# Radiated Sound Output Analysis

Radiated Sound Output can be requested for grid points on the structural surface and in the exterior acoustic field. Grid points are used to represent microphones to record the radiated sound, sound power, and sound intensity.

## Request Radiated Sound Output Guide

The following procedure can be considered as a guide for requesting radiated sound output:

1. Microphones that record sound levels in the acoustic field can be defined as grid point sets using the RADSND (MSET field) Bulk Data Entry.
2. PANELG (TYPE=SOUND/Blank) can be used to define the sound generating panel(s) which are to be considered for radiated sound output calculations.
3. The PANEL continuation line in the RADSND Bulk Data Entry can be used to list the panel IDs of the panels defined using PANELG (TYPE=SOUND/Blank). This allows the definition of the sound generating panels that contribute to the calculation of radiated sound output at the microphones (Grid points) listed in the MSET field of the RADSND Bulk Data Entry.
4. The value of the speed of sound $c$ required to define the wave number and the complex particle velocity vector is input using PARAM, SPLC. The density of the acoustic medium $e$ used in the calculation of the complex acoustic sound pressure and the complex particle velocity vector is defined using PARAM, SPLRHO. An additional scale factor $q$ can be specified using PARAM, SPLFAC in the Sound Pressure Level calculation.
5. Various outputs can be requested for this analysis. SINTENS can be used to request sound intensity and SPL can be used to request sound pressure.

The set up guide for radiated sound output calculation is described in the previous section. The procedure is based on the following set of equations for the calculation of each output type.

## Analytical Background for Radiated Sound Output

The sound radiated from the sound generating panel is reduced to sound generation from discrete point sources. The grid points of the finite element mesh on the surface of the panel are considered as sound sources. Sound power and sound intensity can be requested for both the source grids and the microphone grids.

## At the Microphone Location

### Wave Number

Where,
$c$
Speed of sound defined by PARAM, SPLC
$f$
Frequency of the sound wave in the medium

### Velocity Flux of the Source Grid

The velocity flux of the source grid is the rate at which panel material in an infinitesimal area surrounding the grid point moves through the medium.
For each frequency, it is calculated as:(2) ${v}_{flux}={v}_{s}·\delta A$
Where,
${v}_{s}$
Velocity vector of the source grid.
$\delta A$
Area vector associated with the source grid defined as:(3) $\delta A=A{\stackrel{^}{A}}_{s}$
Where,
$A$
Area associated with the source grid
${\stackrel{^}{A}}_{s}$
Unit area vector normal to the panel surface at the source grid (Figure 2).

### Complex Acoustic Sound Pressure (Requested using SPL)

The complex acoustic sound pressure is the deviation from the ambient atmospheric pressure caused by a sound wave. This is specified by ${p}_{j}$ and is defined as the sound pressure deviation, due to a single sound panel grid $j$ at the microphone location for each frequency as:(4) ${p}_{j}=\frac{f\rho q}{{r}_{j}}{\left({v}_{flux}\right)}_{j}i{e}^{-ik{r}_{j}}$
Total Complex Acoustic Sound Pressure requested by SPL is:(5) $p=\sum _{j=1}^{np}\left(\frac{f\rho q}{{r}_{j}}{\left({v}_{flux}\right)}_{j}i{e}^{-ik{r}_{j}}\right)$
Where,
$f$
Frequency of the sound wave in the medium
$\rho$
Density of the acoustic medium defined by PARAM, SPLRHO
${r}_{j}$
Distance from the acoustic source grid $j$ on the panel to the microphone location grid (Figure 1).
${\left({v}_{flux}\right)}_{j}$
Velocity flux of the source grid
$k$
Wave number as defined in Wave Number.
$i$
Square root of -1
$np$
Number of source grids (Figure 1).
$q$
Value of the scale factor specified using the parameter PARAM, SPLFAC.
The Sound Pressure Level in decibels ($SP{L}_{dB}$ - also requested using SPL) can be calculated using:(6) $SP{L}_{dB}=20.0*{\mathrm{log}}_{10}\left(\frac{|SPL|}{SPLREFDB}\right)$
Where,
$SP{L}_{dB}$
Sound Pressure Level in decibels
$|SPL|$
Magnitude of the acoustic sound pressure
$SPLREFDB$
Reference sound pressure value specified using the parameter PARAM, SPLREFDB

