0 ) = 1 (0)=1} , the full physical expression becomes u ( 0 , t ) = r e { ∑ n = 1 n i p n ′ ρ n ω [ j 0 ( λ n ) − 1 j 0 ( λ n ) ] e i n ω t } \left\^\lefte^\right\}} at the centre line. the measured velocity is compared with the full expression by applying some properties of complex number. for any product of complex numbers ( c = a b ), the amplitude and phase have the relations | c | = | a | | b | and ϕ c = ϕ a + ϕ b =\phi _+\phi _ , respectively. hence, u ~ n = | i p n ′ ρ n ω [ j 0 ( λ n ) − 1 j 0 ( λ n ) ] | ⇒ p n ′ = u ~ n | i ρ n ω [ j 0 ( λ n ) 1 − j 0 ( λ n ) ] | _=\left|\left\right|\quad \rightarrow \quad p'_=_\left|i\,\rho \,n\,\omega \left\right|} and ϕ ~ = 0 = ϕ p n ′ + ϕ u n ⇒ ϕ p n ′ = phase ⁡ ( i ρ n ω [ 1 − j 0 ( λ n ) j 0 ( λ n ) ] ) , =0=\phi _+\phi _\quad \rightarrow \quad \phi _=\operatorname \left(\left\right)\,,} which finally yield 1 ρ ∂ p ∂ x = ∑ n = 1 n u ~ n | i ρ n ω [ j 0 ( λ n ) 1 − j 0 ( λ n ) ] | cos ⁡ { n ω t + phase ⁡ ( i ρ n ω [ 1 − j 0 ( λ n ) j 0 ( λ n ) ] ) } . =\sum _^_\left|i\,\rho \,n\,\omega \left\right|\,\cos \left\ \left(\left\right)\right\}\,.}