spreading of vibration

Collection of information about noise and vibration

• slide ~N/A (on 14-feb)~ https://osf.io/xkan8 (15-feb)
• story https://medium.com/p/617655776d61 (on 14-feb)

source

At position $\vec{r}_s$ there is source of vibration

$$\tag{1} \psi_s(t) = \sum_{i=1}^N A_i \sin(\omega_i t + \varphi_i)$$

which is simplified as superposition of $N$ simple harmonic motion of physical quantity $A_i$ with angular frequency $\omega_i$ and initial phase $\varphi_i$, where $A_i$ migth represent difference between variation of air pressure and normal atmospheric pressure

$$\tag{2} \Delta p = p(t) - p_0$$

or difference between variation of position of parts of solid body, e.g. the ground, to their equilibrium position

$$\tag{3} \Delta \vec{r} = \vec{r}(t) - \vec{r}_0.$$

Sound waves are accomodated by Eqn (2), while waves on solid are by Eqn (3). The first are longitudinal waves, while the later could be longitudinal and transversal waves.

wave

Function represents wave from previous source of vibration

$$\tag{4} \psi({\vec{r}, t}) = \sum_{i=1}^N A_i \sin(\omega_i t + \varphi_i - k_i |\vec{r} - \vec{r}_s|),$$

where wavenumber is

$$\tag{5} k = \frac{2\pi}{\lambda}$$

with wavelength $\lambda$. If $v$ is wave velocity then it is obtained from

$$\tag{6} v = \frac{\omega}{k}$$

or

$$\tag{7} v = \frac{\lambda}{T} = \lambda f.$$

intensity

Intensity at vibration source is proportional to square of amplitude

$$\tag{8} I_s \propto | A |^2$$

and at distance $r$ from the source

$$\tag{9} I(r) = \frac{r_o^2}{r^2} I_s$$

for sound assuming it is spreading as spherical wave, where $r_o$ is size of the source.

resonance

When wave arrives at some position $\vec{r}_j$ it will induce vibration on that position which can be considered as sinusoidal driving force with amplitude $F_0$ and frequency $\omega$. Equation of motion at that placew would be

$$\tag{10} m \frac{d^2 \psi}{dt^2} + b \frac{d\psi}{dt} + k \psi = F_0 \cos (\omega t + \varphi_d)$$

with damping coefficient $b$, spring constant $k$, and mass $m$. Then it can be obtained that

$$\tag{11} A_j = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + 4\gamma^2 \omega^2}}$$

is the amplitude at that position, where

$$\tag{12} \omega_0 = \sqrt\frac{k}{m}$$

and

$$\tag{13} \gamma = \frac{b}{2m}.$$

Then it should be assumed

$$\tag{14} F_0 \propto \sqrt{I(r)},$$

to relate this part with previous one.

velocity

Sound velocity in liquid is obtained from

$$\tag{15} v_{\rm liq} = \sqrt{\frac{B}{\rho}},$$

in solid from

$$\tag{16} v_{\rm sol} = \sqrt{\frac{Y}{\rho}},$$

and in ideal gas from

$$\tag{16} v_{\rm gas} = \sqrt{\frac{\gamma_a R T}{M}}.$$

Sound velocity in water is about 1500 m/s and in air 331 m/s (Gea-Banacloche, 2020), while maximum velocity is about 1250-1730 m/s on various sediment, e.g clay-dominated, silt-dominated, sandy, gas-charged, (Novak et al., 2020).

natural frequency

Earth natural frequencey of Schumann resonance is about 7.83 Hz (Biotonomy, 2017). For building it depend also on soil base, e.g Limetone 7.1 Hz, breccia and debris deposits 5.3 Hz, alluvional deposits 4.6 Hz, year of built 4.0 - 5.8 Hz, and number of floors, e.g. 2-5 floors gives 7.0 - 3.6 Hz (Gangone et al., 2023).

damping ratio

See (Irvine, 2002) for viscous damping ratio.

summaries

• From Eqns (8), (9), and (11) $$\tag{17} A(r + \Delta r) = \frac{\omega^2 A(r)}{\sqrt{[\omega_0^2(r) - \omega^2]^2 + 4\gamma^2(r) \omega^2}} \left( \frac{r}{r + \Delta r} \right)$$ with $A_s = A(r_s)$ at the source and $r_s > 0$, it is not a point source but more to spherical source for simplicity.