Measuring sound intensity in decibel, dB
 I 
β = 10 log 
 I0 
Number of dB, decibel, dimensionless parameter
−12
2 ­ reference value, corresponding to threshold of sensitivity of a normal ear at about optimal frequency
I 0 = 10
W/m
I 
log  = β / 10
 I0 
I = I 0 ⋅ 10
β / 10
Measuring sound intensity in decibel, dB
 I 
β = 10 log 
 I0 
Number of dB, decibel, dimensionless parameter
I = I 0 ⋅ 10
β / 10
­ reference value, corresponding to threshold of sensitivity of a normal 0
ear at about optimal frequency
An example: TV is turned down from 75 dB to 60 dB.
How does its sound intensity change?
I = 10
−12
W/m
2
From I1 = I 0 ⋅10
To I 2 = I 0 ⋅10 = 10
7.5
= 10
6
Goes down by a factor of −12
−12
−5
⋅10 W/m = 3.1 ⋅10 W/m
7.5
2
−6
⋅10 W/m = 10 W/m
6
2
2
101.5 = 31
In general: sound level is going up/down by x dB ­ 10
β / 10
x / 10
x / 10
I ( β + x) = I 0 ⋅ 10
⋅ 10
= I ( β ) ⋅ 10
sound intensity is multiplied by ± x / 10
2
Reflection of a wave on string.
Full or partial reflection always occurs at a boundary the wave cannot cross
For example, with a medium, which does not support the waves of that kind.
Reflection depends a lot on the boundary conditions: a clamped string vs. a freely sliding string.
Reflection of a wave on string.
A partial reflection always occurs if properties of the medium change abruptly.
Abruptly compared to what?
Examples: reflection of sound off a rock, light off a glass window
Two connected strings. What is different between them, F ? µ ?
How do the wave speeds in them compare? What about the amplitudes?
v=
F
µ
1
1
2 2
2
2
P = µ ωA v =
Fω µ A
2
2
A12 / A22 = µ 2 / µ1
Refraction of waver waves…
Refraction – change in the direction of wave propagation at a boundary.
Waves change direction, or refract, as they enter shallow water, because the waves slow down. Reflection of a wave on string.
Full or partial reflection always occurs at a boundary the wave cannot cross
For example, with a medium, which does not support the waves of that kind.
Reflection depends a lot on the boundary conditions: a clamped string vs. a freely sliding string.
Reflection of a wave on string.
A partial reflection always occurs if properties of the medium change abruptly.
Abruptly compared to what?
Examples: reflection of sound off a rock, light off a glass window…
Two connected strings. What is different between them, F ? Μ ?
How do the wave speed in them compare? What about the amplitudes?
v=
F
µ
1
1
2 2
P = µ ωA v =
F ω 2 µ A2
2 2
Refraction of waver waves…
Refraction – change in the direction of wave propagation at a boundary.
Waves change direction, or refract, as they enter shallow water, because the waves slow down. Two waves of the same frequencies and wavelengths propagating in opposite directions interfere to make a composite standing wave.
Unlike the two interfering propagating waves, the standing wave does not seem to go anywhere, just like an oscillation…
But it has all the essential properties of the waves:
spatial structure, wavelength, wave number. It also has amplitude and frequency just as the oscillations do.
A longitudinal standing wave:
How do you create a standing wave? Make a traveling wave interfere with its reflection off some boundary!
There are some points, nodes, N, where the string does not move at all.
There are other points, antinodes, A, where the oscillations have the maximum amplitude.
How far apart are those nodes and antinodes?
We normally consider standing waves in a confined space, like on string clamped on both sides.
It imposes some boundary conditions, which have to be matched by all standing waves, which are “allowed”.
v
f =
λ
A propagating wave on a string can have any frequency/length.
In contrast, stable standing waves on a string can only have some discrete, well defined wavelengths and frequencies.
How many different wavelengths are allowed? What is the smallest/largest one?
Some math:
largest wavelength / lowest frequency
λ
=L
2
λ = 2 L,
v
f =
2L
Higher harmonics, wavelengths:
λ
λ
λ
2 ⋅ = L, 3 ⋅ = L, 4 ⋅ = L
2
2
2
λm
2 L λ1
m⋅
= L, λm =
=
2
m m
Higher harmonics, frequencies:
v
v
1
fm =
=m
= m = mf1
λm
2L
λ1
Some math:
largest wavelength / lowest frequency
v
f1 =
2L
f m = mf1
Fundamental mode
Higher harmonics
More math:
interference
y1 = A cos(kx − ωt )
y2 = − A cos(kx + ωt )
Composite wave:
y = y1 + y2 =
= −2 A sin kx sin ωt
Importantly, unlike a propagating wave, where kx and ω t appear in the argument of one sin (or cos), in a standing wave they are separated between two sin functions. Standing waves in 2D systems: