibrium (y). the wave pattern moves through the medium.

 

ct k

   2   2 

f

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superposition reflection “Standing waves”

y y y

   

 

x

, 

y

1

A

 

cos





kx

2

A

sin 

y

2

 



t



 sin 

t A

cos



kx

 

t



cos 

a



b

  cos

a

cos

b

 sin

a

sin

b

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c



F m L

Stretched string Only vibrations that “fit” in the length of the string will persist. This is an example of resonance. Every physical system has “natural” modes at which it will vibrate. The natural modes depend on the physical properties of the system: mass, elasticity, size.

We saw this same phenomenon with the spring and the pendulum.

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“Beats” In this case, two waves are traveling in the same direction, but with slightly different frequencies.

y y

 

y

1

A

  cos

y

 2 2 

f

1

t

  cos  2 

f

2

t

 

y

 2

A

cos   2 

f

1  2

f

2

t

  cos   2 

f

1  2

f

2

t

  9

Spectrum

y

 

i A i cos

  2  

i x

 2 

f i t

  10

.

Energy While the medium in which the wave propagates does not flow from one place to another, the wave disturbance nonetheless carries energy from one place to another. Each mass element, dm, of the medium executes simple harmonic motion. K is the restoring force constant. It’s related to the frequency by  

K dm E

 1 2

Ky

2  1 2

dmv

2

E

 1 2

dm

 2

A

2 cos 2 

kx

 

t

  1 2

dmA

2  2 sin 2 

kx

 

t



E

 1

dm

 2

A

2  cos 2 

kx

 

t

  sin 2 Over one cycle, the cosine-squared and sine-squared average to 2 1 2 

kx

 

t

  . The total mass of the medium spanning one cycle (or one wavelength) is  , where  is the mass per unit length of the medium.

E

  2

f

2   2

f

2  2  2

f

2 

A

2

f

2 In terms of the wave speed, c,

E

 2  2 

c f A

2 .The energy flux is the power transported through the medium by the wave:

P



fE

 2  2 

c f

2

A

2 The intensity is the power pr unit area through which the power is transported:

I



P a

 2  2 

c f

2

A

2 , were  is the mass per unit volume.

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Pressure waves--Sound

Compression, or longitudinal waves.

Medium oscillates parallel to direction of propagation.

Pressure amplitude, y

p .

Speed of sound waves depends on density, pressure, temperature & elasticity of the medium.

Doppler effect. . . .

deciBels. . . 12

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