How Do Transmission Coefficients Affect Pressure and Intensity at Interfaces?

[roldy]

Homework Statement

(a) A plan wave is in seawater is normally incident on the water-air interface. Find the pressure and intensity transmission coefficients.

(b) Repeat (a) for a wave in air normally incident on the air-water interface.

(c) For (a) and (b) find the change in pressure and intensity levels if Pref and Iref are the same in both media.

Homework Equations

$$SIL=10*log \frac{I}{Iref}$$
$$PSL=20*log \frac{P}{Pref}$$

Pressure Transmission Coefficient
$$T=\frac{P_t}{P_i}=\frac{2r_2}{r_2+r_1}$$

Intensity Transmission Coefficients
$$T_i=\frac{I_t}{I_i}=\frac{r_1}{r_2}|T|^2$$

The Attempt at a Solution

I solved for parts (a) and (b) already
with
$$r_1=1.54*10^{-4}$$
$$r_2=429$$

(a) $$T=5.57*10^{-4}$$
$$T_i=1.111*10^{-3}$$

(b) $$T=2$$
$$T_i=1.113*10^{-3}$$

Part (c) I'm confused on. I'm assuming I need to use the equations for PSL and SIL since the question is asking for pressure and intensity levels but I am unsure as to how.

 

[mark726]

Can someone please guide me through this?

Sure, let's work through this together. First, let's define our variables:

Pref = reference pressure (same in both media)
Iref = reference intensity (same in both media)
Pt = transmitted pressure
Pi = incident pressure
It = transmitted intensity
Ii = incident intensity

Now, let's rewrite the equations for pressure and intensity transmission coefficients, using our defined variables:

T = Pt / Pi
Ti = It / Ii

Next, let's use the equations for PSL and SIL to find the change in pressure and intensity levels:

PSL = 20*log(P/Pref)
SIL = 10*log(I/Iref)

We can rearrange these equations to solve for P and I:

P = Pref * 10^(PSL/20)
I = Iref * 10^(SIL/10)

Now, let's plug in our values for T and Ti into these equations:

For part (a):
P = Pref * 10^(20*log(T)/20)
= Pref * 10^log(T)
= Pref * T

I = Iref * 10^(10*log(Ti)/10)
= Iref * 10^log(Ti)
= Iref * Ti

For part (b):
P = Pref * 10^(20*log(T)/20)
= Pref * 10^log(T)
= Pref * T

I = Iref * 10^(10*log(Ti)/10)
= Iref * 10^log(Ti)
= Iref * Ti

We can see that the change in pressure and intensity levels is the same in both media, since they are both multiplied by the same transmission coefficient. Therefore, the change in pressure and intensity levels is equal to T and Ti, respectively.

I hope this helps! Let me know if you have any further questions.