Standing waves on a string (review)

n=1,2,3...

12nf

L

vn

vf

nn

121 L L

L2

1

21

2222 L L

L

2

22

323 L

3

23

L

12 nL n

Ln

2

Different boundary conditions:•Both ends fixed (see above)•Both ends free (similar to both ends fixed )•One end fixed and on end free (next slide)

1nffn

One end fixed and on end free

343 L

141 L

nnL 4

...5,3,1

4

nn

Ln

nnn

n fT

v nf

L

vnvf

nn 14

n=1

n=3

Standing waves on a string (review)

Standing waves in tubes (longitudinal)

Waves in tubes (pipes) can be described in terms of:•displacement vibrations of the fluid•pressure variations in the fluid

A pressure node is a displacement antinode and vice versa

Open and both ends closed pipes

121 L

222 L

323 L

n=1

n=2

n=3

Closed (both ends): displacement pressureOpen: pressure displacement

nnL 2

...3,2,1

2

nn

Ln

nnn

n fT

v 12

nfL

vnvf

nn

One end open pipes (stopped pipes)

n=1

n=3

displacement pressure

343 L

141 L

nnL 4

...5,3,1

4

nn

Ln

nnn

n fT

v nf

L

vnvf

nn 14

Example: Standing sound waves are produced in a pipe that is 0.6 m long. For the first overtone, determine the locations along the pipe (measured from the left end) of the displacement nodes. The pipe is closed at the left end and open at the right end.

m

mmL

4.02/

8.03

6.04

3

4

3

3

mL 6.0

Nodes at 0.0 m and 0.4 m

SoundAcoustic waves in the range of frequencies: 20Hz -20,000Hz

Sound waves:• can travel in any solid, liquid or gas• travel faster in a medium that is more dense• in liquids and gases sound waves are longitudinal ONLY!• longitudinal and transversal sound waves could propagate in in solids

Sound in air is a longitudinal wave that contains regions of low and high pressure

Vibrating tuning fork

These pressure variations are usually small – a “loud” sound changes the pressure by 2.0x10-5 atm

Pressure sensor

Speed of Sound

F

v Speed of waves in strings (review):

Speed of sound waves:B

v

pBV

dV 1B is bulk modulus, defined as

is density

Speed of sound waves in a gas:

0pB

is ratio of heat capacities,

0p is the equilibrium pressure of the gasM

RTv

Example: Speed of Sound in Air

The speed of sound in an ideal gas is

v RTM

, where

"ratio of heat capacities"

R molar gas constant

= 8.314 J/molKT temperature in K

M molar mass in kg

Air is a diatomic gas, so

1.40 .

The molar mass of dry air, containing

about 20% O2 and 80% N2 , is

M 0.0290 kg/mol .

Let the ambient temperature be

T 22 °C = 72 °F = 295 K .

Then

v (1.40)(8.31 J/molK)(295 K)

0.0290 kg

344 m/s

.

This agrees well with experiments.

Speed of Sound in Some Common Substances

1. Air (20 oC) 344

Substance Speed (m/s)

3. Water 1,140

5. Human tissue 1,540

6. Aluminum 5,100

7. Iron and steel 5,200

4. Lead 1,200

2. Helium 1,006

Pressure variations

x

y

V

dV

pBV

dV 1

x

yBp

tkxBkAtxp

tkxAtxy

sin,

cos,

Example: Displacement Wave Amplitude of Sound in Air

Let the pressure wave amplitude of a sound wave in air at pressure p0 1.00 atm

be pmax 0.030 Pa (a typical value). The frequency of the wave is f 1000 Hz

and the speed of sound in the air is v 344 m/s .

What is the displacement wave amplitude of the sound wave?

Solution :

pmax BAk, so A pmax

Bk (1)

1 atm = 1.013 kPa and = 1.40 for air.

B p0 (1.40)(1.013105 Pa) 1.42 105 Pa. (2)

k v

and 2 f , so k 2 fv

(3)

Putting (3) and (2) into (1) gives

A pmaxv

2Bf

(0.030 Pa)(344 m/s)

2 (1.42 105 Pa)(1000 Hz)1.2 10 8 m 12 nm .

We classify sounds according to their waveforms:

piano note

Properties of Sound

- A “pure tone” is a sound with a sinusoidal waveform.

- This is a sound with a single frequency; produced by tuning fork, etc.

- A “complex tone” is a sound that repeats itself but is not sinusoidal

Most sounds are like this!

A note from a musical instrument will be mostly sinusoidal, but have a character all its own that is specific to the instrument.

“Noise” is sound with a complex waveform that does not repeat

No definite wavelength or frequency

We use three qualities to characterize how we perceive sound:

Perception of Sound

3. The tone quality of a sound is how we distinguish sounds of the same pitch and loudness (how we perceive the qualities of the waveform)

1. The pitch of a sound is how “high” or “low” we perceive a sound to be (directly related to how we perceive frequency); combinations of notes that are “pleasing” to the ear have frequencies that are related by a simple whole-number ratio (Pythagoras)

2. The loudness of a sound is how we perceive the amplitude of the sound wave

1. Pitch. 2. Loudness. 3. Tone quality.

The unit of sound loudness is the decibel (dB)

Quiet: 30 dB; Moderate: 50 dB; Noisy: 70 dB; Very loud: 90 dB; Problems: 120 dB

Anything that causes pressure vibrations creates sound!

Musical Instruments

Stringed musical instruments produce sound by vibrating a wire or string

A plucked string on a guitar produces a different sound than a violin

The tension in the string is used to adjust the wave speed and the frequency

Wind instruments produce sound by pressure

waves in a tube

The sound reflects partly at the open end

Valves change the effective length of the tube

Complex Waves

To produce a pure fundamental tone on a guitar string:

Pull the template away fast. Then the string will vibrate in its fundamental mode.

Pluck a guitar string in the middle:

Fourier Theorem: The initial shape is a superposition of the sinusoidal shapes for the fundamental mode and for higher harmonics.

The sound produced is the fundamental frequency plus higher harmonics. Different musical instruments sound different even when playing the same tone partly because the harmonic contents are different.

Fourier Analysis

Fourier Theorem for an Even Function

y(x, t 0) An cos(knx)n=0

Two fixed ends: kn 2n

and n 2L

n, so

y(x, t 0) An cosnLx

n=0

.

Fourier Analysis

"Fourier analysis" allows the amplitudes Anto be determined from y(x, t 0) (see text

supplement "Waves"). For our y(x, t 0),

one obtains:

An 8A

2n2 if n is odd

An 0 if n is even.