# 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.

## 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

### 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.

### 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.
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.

## 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

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
${\begin{bmatrix}T_{i}\end{bmatrix}}$=${\begin{bmatrix}cos(kL_{i})&jsin(kL_{i}){\frac {\rho c}{S_{i}}}\\jsin(kL_{i}){\frac {\rho c}{S_{i}}}&cos(kL_{i})\end{bmatrix}}$
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

[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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