At a large enough distance from the reflective material, there is no interference left anymore. And as interference decreases, so does the phase difference between sound pressure and particle velocity. As the power of the reflective wave decreases compared to the power of the incident wave, interference also decreases.
As a consequence, the local pressure in the near field is doubled, and the particle velocity becomes zero.Īttenuation causes the reflected wave to decrease in power as distance from the reflective material increases. If a travelling wave is reflected, the reflected wave can interfere with the incident wave causing a standing wave in the near field.
As pressure is maximum at the ends while velocity is zero, there is a 90 degrees phase difference between them.Īn acoustic travelling wave can be reflected by a solid surface. Pressure however doubles at the ends because of interference of the incident wave with the reflective wave. MATLAB code is also included for matrices used in the method of moment with piecewise constant and piecewise linear functions.F = N c 2 d N ∈ is the length of the tube in mĪt the ends particle velocity becomes zero since there can be no particle displacement. Detailed algorithms are included in appendices with verified MATLAB code to compute many special functions such as Struve functions and successive primitives of Bessel and Hankel functions. Approximate solutions, residuals and error bounds are illustrated in many color plots. A variational method derives novel approximate solutions of the above scattering problems that minimize the error bounds. Bounds are also derived using values of the residuals solely on the strip or plate where they are defined. Error bounds are expressed with norms in Sobolev spaces of order 1/2 and 1. In each case the scattering problem is solved by the Galerkin method (method of moments with identical sets of basis and testing functions) and an error bound is derived for the approximation. Electromagnetic and Acoustic Wave Scattering, Method of Moments and Error Bound studies examples of transverse magnetic (TM) and transverse Electric (TE) electromagnetic scattering, and acoustic scattering by a sound soft surface leading to Fredholm integral equations of the first kind.
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