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# Acoustic fundamental. Part 1: Real and complex part of Pressure

## Acoustic fundamental. Part 1: Real and complex part of Pressure

(OP)
Hi,

There is a series of acoustic concept I would like to review. My background is rather in the fields of elasticity and CFD.

I read good references explaining the maths. I can understand them, but most of those references don't link it with practical application.

1) Complex and Real part of pressure

When solving the Helmholtz equation, the acoustic pressure has a REAL and IMAGINARY parts.

1.1) What does the COMPLEX part represent? I know it's the "phase", but what it is actually? The difference in frequency between the source and a point of the mesh?

1.2) In many acoustic FEA problem that I tried, the complex part was 0. Why?
An idea: in dynamic, the modes (natural frequency of vibration) of a structure will have a complex part only if damping is considered. Maybe it is the case here?
I will see a non-zero complex part if I add impedance to a wall of my model?

1.3) The FEA returns a vector with two elements: P=[real, complex]
Supposing both part are not zero.
I have a solid submerged in a bath of water. Waves are generate inside the bath.
I am concern about the mechanical force (or pressure) on the walls of the submerged solid.
Which value will give me the "mechanical" pressure: real part of P, or the norm of P?

Thanks

### RE: Acoustic fundamental. Part 1: Real and complex part of Pressure

Of course you are dealing with wave propagation and attenuation...boundary conditions, etc.

As to the forces, you have to decide what you need for the specific problem at hand. For the steady-state case, root mean square amplitude gives you the total amplitude (not the peak amplitude) with the real part: the elastic component and the imaginary component: the losses to the system modeled.

### RE: Acoustic fundamental. Part 1: Real and complex part of Pressure

1.1 (b) No

1.2 (a) Because you had no damping and an infinite sound field at a guess

1.2 (c) Yes

1.3 The magnitude of the vector

Cheers

Greg Locock

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### RE: Acoustic fundamental. Part 1: Real and complex part of Pressure

(OP)
Hi,

#### Quote (Greg)

1.1 (b) No

Ok...but how to interpret it then? Maybe it does not have a physical interpretation itself, but has sense only for computing the magnitude of P.

#### Quote (Greg)

1.2 (a) Because you had no damping and an infinite sound field at a guess
Yes, I confirmed with a simple test.
I work with lossless flow. I have a source of pressure in a closed box. If all the walls are rigid, the imaginary part is zero.
If I add impedance to a wall, that creates damping and I see a non-zero imaginary part.

#### Quote (Greg)

1.3 The magnitude of the vector

Ok, thanks. What confuses me is a sentence found in a Comsol document.

I attached an image showing the sentence.

Thanks.

### RE: Acoustic fundamental. Part 1: Real and complex part of Pressure

if wanting a solution in time, p(t), you take the real part...the amplitude of the time solution in the dynamic steady-state case is root mean square of the real and imaginary parts.

p(t) = Amplitude *Cos(omega*t) = Real Part[ Amplitude*Exp(j*omega*t)]

= Real Part[ Amplitude*Cos(omega*t)+j*Amplitude*Sin(omega*t)], courtesy of Euler....

### RE: Acoustic fundamental. Part 1: Real and complex part of Pressure

(OP)
No GregLocock, I think hacksaw agrees with you (see previous answers).

To summarize hacksaw's answers (correct me if I am wrong):

- When working in the frequency domain (with Helmholtz equation): physical pressure is the magnitude of the pressure vector {Re,Im}
- When working in time (directly with the wave equation): physical pressure is the real part of the pressure vector {Re,Im}

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