# Engineering Acoustics/Car Mufflers

## Introduction

A car muffler is a component of the exhaust system of a car. The exhaust
system has mainly 3 functions:

1\) Getting the hot and noxious gas from the engine away from the
vehicle

2\) Reduce exhaust emission

3\) Attenuating the noise output from the engine

The last specified function is the function of the car muffler. It is
necessary because the gas coming from the combustion in the pistons of
the engine would generate an extremely loud noise if it were sent
directly in the ambient surrounding through the exhaust valves. There
are mainly 2 techniques used to dampen the noise: the absorption and the
reflection. Each technique has its advantages and inconveniences.

!Muffler type \"Cherry
bomb\"\|right{width="150"}

## The absorber muffler

The muffler is composed of a tube covered by sound absorbing material.
The tube is perforated so that some part of the sound wave goes through
the perforation to the absorbing material. The absorbing material is
usually made of fiberglass or steel wool. The dampening material is
protected from the surrounding by a supplementary coat made of a bent
metal sheet.

The advantages of this method are a low back pressure and the relatively
simple design. The inconvenient aspect of this method is a low sound
damping compared to the other techniques, especially at low frequency.

Mufflers using the absorption technique are usually installed on sports
vehicles to increase the performances of the engine because of their low
back pressure. A trick to improve their
muffling ability consist of lining up several \"straight\" mufflers.

## The reflector muffler

Principle: Sound wave reflection is used to create a maximum amount of
destructive interferences

!Destructive
interference\|right{width="400"}

### Definition of destructive interferences

Let\'s consider the noise a person would hear when a car drives past.
This sound would physically correspond to the pressure variation of the
air which would make his ear-drum vibrate. The curve A1 of the graph 1
could represent this sound. The pressure amplitude is a function of the
time at a certain fixed place. If another sound wave A2 is produced at
the same time, the pressure of the two waves will add. If the amplitude
of A1 is exactly the opposite of the amplitude A2, then the sum will be
zero, which corresponds physically to the atmospheric pressure. The
listener would thus hear nothing although there are two radiating sound
sources. A2 is called the destructive interference. !Wave
reflection\|right{width="250"}

### Definition of the reflection

The sound is a travelling wave i.e. its position changes in function of
the time. As long as the wave travels in the same medium, there is no
change of speed and amplitude. When the wave reaches a frontier between
two mediums which have different impedances, the speed, and the pressure
amplitude change (and so does the angle if the wave does not propagate
perpendicularly to the frontier). The figure 1 shows two medium A and B
and the 3 waves: incident transmitted and reflected.

### Example

If plane sound waves are propagating across a tube and the section of
the tube changes at a point x, the impedance of the tube will change. A
part of the incident waves will so be transmitted in the part of the
tube with the new section value and the other part of the incident waves
will be reflected.

Animation

Mufflers using the reflection technique are most commonly used because
they damp the noise much better than the absorber muffler. However, they
often create higher back pressure which can
lower the performance of the engine at higher rpm\'s. While some engines
develope maximum horsepower at lower rpm\'s (say, under 2800 rpm), most
do not and would thus yield a greater net horsepower (at the higher
rpm\'s)with no muffler at all.
!Schema{width="400"}

The upper right image represents a Car Muffler typical architecture. It
is composed of 3 tubes. There are 3 areas separated by plates, the part
of the tubes located in the middle area are perforated. Small quantity
of pressure \"escapes\" from the tubes through the perforation and
cancel one another.

Some high-end mufflers use the reflection principle together with a
cavity(shown in red below) known as a Helmholtz
Resonator
to further dampen the noise.

![](Muffler_resonator.png "Muffler_resonator.png"){width="300"}

## Back pressure

Car engines are 4 stroke cycle engines. Out of these 4 strokes, only one
produces the power, this is when the explosion occurs and pushes the
pistons back. The other 3 strokes are necessary evil that don\'t produce
energy. They on the contrary consume energy. During the exhaust stroke,
the remaining gas from the explosion is expelled from the cylinder. The
higher the pressure behind the exhaust valves (i.e. back pressure), and
the higher effort necessary to expel the gas out of the cylinder. So, a
low back pressure is preferable in order to have a higher engine
horsepower.

## Muffler Modeling by Transfer Matrix Method

This method is easy to use on computer to obtain theoretical values for
the transmission loss of a muffler. The transmission loss gives a value
in dB that correspond to the ability of the muffler to dampen the noise.

### Example

!Muffler working with waves
reflections{width="500"}

P stands for Pressure \[Pa\] and U stand for volumetric flowrate\[m3/s\]

$\begin{bmatrix} P1 \\ U1 \end{bmatrix}$=$\begin{bmatrix}  T1 \end{bmatrix}
\begin{bmatrix} P2 \\ U2 \end{bmatrix}$ and
$\begin{bmatrix} P2 \\ U2 \end{bmatrix}$=$\begin{bmatrix}  T2 \end{bmatrix}
\begin{bmatrix} P3 \\ U3 \end{bmatrix}$ and
$\begin{bmatrix} P3 \\ U3 \end{bmatrix}$=$\begin{bmatrix}  T3 \end{bmatrix}
\begin{bmatrix} P4 \\ U4 \end{bmatrix}$

So, finally: $\begin{bmatrix} P1 \\ U1 \end{bmatrix}$=
$\begin{bmatrix} T1 \end{bmatrix}
\begin{bmatrix} T2 \end{bmatrix}
\begin{bmatrix} T3 \end{bmatrix}
\begin{bmatrix} P4 \\ U4 \end{bmatrix}$

with

$\begin{bmatrix} T_i \end{bmatrix}$=$\begin{bmatrix}  cos (k L_i) & j sin (k L_i) \frac{\rho c}{S_i}  \\ j sin (k L_i) \frac{\rho c}{S_i} & cos (k L_i) \end{bmatrix}$

Si stands for the cross section area

k is the angular
velocity

$\ \rho$ is the medium density

c is the speed of sound of the medium

### Results

!Schema{width="400"}

https://commons.wikimedia.org/wiki/File:Transmission_loss.png#Source_code
Matlab
code
of the graph above.

### Comments

The higher the value of the transmission loss and the better the
muffler.

The transmission loss depends on the frequency. The sound frequency of a
car engine is approximately between 50 and 3000 Hz. At resonance
frequencies, the transmission loss is zero. These frequencies correspond
to the lower peaks on the graph.

The transmission loss is independent of the applied pressure or velocity
at the input.

The temperature (about 600 Fahrenheit) has an impact on the air
properties : the speed of sound is higher and the mass density is lower.

The elementary transfer matrice depends on the element which is
modelled. For instance the transfer matrice of a Helmholtz Resonator is
$\begin{bmatrix} 1 & 0 \\ \frac{1}{Z} & 1 \end{bmatrix}$ with
$\ Z = j \rho ( \frac{\omega L_i}{S_i} - \frac{c^2}{\omega V})$

## Links

More informations about the Transfer Matrice Method :
www.scielo.br/pdf/jbsmse/v27n2/25381.pdf

General informations about filters: Filter Design &
Implementation

General information about car mufflers:
<http://auto.howstuffworks.com/muffler.htm>

Example of car exhaust manufacturer
<http://www.performancepeddler.com/manufacturer.asp?CatName=Magnaflow>

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