Understanding sound transmission loss (TL)

[Pete_L]

In considering a sound wave (in air) incident on a wall, the transmission loss (TL) is calculated by the equation

TL (dB) = 10*Log (W1/W2)
where
W1 = sound power incident on wall
W2 = sound power transmitted through wall

Based on this understanding of TL, I wish to determine the extent to which TL diminishes sound power of the reflected sound wave. Understanding that W1/W2 is the ratio of sound powers, I calculated

Lt (dB) = 10*Log [ 1- (W2/W1)]
where
Lt = the extent to which power of the reflected wave is diminished relative to that of the incident wave due to some of the power transmitted through the wall

This might become part of an article that I hope will be published. As I am not a professional, I just wanted to check that my derivation from the standard equation for transmission loss is correct. So thanks if someone who is knowledgeable concerning acoustics can confirm that this is correct or not.

-Pete

 

[Pete_L]

To make this easier to respond to, here is how I derived the equation for Lt.
Consider the fraction below,

[(W1/W2) -1] / (W1/W2)

If for example W1/W2 equals 10, this means 10 parts incident sound power to 1 part transmitted power. In the numerator, 10 minus 1 equals 9. The denominator is the original ratio of 10 without subtracting the 1 part of transmitted sound power. Thus,

10*log [((W1/W2) - 1 ) / (W1/W2)] = 10*log 0.9 = -0.46 dB

This tells me that power in the reflected wave with respect to power of the incident wave has been diminished by about 0.5 dB as a result of some of the energy transmitted through the wall.

Simplifying the fraction,

(W1/W2) -1 = (W1 - W2) / W2

[(W1- W2) / W2] * 1/(W1/W2) = (W1- W2)/ W1

(W1 - W2)/ W1 = 1 - (W2 / W1)
Then
Lt = 10*Log [1- (W2 / W1)]

Please tell me if this is a false line of reasoning, or not. Thank you in advance if anyone can.
-P

 

[tech99]

I think the simple model for reflection you are using assumes that the wall is acting like a piece of mismatched transmission line.
But W2 will be affected by the heat losses in the wall. You would need to define W2 as the sound power entering the wall rather than the sound power radiated on the far side.
An analogue might be the power reflected from a mismatched attenuator in a transmission line.

 

[Pete_L]

I think the simple model for reflection you are using assumes that the wall is acting like a piece of mismatched transmission line.
But W2 will be affected by the heat losses in the wall. You would need to define W2 as the sound power entering the wall rather than the sound power radiated on the far side.
An analogue might be the power reflected from a mismatched attenuator in a transmission line.

My approach is to separately calculate loss due to absorption and transmission. I came across an equation for approximating transmission loss only. Then I also can approximate absorption knowing what the material of the wall is and knowing the absorption coefficient for that material.

But my question is- Is my method of deriving an equation for attenuated power of the reflected sound wave correct? Assume that I have a correct value of W2, whatever that is.

-P

 

[tech99]

My approach is to separately calculate loss due to absorption and transmission. I came across an equation for approximating transmission loss only. Then I also can approximate absorption knowing what the material of the wall is and knowing the absorption coefficient for that material.

But my question is- Is my method of deriving an equation for attenuated power of the reflected sound wave correct? Assume that I have a correct value of W2, whatever that is.

-P

Yes, correct I believe.

 

[CWatters]

Isnt all this just restating conservation of energy? Eg...

Incident power - transmitted power - absorbed power - reflected power = 0

 

[Pete_L]

Yes, correct I believe.

Thank you, I'm not always as rigorous as I should be in solving a problem.
-P

 

[Pete_L]

Isnt all this just restating conservation of energy? Eg...

Incident power - transmitted power - absorbed power - reflected power = 0

What I need to solve is dB of loss, or reflected power relative to incident power in dB, without regard to the amount of acoustic power involved. So regardless of any particular incident power, the reflected power is reduced by some dB amount.

 

[tech99]

Isnt all this just restating conservation of energy? Eg...

Incident power - transmitted power - absorbed power - reflected power = 0

Yes, but I think the OP question related to the use of decibels, which can be tricky when subtraction or addition are involved.

 

[nasu]

In considering a sound wave (in air) incident on a wall, the transmission loss (TL) is calculated by the equation

TL (dB) = 10*Log (W1/W2)
where
W1 = sound power incident on wall
W2 = sound power transmitted through wall

Based on this understanding of TL, I wish to determine the extent to which TL diminishes sound power of the reflected sound wave. Understanding that W1/W2 is the ratio of sound powers, I calculated

Your first equation assumes zero reflection. As you assume zero reflection, it seems odd to use this formula to find something about reflected wave.
The complete formula should be
$L_t=10 log \frac{W1(1-R)}{W2}$
where R is the coefficient of reflection (intensity) at the interface.
The intensity of the reflected wave depends on the impedance mismatch at the interface. You want to study deviations from Frenel's equations predictions due to absorption in the second medium?