University:
Boston University
Course:
CAS PY 105 | Physics 1
Academic year:

2021
Views:

249

Pages:

10
Author:

shacuasseh8jm

Acoustically compact source Alternative method of defining integral form of wave eq. Meaning of compact Given: If : Characteristic length scale of the source region : L Wavelength of the sound : λ L/λ << 1 we say that the source region is acoustically compact 22 Consider alternative Green’s function Integral form of solution to analogy was formed using the free-space Green’s function For the case of solid boundaries in the fluid, surface integrals that involved the normal derivative of the Green’s function We can construct a Green’s function such that on the surface, A straight forward method exists to construct such functions when the source region is acoustically compact This method is closely related to the method of matched asymptotic expansions: Solve the Laplace equation not the Helmholtz equation. Construction done in frequency domain Transform of the Green’s function wave equation gives Added constraint. G must still be causal. Reciprocal relation Source x Receiver x Source y Receiver y 23 Construct the Compact Green’s function Assume the form free space correction Far-field expansion + Compact assumption Note this looks like the potential function for freestream flow The correction term could then be analogous to the perturbation potential required to create potential flow past the object (reciprocal idea) However, this means that the correction function satisfies Laplace Eq. and by its definition it satisfies the normal condition It is argued that the main equation because is still satisfied to 1st order Construction cont. Add the perturbation potential to get first order approx to compact Green’s function Surface normal condition gives: Therefore, φ∗j signifies the perturbation potential necessary to produced the appropriate geometry when there is a unit freestream flow in the jth direction impinging on the body Final form of the compact Green’s function Kirchhoff vector 24 Example: Circular Cylinder ¾Source of sound near a circular cylinder ¾Source is located near the y1-y2 plane ¾No “perturbation potential” needed in the y3 direction: ¾Need to calculate the required perturbations in the other two directions Consider potential flow past a cylinder from elementary fluid mechanics y Cylinder formed by superposition of freestream and doublet x freestream in x-direction doublet at the origin necessary strength relation 2D Potential flow for flow past a cylinder freestream in x-direction doublet at the origin + = flow past cylinder 25 2D Potential flow for flow past a cylinder From this 2D solution We infer the perturbation parts of the Kirchhoff vectors Recall: Compact Green’s function for the cylinder Example: Dipole near rectangular wing ¾Dipole near rigid strip ¾Dipole is located near trailing edge centerline ¾No “perturbation potentials” needed in the y1 & y3 directions: ¾Need to calculate the required perturbations in the other direction Governing wave equation (single frequency disturbance amplitude f2 at x1 = L) General solution: Simplifies to 26 Example: Dipole near rectangular wing (cont.) Details for defining the potential flow solution are given in the notes. The transformation (in the complex plane) from the rectangle to a cylinder is used. Kirchhoff vector component: Solution for the dipole near the rectangular wing has the form: where the compact Green’s function is Example: Dipole near rectangular wing (cont.) Final solution: Dipole in free space radiates similarly, but no factor: 27 Other forms of the compact Green’s function Symmetric form Time domain form Exercise: Compact Green’s function for sphere Coordinates x2 x1= r sinϑ cosθ x2= r sinϑ sinθ ϑ r θ x3= r cosϑ x1 x3 Potential function for a freestream flow in x3 direction Potential function for a sphere with flow in the x3 direction: For sphere with radius a 28 Exercise: Compact Green’s function for sphere Coordinates x2 x1= r sinϑ cosθ x2= r sinϑ sinθ ϑ r θ x3= r cosϑ x1 x3 Third component of Kirchhoff vector is : Exercise: Dipole near sphere x2 Dipole in x1 direction Oscillating hamonically with radial frequency ω f1 x1 x3 On x1-axis at x1=L Find the far-field sound 29 Exercise: Dipole near sphere x2 f1 x1 x3 Recall representation of dipole from previous example (in x2 direction) Exercise: Dipole near sphere x2 f1 x1 x3 30 31

Advances in Aeroacoustics Acoustically Compact Source

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