2/23/00
More on Sound Transmission
In a sound transmission
system, there are five possible outcomes we will consider:
1. Sound transmission =
2. Sound reflection =
3. Sound absorption =
4. Sound diffraction =
5. Sound refraction =
Types of sound fields
1. Free field =
2. Reverberant field =
3. Diffuse field =
Exponents
The general expression for
an exponential series is:
is the base number which is used (the exponent) times in multiplication resulting in the
answer .
is said to be raised to the
th power.
If X= and n=
, then Y=
because
If X= and n=
, then Y=
because
If X= and n= , then Y=
because
What if the exponent n is a
negative number?
If X= and n=
, then Y=
because
If X= and n=
, then Y=
because
If X= and n= , then Y=
because
What if the exponent n is 1?
Any base X raised to the power of 1 equals
What if the exponent n is
zero?
Any base X raised to the zero power equals
Laws of Exponents :
1. Xm x Xn =
Example:
2. Xm
---- =
Xn
Example:
3. (Xm) n =
Example:
Logarithms
The general expression for a
logarithm is:
When you take the log of a
number, you are solving for the
For our purposes, we will
always use base 10
For odd logs like log10
21, you have to use
Laws of Logarithms :
1. log (ab) =
Example:
2. a
log ---- =
b
Example:
3. log (a) b =
Example:
4. 1
log ---- =
a
Example:
Decibels
The range of human hearing
from threshold (just
hearing) to pain threshold (when sound becomes
painful) is
The difference between the sound pressure of a
just detectable sound to the loudest sound
tolerated is
about !
Having to do calculations with these numbers
would be very cumbersome! (in other words –
! )
In order to make calculations easier and quicker,
we use a
scale of measurement instead of
a scale
of measurement
interval scale =
ratio scale
(logarithmic scale) =
Decibel is the logarithm of
the ratio of
Bel (named after ) was
the unit named for these calculations; however,
we typically use the decibel
( ) as
the bel is still too large to use easily
The symbol for the decibel
is
Ratio is a way of telling us
how one thing relates
to another
We use a in our ratio
Ratios are measures, not
(meaningless if you do
not know the )
In our course, we will consider two different
values
dependent upon what we are
measuring:
dB IL =
dB SPL =
These two are not the same
dB IL
is a ratio
The reference
This is equal to
Calculation:
dB IL =
Example:
If the output of a loudspeaker is 10 –6 W/m2,
what is its output in dB IL?
dB IL =
What if the output is equal to the reference?
dB IL =
=
=
=
=
Thus
dB SPL
is a ratio
The reference is
This is equal to
which is the
same as
which is the
same as
which is the
same as
which is the
same as
The calculation for dB SPL is different due to the
relationship between pressure and intensity
Intensity is
If we substitute pressure into the intensity
calculation, we get:
Calculation:
dB SPL =
Using the laws of logs, we can simplify this
equation to:
dB SPL =
=
Example:
If the output of a loudspeaker is 2 x 10 –2 Pa,
what is its output in dB SPL?
dB SPL =
=
=
=
=
What if the output is equal to the reference?
dB SPL =
=
=
=
=
Thus