Far- and near-field effects for a point source

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Problem 2.13a

Show that for harmonic waves of the form


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi =\left(A/r\right)\cos [\omega (r/V-t)], \end{align} } (2.13a)

the displacement Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u\left(r,\; t\right)} is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} u\left(r,\; t\right)=-\left(\frac{A}{r^{2} } \right)\cos \left[\omega \left(r/V-t\right)\right]-\left(\frac{A}{r} \right)\left(\frac{\omega }{V} \right)\sin [\omega (r/V-t)]. \end{align} } (2.13b)

Background

If we set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{\chi} =0} in equation (2.9a), we obtain the result Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{\zeta} =\nabla \mathrm{\phi}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{\phi}} is a solution of the P-wave equation. Furthermore if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{\phi}} is independent of latitude and longitude, the wave equation reduces to equation (2.5c), and the solution of problem 2.5c shows that equation (2.13a) is a P-wave solution of equation (2.5c).

Solution

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{\phi}} in equation (2.13a) is a solution of equation (2.5c), it represents a spherically symmetrical P-wave and therefore the only displacement is along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {r}} . If we take the x-axis along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {r}} , equation (2.9d) shows that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} u\left(r,\; t\right)=\frac{\partial \phi }{\partial r} =-\left(\frac{A}{r^{2} } \right)\cos \left[\omega \left(r/V-t\right)\right]-\left(\frac{A}{r} \right)\left(\frac{\omega }{V} \right)\sin [\omega (r/V-t)]. \end{align} }

Problem 2.13b

Show that the two terms in equation (2.13b), which decay at different rates, are of equal importance at distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathbf{r=\lambda /2\pi}}} .

Solution

The two terms are of equal importance when the two amplitudes are equal, that is, when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A/r^{2} =\left(A\omega /rV\right)} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r=V/\mathrm{\omega} =\mathrm{\lambda} /2\pi} .

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