Mechanical Waves

Chapter 16

Expectations

After this chapter, students will: know what a mechanical wave is. distinguish between transverse and longitudinal

waves. know how wavelength, period, and velocity are

related for periodic waves. identify the frequency, wavelength, amplitude,

and direction of travel of a wave from the wave equation.

Expectations

After this chapter, students will: calculate the speed of a wave on a string. recognize sound as a longitudinal wave. relate power to sound intensity and intensity

levels. apply Doppler effect calculations to situations

involving moving sources of sound, or moving observers.

Waves: What Are They?

A wave is a travelling condition or disturbance. Energy travels from one place to another by means of a wave.

Transverse wave: disturbance is perpendicular to travel direction.

Longitudinal wave: disturbance is parallel to travel direction.

Periodic Waves

If the source of the disturbance produces it repeatedly, at equal time intervals, the resulting wave is called periodic.

Like anything else periodic, these waves are characterized by an amplitude, a period, and a frequency.

Periodic Waves

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

0.00 5.00 10.00 15.00 20.00 25.00 30.00

time

surf

ace

hei

gh

tT

Periodic Waves

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

0.00 2.00 4.00 6.00 8.00 10.00 12.00

location

surf

ace

he

igh

t

l

Periodic Waves

Amplitude: maximum magnitude of disturbance

Period: time required for one complete cycle

Wavelength: distance required for one complete cycle

Frequency: number of cycles per second of time

Periodic Waves

Relationships:

f

vvf

fvT

f

ll

l

1

The Wave Equation

We can write an expression for the disturbance as a function of both position and time:

This is called the wave equation.

l x

ftAy2

2sin

“+” if wave travels toward –x

“-” if wave travels toward +x

The Wave Equation

Follow the point on the wave where y = 0: we see that this wave is moving toward the right (+x).

The Wave Equation

The wave equation for this rightward-moving wave:

If we freeze time (constant t, “snapshot”):

then we have y as a function of position (x), and t is a constant phase angle whose value depends on the time at which we stopped the clock.

l x

ftAy2

2sin

l x

Ay t

2sin

The Wave Equation

The wave equation for this rightward-moving wave:

Now, if we choose just one location (constant x):

then we have y as a function of time (t), and x is a constant phase angle whose value depends on the x we chose.

l x

ftAy2

2sin

xftAy 2sin

The Wave Equation

If we see an equation that looks like:

... we can write down the amplitude, frequency, velocity, and wavelength of the wave it describes.

xty m 5796.4 s 8.1570sinm15.0 -1-1

direction.x in the m/s, 343

m 1.372 m 5796.42

Hz 250 s 1570.8 2 m 15.0

22sin

1-

1-

l

ll

l

fv

ffA

xftAy

Speed of a Wave on a String

A transverse wave on a string (or wire, rope, cable, etc.) depends on the tension in the string, as well as its diameter and the material from which it is made:

LmF

v tension force

(string mass / string length)

Sound

Sound is a longitudinal wave in which the disturbance is a change in the pressure in the air (or other medium).

Sound

Like any wave, sound is characterized by a velocity and a wavelength.

Sound

As with any wave, the disturbance travels, and energy travels, but the material (air) “sloshes back and forth” mostly in one place.

Sound: Speed

The speed of a sound wave depends on the mechanical properties of the material through which it moves.

Gas:

Liquid:

Solid:

mc

Tkcv

V

P

adBv

Y

v

Sound: Energetics

The energy carried by a sound wave per second is its power:

Power has SI units of J/s = W (watts)

t

EP

Sound: Energetics

We define the intensity of a sound wave as the power it carries perpendicularly through a surface, divided by the area of the surface:

Intensity has SI units of W/m2.

A

PI

Intensity decreases from surface 1 to surface 2.

Sound: Energetics

If the source of the sound wave radiates waves equally in all directions (spherically symmetric):

24 r

PI

sphere area

Sound: Energetics

We can compare the intensities of two sound waves in terms of intensity levels:

is dimensionless, but is labeled with units of decibels (dB).

I0 is a reference level: usually the “threshhold of hearing,” 1.0×10-12 W/m2 .

0

logdB 10I

I

The Doppler Effect

The Dopeler Effect is what happens when a stupid idea seems like a good idea because it comes at you really fast.

But wait: we wanted to talk about the Doppler Effect, instead.

The Doppler Effect

The Doppler Effect is the change in observed frequency of a sound wave (other sorts of waves, too) because of the movement of either the source, or the observer, or both, relative to the air through which the sound is traveling.

The Doppler Effect

The Doppler Effect is the change in observed frequency of a sound wave (other sorts of waves, too) because of the movement of either the source, or the observer, or both, relative to the air through which the sound is traveling.

The observer’s motion causes him to intercept more waves per second than he would if he were standing still.

The Doppler Effect

Equations for a stationary source and moving observer:

v

vff OSO 1

observer moves toward source

observer moves away from source

v

vff OSO 1

The Doppler Effect

General case (both source and observer move relative to the air):

vvvv

ffS

O

SO

1

1

“+” if observer moves toward source; “-” if observer moves away from source

“-” if source moves toward observer; “+” if source moves away from observer

Ch. 16 Takeaways

Wavelength, frequency, period, velocity:

Wave equation:

f

vvf

fvT

f

ll

l

1

l x

ftAy2

2sin

Ch. 16 Takeaways

Transverse wave on string:

Sound intensity: Spherically symmetric source:

Sound intensity level:

LmF

v

A

PI 24 r

PI

0

logdB 10I

I

Ch. 16 Takeaways

Doppler effect: moving source, stationary observer:

v

vff

SSO

1

1

v

vff

SSO

1

1

source moves toward observer

source moves away from observer

speed of sound

speed of source

observed frequency

source frequency

Ch. 16 Takeaways

Doppler effect: moving observer, stationary source:

v

vff OSO 1

observer moves toward source

observer moves away from source

v

vff OSO 1

Ch. 16 Takeaways

Doppler effect, general (both source and observer move):

vvvv

ffS

O

SO

1

1

“+” if observer moves toward source; “-” if observer moves away from source

“-” if source moves toward observer; “+” if source moves away from observer