# Acoustics/Car Mufflers

![](Acoustics_car_muflers.JPG "Acoustics_car_muflers.JPG")

## 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 air surrounding engine through the exhaust
valves. There are 2 techniques used to dampen the noise: absorption and
reflection. Each technique has its advantages and disadvantages.

!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 bend
metal sheet.

The advantage of this method is low back pressure with a relatively
simple design. The inconvenience of this method is low sound damping
ability compared to the other techniques, especially at low frequency.

The mufflers using the absorption technique are usually sports vehicle
because they increase the performances of the engine because of their
low back pressure. A trick to improve their
muffling ability consists 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 traveling 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, the impedance of the tube will change. Part of the
incident waves will be transmitted through the discontinuity and the
other part will be reflected.

Animation

Mufflers using the reflection technique are the most commonly used
because they dampen the noise much better than the absorber muffler.
However they induce a higher back pressure,
lowering the performance of the engine (an engine would be most
efficient or powerful without the use of a muffler).

!Schema{width="400"}

The upper right image represents a car muffler\'s 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. A small
quantity of pressure \"escapes\" from the tubes through the perforation
and cancel one another.

Some mufflers using the reflection principle may incorporate cavities
which dampen noise. These cavities are called Helmholtz
Resonators
in acoustics. This feature is usually only available for up market class
mufflers.

![](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), the
higher the 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 volume velocity \[m3/s\]

$\begin{bmatrix} P1 \\ U1 \end{bmatrix}$=$\begin{bmatrix}  T1 \end{bmatrix}
\begin{bmatrix} P2 \\ U2 \end{bmatrix}$ 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

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

Si stands for the cross section area

k is the wave
number

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

The transmission loss and the insertion loss are different terms. The
transmission loss is 10 times the logarithm of the ratio output/input.
The insertion loss is 10 times the logarithm of the ratio of the
radiated sound power with and without muffler.

## Links

-   General Information about Filter Design &
    Implementation
-   More information about the Transfer Matrix
    Method
-   General information about car
    mufflers

