Only the real part has any physical meaning like all

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• Only the real part has any physical meaning. • Like all complex representations, require complex conjugate rather than straight square for squared terms. • e.g. 7 u 2 = uu * complex conjugate 2.2.1 Harmonic solutions
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2. 1D wave propagation 8 Wavelength For a single frequency, the wavelength is the distance between two points with the same phase. ω t - x 1 c - ω t - x 2 c = 2 π Wavelength = λ = x 1 - x 2 = 2 π c ω = c f 2.2.1 Harmonic solutions
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2. 1D wave propagation 9 Wavenumber The wavelength, λ , appears in the solution to the wave equation as 2 π x/ λ Hence we define the quantity 2 π / λ as the wavenumber Wavenumber = k = 2 π λ = ω c 2.2.1 Harmonic solutions
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2. 1D wave propagation 2.2.2 Relationship between variables • Sound waves characterised by three variables: – acoustic pressure, p / – acoustic density, ρ / – acoustic velocity, u / • These variables are related by the equations of motion of fluids. • Have already seen that: 10 δ p = p ∂ρ δρ δ p = (¯ c 2 ) δρ p = (¯ c 2 ) ρ ¯ c 2 = p ∂ρ s = γ RT
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2. 1D wave propagation • We now consider the momentum equation: 11 u t = - 1 ¯ ρ p x but u and p have harmonic solutions so we can write: u t = i ω u - 1 ¯ ρ p x = - i ω c p - 1 ¯ ρ p x = i ω c p Right-travelling wave Left-travelling wave Hence: p = +¯ ρ cu p = - ¯ ρ cu Right-travelling wave Left-travelling wave 2.2.2 Relationship between variables
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2. 1D wave propagation • Finally, combining these relationships together we have: 12 p = ¯ c 2 ρ = ¯ ρ ¯ cu ρ = ¯ ρ u ¯ c Right-travelling wave ρ = - ¯ ρ u ¯ c similarly: Left-travelling wave 2.2.2 Relationship between variables
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2. 1D wave propagation 2.2.3 The un-importance of viscosity • So far we have assumed that sound propagation is inviscid • Here we justify this assumption • For 1D waves, there is no variation normal to the wave direction so the viscous effects are given only by the dilation term : 13 τ xx = μ u x = ± μi ω ¯ c u τ xx p = μ ω ¯ c u p = μ ω ¯ c u ¯ ρ
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  • Summer '16
  • Energy, Wavenumber, B ∗

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