Interference of sound waves
Sound II.
Two sound waves superimposed
Constructive Interference
Interference of sound waves
Standing waves
Complex sound waves
Destructive Interference
Interference due to path
difference
Noise canceling headphones
path difference =δ =r2 – r1
r1
Noise
Wave 1
Wave 1
A
r2
Wave 2
Wave 2
Anti-noise
In phase at x=0
Condition for constructive interference
Condition for destructive interference
where m is any integer
Interference
δ=0
Constructive Interference
Superposition of waves
at A shows interference
due to path differences
δ = mλ
1
2
δ = (m + )λ
m = 0 + 1, + 2,….
Interference
λ
δ= 2
Destructive Interference
Sum
λ
1
Interference
Interference of sound waves
Phase shift due to path differences
δ=λ Constructive Interference
x
When
r2 –r1 =mλ
When r2 – r1 = (m+½) λ
m is any integer
Constructive Interference
Destructive Interference
Example 14.6 Path difference for two sources.
Example
An experiment is performed to measure the speed of sound
using by separating the sound from a single source along
two separate paths with different path lengths and
combining them at the detector. For a frequency of 3.0
kHz (assume vsound =340 m/s);
A) What would the smallest path difference be to observe a
minimum in intensity
v
340m / s
r2 – r 1 = λ
=
=
= 5.7 x10−2 m = 5.7cm
2f 2(3 x103 s −1 )
2
B) What would the smallest (non-zero) path difference be
to observe a maximum in intensity.
r2 – r 1 =
λ = 11cm
r2-r1 =0.13 m
At position P the listener hears the first minimum in
sound intensity. Find the frequency of the oscillation.
vsound =340 m/s
At position P the path difference is equal to λ/2. (first
minimum) destructive interference.
λ
= r2 − r1 = 0.13m
2
λ = 2(0.13) = 0.26m
v 340m / s
f= =
= 1.31x103 Hz
λ
0.26m
Standing Wave
Standing Waves
Standing waves (waves on a
string)
Standing waves in air columns.
• A standing wave is formed by reflections
back and forth at the boundaries of a
media.
• The standing wave does not carry energy
but serves to store energy.
• The standing wave stores energy of waves
with specific wavelengths.
2
Standing Wave with 3 nodes
– At different times.
Standing wave on a string
Nodes
The Standing wave
doesn’t
“move” it just stands in
one place
Anti-nodes
Fixed end
Standing Wave Conditions
Standing Waves
One node at each end with
additional nodes only at specific
positions
L
• A standing wave is generated by superposition
of two waves with the same frequency and
wavelength traveling in opposite directions.
n=1
Simulation of a standing wave.
n=2
Distance between nodes = λ/2
A string with length L can support
standing waves of only at certain
wavelengths.
L=
http://www.walter-fendt.de/ph14e/stwaverefl.htm
n=3
Standing wave frequencies and
wavelengths
n=1
n=2
n=3
f1 =
where n=1, 2, 3… integer values
n=1 called the fundamental
n=2 called the second harmonic etc.
Example
Find the fundamental and second harmonics of a steel
wire 1.0m long fixed at both ends. The speed of the wave
in the string is v=200 m/s
v
2L
f2 =
v
= 2f1
L
f3 =
3v
= 3f1
2L
Generally
fn =
v
v
=
n = nf1
λ 2L
λ
n
2
λ1 = 2L
f1 =
v
2L
λ2 = L
f2 =
v
L
=
200
= 100Hz
2(1)
=2f1 =200 Hz
L
where n = 1, 2, 3......
3
Standing waves in air columns
Fundamental Frequency
N
L
L=
one end open
one end closed
2 ends open
2 ends closed
N
A
λ1
2
A
L=
λ1= 2L
N
λ1
2
F1=
A
L=
λ1
4
λ1= 4L
λ1= 2L
v
F1=
2L
Cylinder open at both ends
Harmonics
v
2L
F1=
v
4L
fn = n f1
F1 lower by a
factor of 2
v is the speed of sound in air
Cylinder open at one end closed at
one end - Harmonics
n = 1, 2, 3, 4 ....... All harmonics
Summary
For a cylinder with the same length
5f1
Frequency
4f1
3f1
2f1
f1
0
fn =nf1
n= 1, 3, 5, 7 .... Only odd harmonics
v
n
2L
fn = f1n
7f1
5f1
fn =
n=1,2,3...
3f1
f1
n=1, 3, 5.......
only odd harmonics
fn =
all harmonics
both ends
open/ closed
v
n = f1n
4L
0
one open
one closed
Example
A cylinder 5.0 cm in length is closed at one end and
open at the other end. Find the frequency of the third harmonic
of the standing wave in the column. vair =340 m/s
L
v
v
λ1 = 4L
f1= λ = 4L
1
λ 3=
4L
3
f3 = 3f1 =
3v
4L
=
3(340)m / s
= 5.1x103 Hz
4(0.05)m
Musical Instruments
String Instruments
Frequency due to standing waves on the string.
The body of the instrument acts as a resonator to move
air to amplify the sound.
4
Musical Instruments
Wind instruments
The sound is produced by vibrating element (reed,
vocal chord) and certain specific frequencies are
enhanced by resonance in the air column
Complex waves
• In general sound waves are a combination
of different frequencies.
• The superposition of waves with different
frequencies gives rise to the characteristic
quality (timbre) of the sound.
• The different frequencies can be
determined by mathematical procedure
called a Fourier Transform.
Complex waves consist of different frequency
components , i.e. harmonics.
displacement
Time
relative
amplitude
Frequency
5