VRIJE UNIVERSITEIT BRUSSEL Acoustics - the Dept. of ...

6 CHAPTER 1. FUNDAMENTAL CONCEPTS OF ACOUSTICSthe wave equation by differentiating two times with respect to the time andrepeating the same with respect to the position x :∂ 2 u= −ω 2 u(t,x)∂t 2∂ 2 u= − ω2∂x 2 c2u(t,x) (1.4)and filling these expressions in the wave equation.If damping is present due to absorption of the medium, the solution hasthe following form:U = U 0 exp(−αu) (1.5)Note that the wave can be considered undamped if propagation of soundoccurs in an (unconfined) air volume. Indeed, the damping of sound at 1000Hz is only 5 decibel per km. One can show that the damping is proportionalto the square of the frequency according to :α = ω2 τc(1.6)With τ the relaxation time (around 0.2 ns for monatomic gases). Consequentlyfor high frequencies damping in air may not be neglected. Concerningthe periodic character, we know that the exponential function withimaginary exponent has a periodicity equal to 2π :1. Time periodicity: u(t + T,x) = u(t,x) with period T. From whichfollows that ωT = 2π and further ω = 2π = 2πf with f = 1 theT Tfrequency.2. Space periodicity: u(t,x + λ) with wavelength λ, from which followsthat λk = 2π. Replacing k = ω c yields : ωλ c = 2π and finally ω = 2πλ c3. Combining 1. and 2. gives us the relation between the spatial andfrequency domain wave propagation parameters :with c the speed of sound.1.2.2 The speed of soundλf = c (1.7)The speed of sound is the velocity at which a perturbation, a wave front,propagatesinthegivenmedium. Itdependsonthepropertiesofthismedium,