Table A3.

Lumped parameter model for the acoustic radiation impedance Z_r and acoustic impedance of a protective diaphragm Z_p,d.

Model for the acoustic radiation impedance Zr [kg/s/m4], [23]: Transducer represents a piston at the end of a long tube (piston constitutes the side of a box). Model requirements: k·rp < ½ ≥ radiation is omnidirectional (for 10 kHz rp < 3 mm in air and < 12 mm in water).
$R_r=\frac{\pi }{4}\rho c{k}^2{r}_p^4$	Rr: radiation resistance [kg/s]	(A4)
$X_r=1.9\omega \rho r_p^3$	Xr: acoustic radiation mass (reactance) [kg/s]	(A5)
$Z_r=\frac{1}{A_{d,p}^2}\left(R_r+\mathrm{i}{X}_r\right)$	Zr: acoustic radiation impedance [kg/s/m4]	(A6)
Model for the acoustic impedance of the protective, square diaphragm Zd,p [kg/s/m4]
$p=\frac{E}{\left(1-v^2\right)}{\left(\frac{t_p}{a_p}\right)}^4\left[67.2\left(\frac{{\mathrm{w}}_{\mathrm{c}}}{{\mathrm{t}}_{\mathrm{p}}}\right)+25.28{\left(\frac{{\mathrm{w}}_{\mathrm{c}}}{{\mathrm{t}}_{\mathrm{p}}}\right)}^3\right]$	Relationship between pressure load p [Pa] and centre deflection wc [m] (large deflections) of a square diaphragm with no intrinsic stress, [32].	(A7)
$C_{d,p}=\frac{1}{A_{d,p}}\frac{d{w}_c}{dp}=\frac{1}{A_{d,p}}{S}_{m,d}$	Cd,p [m/N], Sm,d [m3/N]: mechanical compliance and sensitivity of the protective diaphragm.	(A8)
$m_{d,p}=\frac{1}{C_{d,p}{\left(2\pi f_r\right)}^2}$	md,p: equivalent mass of the diaphragm [kg]	(A9)
$f_r=f_{r0}{\left[1+1.18{\left(\frac{w_c}{t_p}\right)}^2\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$	fr: resonance frequency of a square diaphragm (fundamental mode) for any static deflections [Hz], [33].	(A10)
$f_{r0}=\frac{35.99}{2\pi a_p^2}{\left[\frac{E{t}_p{}^2}{12{\rho }_s\left(1-v^2\right)}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$	fr0: resonance frequency of a square diaphragm at zero deflection [Hz], [34].	(A11)
$Z_{d,p}=\frac{1}{A_{d,p}^2}\left(\mathrm{i}\omega m_{d,p}+\frac{1}{\mathrm{i}\omega C_{d,p}}\right)$	Acoustic impedance of the protective, square diaphragm Zd,p [kg/s/m4]	(A12)
$Z_{1,2,3,4}=Z_r+Z_{d,p}$		(A13)

k: angular wave number [rad/m], c: speed of sound [m/s], ω: angular frequency [rad/s], r_p: radius of the vibrating surface [m] (for square protective diaphragm A_dp; r_p = (A_dp/π)^−1/2, a_p: side length of a square diaphragm [m], t_p: diaphragm thickness [m], ρ: fluid density [kg/m³], ρ_s: density of the diaphragm material [kg/m³], E: Young’s modulus [N/m²] and v: Poisson’s ratio [–].