### Complex Particle Velocity Vector

The complex particle velocity vector is the velocity of a particle in a medium measured as a wave passes through it. The particle velocity is not the velocity of the wave itself; rather it is the velocity of a particle as it oscillates about a mean position, due to the passage of the wave. It is specified by ${v}_{j}^{p}$ at the location of the microphone, due to the source grid $j$ (Figure 1) and is defined for each frequency as:(7) ${v}_{j}^{p}=\frac{{p}_{j}{\stackrel{^}{r}}_{j}}{\rho c}\left(1-\frac{i}{k{r}_{j}}\right)$
Where,
${p}_{j}$
Complex acoustic pressure, due to source grid, $j$ at the microphone location.
${\stackrel{^}{r}}_{j}$
The unit vector from the source grid $j$ to the microphone grid (Figure 1).
$\rho$
Density of the acoustic medium defined by PARAM, SPLRHO.
$c$
Speed of sound defined by PARAM, SPLC
$k$
Wave number as defined in Wave Number
${r}_{j}$
Distance from the acoustic source grid $j$ on the panel to the microphone grid (Figure 1).
$i$
Square root of -1

### Total Complex Intensity Vector (Requested using SINTENS)

The total complex intensity vector is the sound power per unit area. The sound intensity can be defined as a product of sound pressure and the particle velocity vector. For multiple source grids, the total sound intensity at a microphone location for each frequency is given as follows:(8) $i=\frac{1}{2}\sum _{j=1}^{np}\text{Real}\left({p}_{j}{\left({v}_{j}^{p}\right)}^{*}\right)$

Where, ${p}_{j}$ is the acoustic pressure at the microphone location due to the sound generated at the source grid $j$ and ${\left({v}_{j}^{p}\right)}^{*}$ is the complex conjugate of ${v}_{j}^{p}$, which is the complex particle velocity vector at the microphone location, due to the sound generated at the source grid $j$.

### Source Grid Location

#### Wave Number

The vector addition operation for displacement vectors from Figure 3 is as:(9) ${x}_{s}=x+{x}_{r}$

Where, $x$ is the vector from a source grid (1) to the source grid (2) of interest.

${x}_{r}$ is defined as:(10) ${x}_{r}=\frac{1}{2}\sqrt{\frac{A}{\pi }}{\stackrel{^}{x}}_{r}$

Where, $A$ is the area associated with a source grid and ${\stackrel{^}{x}}_{r}$ is the unit normal to the area, $A$ associated with a source grid.

#### Complex Acoustic Sound Pressure [at the source grid]

The complex acoustic sound pressure is the deviation from the ambient atmospheric pressure caused by a sound wave. This is specified using ${\left({p}_{j}\right)}_{s}$ and is defined at the source grid for each frequency as:(11) ${\left({p}_{s}\right)}_{j}=\frac{f\rho q}{{\left({r}_{s}\right)}_{j}}{\left({v}_{flux}\right)}_{j}i{e}^{-ik{\left({r}_{s}\right)}_{j}}$
Total Complex Acoustic Sound Pressure at a source grid requested by SPL is:(12) ${p}_{s}=\sum _{j=1}^{np-1}\left(\frac{f\rho q}{{\left({r}_{s}\right)}_{j}}{\left({v}_{flux}\right)}_{j}i{e}^{-ik{\left({r}_{s}\right)}_{j}}\right)$

Where,

$f$ is the frequency of the sound wave in the medium.

$\rho$ is the density of the acoustic medium defined by PARAM, SPLRHO.

${\left({r}_{s}\right)}_{j}$ is equal to $|{x}_{s}|$, for each grid, $j$ ($j$=1 to $np$), as defined in At the Source Grid Location (Figure 3).

${\left({v}_{flux}\right)}_{j}$ is the velocity flux of the source grid, $j$ (Figure 3)

$k$ is the wave number as defined in Wave Number.

$i$ is the square root of -1

$np$ is the number of source grids (Figure 2).

$q$ is the value of the scale factor specified using the parameter PARAM, SPLFAC.