Lesson 15: Standing waves, Sound (11.10-12.4)

Lesson 15, page 1

Interference

Interference is a consequence of the principle of superposition.

The waves need to be coherent – same frequency and maintain a constant phase relationship.

(For incoherent waves the phase relation varies randomly.)

Constructive interference occurs when the waves are in phase with each other. The

amplitude of the resulting wave is the sum of the amplitudes of the two waves, |A1 + A2|

Destructive interference occurs when the waves are 180o out of phase with each other.

The amplitude of the resulting wave is the difference of the amplitudes of the two waves,

|A1 – A2|

Otherwise the wave has amplitude between |A1 – A2| and (A1 + A2).

The two rods vibrate and down in phase and produce

circular water waves. If the waves travel the same

distance to a point they arrive in phase with each other

and interfere constructively. At other points, the phase

difference is proportional to the path difference. Since

one wavelength of path difference corresponds to a phase

difference of 2 radians,

rad2

differencephase21

dd

Lesson 15: Standing waves, Sound (11.10-12.4)

Lesson 15, page 2

If the path difference d1 ‒ d2 = n (n is any integer) the phase difference is 2n rad and

constructive interference occur at P. If the path difference d1 ‒ d2 = , etc., the

phase difference is , 3, 5, etc. and destructive interference occurs.

When coherent waves interfere, the amplitudes add for constructive interference and subtract for

destructive interference. Since intensity is proportional to the square of the amplitude, you

cannot simply add or subtract the intensities of the coherent waves when they interfere. For

incoherent waves, there is no fixed phase relation. The total intensity is the sum of the intensities

of the individual waves.

Diffraction

Diffraction is the spreading of waves around an obstacle. The obstacle must be similar in size to

the wavelength of the wave for the effect to be noticeable.

Many animations are on the web. http://www.youtube.com/watch?v=uPQMI2q_vPQ

Standing Waves

Standing waves occur when a wave is reflected at a boundary and the reflected wave interferes

with the incident wave so that the wave appears not to propagate. A wave propagating in the +x-

direction is described by

)sin(),( kxtAtxy

The inverted reflected wave is

)sin(),( kxtAtxy

(Why are there sines and not cosines?) The waves interfere and

kxtAtxy sincos2),(

This looks like

The places that are stationary are called nodes. Midway between the nodes are antinodes.

Suppose a string is held at both ends.

Lesson 15: Standing waves, Sound (11.10-12.4)

Lesson 15, page 3

The first four possible patterns are given. Higher orders are possible, but become less important.

(Bending the string takes energy. More bending requires more energy.)

For the top pattern = 2L and the frequency is

12

fL

vvf

The second pattern = L

12 22

2 fL

v

L

vvf

The third pattern 1.5 = L and

13 32

35.1

fL

v

L

vvf

The possible frequencies are multiples of the lowest frequency f1 which is called the fundamental

frequency. These are the natural frequencies or resonant frequencies of the string. We will

find a similar situation when discussing standing sound waves in a pipe.

To begin our discussion of music I give you the amazing Vi Hart:

http://www.youtube.com/watch?v=i_0DXxNeaQ0

Chapter 12 Sound Waves

We study the properties and detection of a particular type of wave – sound waves. A speaker

generates sound. The density of the air changes as the wave propagates.

Lesson 15: Standing waves, Sound (11.10-12.4)

Lesson 15, page 4

Notice that the displacement maxima and minima occur where the pressure variation is zero and

the pressure variation maxima and minima occur where the displacement is zero.

The range of frequencies that can be heard by humans is typically taken to be between 20 Hz and

20,000 Hz. Most people struggle to hear the highest frequencies and that ability lessens with

age.

Speed of Sound

Recall

Inertia

ForceRestoringv

In fluids

Bv

The speed of the wave in a fluid (especially air) depends on temperature. In solids

Lesson 15: Standing waves, Sound (11.10-12.4)

Lesson 15, page 5

Yv

Amplitude and Intensity of Sound Waves

A sound wave can be described either by talking about pressure or displacement. Since the

displacement creates the pressure change, there is a relationship between the amplitude of the

pressure p0 and the amplitude of the displacement s0. For a harmonic sound wave the relation is

00 svp

A larger amplitude wave appears louder, but the relation between amplitude and loudness is very

complicated. Loudness is subjective and depends on the response of the ear and the brain.

Usually the intensity and not the amplitude is used for loudness. Again for a harmonic wave

v

pI

2

2

0

“The most important thing to remember is that intensity is proportional to the amplitude

squared, which is true for all waves, not just sound.” (p 437)

Decibel Scale

The perception of hearing is roughly proportional to the logarithm of the intensity. The lowest

intensity of sound that can be heard by most people is

212

0 W/m100.1 I

Lesson 15: Standing waves, Sound (11.10-12.4)

Lesson 15, page 6

I0 is called the threshold of hearing. It is used as the reference level for measuring sound

intensity. The sound intensity level in decibels is defined as

0

10log)dB10(I

I

(Be sure to practice with the decibel scale. Logarithms can be tricky.) An intensity level of 0 dB

corresponds to the threshold of hearing.

For incoherent sound waves with intensities I1 and I2, the total intensity is

21 III

If the sound waves are coherent, the waves can interfere and the intensity is between |I1 – I2| and

I1 + I2, depending on the phase relationship between the two waves.

Decibels can be used in a relative sense. The difference in two dB readings

1

210

0

110

0

21012

log)dB10(

log)dB10(log)dB10(

I

I

I

I

I

I

Lesson 15: Standing waves, Sound (11.10-12.4)

Lesson 15, page 7

is related to the ratio of the intensities.

Standing Sound Waves

Recall that a standing wave is the superposition of two traveling waves. The wave reflects at the

boundary of the wave.

Pipe open at Both Ends

The boundary conditions are the same at both ends. Since the end is open to the atmosphere, the

pressure at the ends can not deviate much from atmospheric pressure. The ends are pressure

nodes. Pressure nodes are displacement antinodes.

From the diagram, the wavelengths satisfy

n

Ln

2

The frequencies

12

nfL

vn

vf

n

n

The index n is an integer and it can vary from 1, 2, etc.

Pipe Open at One End

The situation is different from the pipe opened at both ends. The closed end is a pressure

antinode. The air at the closed end is isolated from the atmosphere and the pressure can deviate

far from atmospheric. The air at the closed end is a displacement node since the rigid wall

prevents the air from moving.

Lesson 15: Standing waves, Sound (11.10-12.4)

Lesson 15, page 8

From the diagram, the wavelengths satisfy

n

Ln

4

The frequencies

14

nfL

vn

vf

n

n

This time n has odd values only (1, 3, 5, etc.)

Problem 35 Two tuning forks A and B, excite the next-to-lowest resonant frequencies in two air

columns of the same length, but A’s column is closed at one end and B’s column is open at both

ends. What is the ratio of A’s frequency to B’s frequency.

Since A excites the pipe open at one end, only the odd harmonics are possible

Lesson 15: Standing waves, Sound (11.10-12.4)

Lesson 15, page 9

14

nfL

vn

vf

n

n

Where n = 1, 3, 5, etc. Next to lowest resonant frequency refers to the second frequency. Here

that mean n = 3 and

143

4 L

v

L

vnf

A

For B, all the harmonics are possible since it is exciting a pipe open at both ends.

12

nfL

vn

vf

n

n

n = 1, 2, 3, etc. Next to lowest in this sequence corresponds to n = 2,

L

v

L

vnf

B2

22

Forming a ratio

4

3

2

2

4

3

22

43

v

L

L

v

L

vL

v

f

f

B

A