## 2 Methods  
In the ex vivo lung model, a 0.1 s harmonic vibration was generated on the surface of the lung using a small vibrator. The surface wave propagation on the lung was measured using an ultrasound probe. The vibration excitation signal was generated using a function generator (Model 33120A, Agilent Inc., Santa Clara, CA, USA). The signal was then amplified using a power amplifier (Model PYLE PROPCA4, PYLE PRO Service Center, 1600 63 rd Street, Brooklyn, NY 11204, USA). The excitation signal was used to excite a small vibrator shaker (Model: FG-142, Labworks Inc., Costa Mesa, CA 92626, USA). The shaker generated a local small vibration on the surface of lung using an indenter with a plastic ball with a diameter of 5 mm.  
A Verasonics ultrasound system (Verasonics, Inc., Kirkland, WA 98034, USA) was used to acquire the ultrasound data. An ultrasound probe with a central frequency of 6.4 MHz L11-5 was used in the experiments. The in-phase/quadrature (IQ) data of ultrasound signals were analyzed by demodulation of the radio-frequency (RF) data of the ultrasound echo. The particle velocity \(v\) due to the external vibration generation at a pixel in the axial direction of ultrasound beam was analyzed. \(v\) was calculated using an autocorrelation method [12; 13]. High frame imaging was obtained using the plane-wave transmission technique. Pulse repetition frequency (PRF) was 2000 pulses per second. Central frequency (\(f\)) was 6.4 MHz. Three pixels in the axial direction and two sampling points in the slow time direction were averaged for data processing. Prior to window selection, the whole autocorrelation matrix for each lateral direction was calculated as follows:  
\[v=\ \frac{c}{4\pi fT_{s}}\left\{\frac{\sum_{m=0}^{M-1}\sum_{n=0}^{N-2}[Q(m,n)I( m,n+1)-I(m,n)Q(m,n+1)]}{\sum_{m=0}^{M-1}\sum_{n=0}^{N-2}[I(m,n)I(m,n+1)+Q(m,n)Q(m,n+1)]}\right\} \tag{1}\]  
where the pulse repetition period is \(T_{s}=\frac{1}{PRF}\), c is the speed of sound of tissue and is assumed to be 1540 m/s, \(I(m,n)\) and \(Q(m,n)\) are the real and imaginary \(IQ\) date at a pixel (\(m,n\)). A \(3\times 3\) pixel spatial median-filter was used to improve the imaging.  
In this ex vivo lung model, the surface wave speed was measured by analyzing the surface lung motion among several locations. \(v_{1}(t)\) and \(v_{2}(t)\) represent the tissue motion at two locations on the lung surface. The cross-spectrum \(S(f)\) of two signals \(v_{1}(t)\) and \(v_{2}(t)\) at the excitation frequency f was analyzed by [14]:  
\[S(f)=S_{1}^{*}(f)\cdot S_{2}(f)=\left|S_{1}(f)\cdot S_{2}(f)\right|e^{-j\Delta \varphi(f)} \tag{2}\]  
where \(S_{1}(f)\) and \(S_{2}(f)\) are the Fourier transforms of \(v_{1}(t)\) and \(v_{2}(t)\), respectively; \({}^{*}\) denotes the complex conjugate; and \(\Delta\varphi(f)\) is the phase change between \(v_{1}(t)\) and \(v_{2}(t)\) over distance between the two locations.  
The change of surface wave phase with distance was analyzed to calculate the surface wave speed:  
\[c_{s}(f)=2\pi f|\Delta r/\Delta\varphi| \tag{3}\]  
where \(\Delta r\) is the distance between the two locations. The estimation of surface wave speed can be improved by measuring the phase change over multiple locations [15].