Question

Calculate acoustic impedance of air and water at STP using the data 3313m kg 10,ms 332,m kg 29.1---=r=n=rwaterairairand .ms 15001-=nwater Derive the formula used.

Answer 1

Derivation of the formulas:

   (Specific) Acoustic impedance z of sound wave in a fluid medium is the resistance the particles of the medium offer to the propagation of sound wave.  Impedance is inversely proportional to the wave number k = ω/v, where ω = angular frequency and v = velocity of the wave.

   The more the density ρ and stiffness K in the medium are, the more the speed v of propagation of sound.

       z defined as √(K*ρ)  = √(B * ρ)
               K = stiffness constant = Modulus of elasticity
               B = Bulk modulus for fluids = K
               ρ = density of the medium.

     For fluids (air and water) B = - P / (ΔV/V) = stress / volume strain
           B = excess pressure / volume strain

    We know that sound waves are the result of SHM of the particles of the medium.  Using this principle, the sound modeled as a longitudinal excess pressure P wave along  with longitudinal displacements s of particles, we derive that : 
       P = B s₀ k cos (ω t - k x)   for   s = s₀ sin(ωt - k x)
      
   Then using the excess pressure model, we find the excess force ΔF on an element of volume A Δx of the medium, by Pressure P times area A. That will be   ΔF = A * ΔP = - B s₀ k²  sin (ω t - k x) Δx

We get the acceleration by dividing by mass Δm = A ρ Δx .  We know for given expression for s, acceleration is d²s/dt². Comparing both expressions we get that:
 
               v = √(B / ρ)     or ,    B = v² ρ

 Substituting this in the definition of z, we get:

   Acoustic Impedance z = √(v² ρ²) = v ρ
============
at STP, we have 
Density of air = 1.29 kg/m³       speed of sound in air = 332 m/s
Density of water = 1,000 kg/m³        Speed of sound in water = 1500 m/s
Refractive index of air = 1                refractive index of water = 1.33

So acoustic impedance of air = z = 1.29 * 332 
                =  428.28  kg/m²-s   or  428.28 Rayl  or  Ry

       z of water = 1500  * 1000
                        = 1.5 * 10⁶ kg/m²-s  or  Ry
==================
Another derivation for acoustic impedance.

Acoustic impedance z = Acoustic pressure p / acoustic fluid velocity u

     s (x,t) = s₀ Sin(ω t - k x)             -- displacement of particles
     u (x,t) = particle speed = ds/dt = s₀ ω Cos (ωt - k x)
     p(x,t)  = - K ds/dx = - B s₀ k Cos (ωt - k x)
     v(t) = speed of the wave at any x.

Thus  z = B k/ω = B / v  = ρ v         as   B = ρ v²   for sound.