RMS of complex waveform (acoustics)

[mattattack900]

Hey everyone, just got a quick question in acoustics. I am mainly looking for a mathematical understanding

Homework Statement

Consider two harmonic plane progressive waves of the form

$\tilde{P(x)}$ = $\tilde{A}$e^-jkx

and$\tilde{P(x)}$ = $\tilde{B}$e^+jkx

traveling in opposite directions. Showing all workings, derive expressions for:

1) Total acoustic pressure
2) Total mean squared pressurein the above expressions $\tilde{P(x)}$ represents the acoustic pressure and $\tilde{A}$ and $\tilde{B}$ are complex amplitudes

Homework Equations

The Attempt at a Solution

my solution for part 1 is due to linear superposition:

$\tilde{P(x)}$ = $\tilde{A}$e^-jkx + $\tilde{B}$e^+jkx

i know that the SOLUTION to the second part is:

|$\tilde{P(x)}$|² = |$\tilde{A}$e^-jkx|² + |$\tilde{B}$e^+jkx|² + 2Re{ $\tilde{A}$$\tilde{B}$*}cos(kx)

where Re{} denotes the real part ( I couldn't find the actual symbol ), * denotes the complex conjugate and k is the wave number ( k= ω/c )

like i said above i am trying to get a mathematical understanding of the second part. I do not understand how this solution is derived. Thanks

 

[Simon Bridge]

Note: \Re gives $\Re$ ... it's easier to just type out the LaTeX than use the equation editor.

What do the tildas indicate here?

$$\tilde{P}\!_A(x) = \tilde{A}e^{-jkx}\\

\tilde{P}\!_B(x)=\tilde{B}e^{jkx}\\

\tilde{P}(x)=\tilde{A}e^{-jkx}+\tilde{B}e^{jkx}\\$$

You want to understand this:
$$\left | \tilde{P}(x) \right |^2=\left | \tilde{A}e^{-jkx}\right |^2+\left | \tilde{B}e^{jkx}\right |^2 = \left |\tilde{A}\tilde{B}^\star\right | \cos(kx)$$

ABcosθ would normally be a scalar product right - so how does that work if A and B are complex valued?

What happens if you expand the complex amplitudes into $\tilde{A}=a+jb\; , \; \tilde{B}=c+jd$ and expand the exponentials into trig functions?

 

[mattattack900]

Hey, thanks for the reply.

i believe the tilda is the nomenclature used to represent a Complex number.
I will try what you have suggested

 

[Simon Bridge]

You may not need to though:

If: $z=a+jb$

Then: $|z|^2 = a^2+b^2 = z\cdot z^\star$

 

[rezeew]

for the question, and great job on attempting the solution for the first part!

The second part of this question is asking for the total mean squared pressure, which is a measure of the average pressure of the complex waveform over time. In order to understand how this solution is derived, we need to break it down step by step.

First, we have the expression for the total acoustic pressure, which is the sum of the two harmonic plane progressive waves traveling in opposite directions. This is known as linear superposition, which states that the total pressure at any point is the sum of the individual pressures at that point.

Next, we have the expression for the total mean squared pressure, which takes into account the magnitude and phase of the complex amplitudes \tilde{A} and \tilde{B}. The squared magnitude of a complex amplitude represents the intensity of the wave, while the phase represents the angle at which the wave is traveling. The squared magnitude is then multiplied by the complex conjugate of the other amplitude, and then multiplied by the cosine of the wave number multiplied by the distance traveled.

This may seem complicated, but it is essentially taking into account the interference patterns of the two waves to find the average pressure over time. I hope this helps to provide a mathematical understanding of the solution.