General solutions of the wave equation

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Problem 2.5a

Verify that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi =f\left(x-Vt\right) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi =g\left(x+Vt\right) are solutions of the wave equation (2.5b).

Background

When unbalanced stresses act upon a medium, the strains are propagated throughout the medium according to the general wave equation


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\nabla} ^{2} \psi =\frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} x^{2} } +\frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} y^{2} } +\frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} z^{2} } =\frac{1}{V^{2} } \frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} t^{2} }, \end{align} (2.5a)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi being a disturbance such as a compression or rotation. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi is propagated with velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V (see Sheriff and Geldart, 1995, Section 2.2). The disturbance is the result of unbalanced normal stresses, shearing stresses, or a combination of both. When normal stresses create the wave, the result is a volume change and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi is the dilitation [see equation (2.1e)], and we get the P-wave equation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V becoming the P-wave velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\alpha}} . Shearing stresses create rotation in the medium and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi is one of the components of the rotation given by equation (2.lg) ; the result is an S-wave traveling with velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\beta}} . Various expressions for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\alpha}} and $ {\mathrm {\beta } } $ in isotropic media are given in Table 2.2a.

In one dimension the wave equation (2.5a) reduces to


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} x^{2} } =\frac{1}{V^{2} } \frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} t^{2}}. \end{align} (2.5b)

Solution

We use subscripts to denote partial derivatives and primes to denote derivatives with respect to the argument of the function. Then, writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \zeta =\left(x-Vt\right) , we have


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {{{\mathrm{\psi}} _x} = \frac{{\partial {\mathrm{\psi}} }}{{\partial x}} = \left( {\frac{{df}}{{d{\mathrm{\zeta}} }}} \right)\left( {\frac{{\partial {\mathrm{\zeta}} }}{{\partial x}}} \right) = \frac{{df}}{{d{\mathrm{\zeta}} }} = f',}\\ {{{\mathrm{\psi}} _{xx}} = \left( {\frac{{df'}}{{d{\mathrm{\zeta}} }}} \right)\left( {\frac{{\partial {\mathrm{\zeta}} }}{{\partial x}}} \right) = f'',}\\ {{{\mathrm{\psi}} _t} = \left( {\frac{{df}}{{d{\mathrm{\zeta}} }}} \right)\left( {\frac{{\partial {\mathrm{\zeta}} }}{{\partial t}}} \right) = - Vf',}\\ {{{\mathrm{\psi}} _{tt}} = - V\left( {\frac{{df'}}{{d{\mathrm{\zeta}} }}} \right)\left( {\frac{{\partial {\mathrm{\zeta}} }}{{\partial t}}} \right) = {V^2}f''} \end{align}

Substituting in equation (2.5b), we get the identity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f''=f'' so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f\left(x-Vt\right) is a solution. We get the same result when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi =g\left(x+Vt\right) . A sum of solutions is also a solution, so $ f\left(x-Vt\right)+g\left(x+Vt\right) $ is a solution.

Problem 2.5b

Verify that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\psi =f\left(\ell x+my+nz-Vt\right)+g\left(\ell x+my+nz+Vt\right)} is a solution of equation (2.5a), where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\left(\ell, {\rm \; }m,\; n\right)} are direction cosines.

Solution

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(\ell x\; +my+nz-Vt\right)=\zeta , \left(\ell x\; +my+nz+Vt\right)=\xi . We now must show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi =f\left(\zeta \right)+g\left(\xi \right) is a solution of equation (2.5a). Proceeding as before, we have


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {{\mathrm{\psi}} _{x} =\left(\frac{df}{d\zeta } \right)\left(\frac{{\mathrm{\partial}} \zeta }{{\mathrm{\partial}} x} \right)+\left(\frac{dg}{d\xi } \right)\left(\frac{{\mathrm{\partial}} \xi }{{\mathrm{\partial}} x} \right)=\ell (f'+g'),}\\ {{\mathrm{\psi}} _{xx} =\ell ^{2} (f''+g'').} \end{align}

In the same way we get


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \psi _{yy} =m^{2} (f''+g''),\;{\quad}\psi _{zz} =n^{2} (f''+g''). \end{align}

But Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(\ell ^{2} +m^{2} +n^{2} \right)=1 (see Sheriff and Geldart, 1995, problem 15.9a), so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(\psi _{xx} +\psi _{yy} +\psi _{zz} \right)= (f''+g'') .

Following the same procedure we find that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(1/V^{2} \right)\psi _{tt} =(f''+g'') thus verifying that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi =f\left(\ell x+my+nz-Vt\right)+g\left(\ell x+my+nz+Vt\right) is a solution of equation (2.5a).

Problem 2.5c

Show that


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \psi \left(r,\; t\right)=\left(1/r\right)f\left(r-Vt\right)+\left(1/r\right)g\left(r+Vt\right) \end{align}

is a solution of the wave equation in spherical coordinates (see problem 2.6b) when the wave motion is independent of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\theta} and $ \phi $:


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \frac{1}{V^{2} } \frac{{\mathrm{\partial}} ^{2} {\mathrm{\psi}} }{{\mathrm{\partial}} t^{2} } =\frac{1}{r^{2} } \left[\frac{{\mathrm{\partial}} }{{\mathrm{\partial}} r} \left(r^{2} \frac{{\mathrm{\partial}} {\mathrm{\psi}} }{{\mathrm{\partial}} r} \right)\right]. \end{align} (2.5c)

Solution

The wave equation in spherical coordinates is given in problem 2.6b. When we drop the derivatives with respect to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\theta}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\phi}} , the equation reduces to equation (2.5c). Writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\zeta}} = (r-Vt) , we proceed as in part (a). Starting with the right-hand side, we ignore Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): g\left(r+Vt\right) for the time being and obtain


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\psi}} _{r} =-\left(1/r^{2} \right)f\left({\mathrm{\zeta}} \right)+\left(1/r\right)f'\left({\mathrm{\zeta}} \right), \\ r^{2} {\mathrm{\psi}} _{r} =-f\left({\mathrm{\zeta}} \right)+rf'\left({\mathrm{\zeta}} \right), \\ \frac{{\mathrm{\partial}} }{{\mathrm{\partial}} r} \left(r^{2} {\mathrm{\psi}} _{r} \right)=-f'\left({\mathrm{\zeta}} \right)+f'\left({\mathrm{\zeta}} \right)+rf''\left({\mathrm{\zeta}} \right)=rf''\left({\mathrm{\zeta}} \right), \\ \left(1/r^{2} \right)\frac{{\mathrm{\partial}} }{{\mathrm{\partial}} r} \left(r^{2} {\mathrm{\psi}} _{r} \right)=\left(1/r\right)f''\left({\mathrm{\zeta}} \right), \\ \psi _t =-\left(V/r\right)f'; {\quad}{\mathrm{\phi}} _{tt} =\left(V^{2} /r\right)f''. \end{align}

Substitution in equation (2.5c) shows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(1/r\right)f\left({\mathrm{\zeta}} \right) is a solution. In the same way we can show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(1/r\right)g\left({\mathrm{\xi}} \right) is also a solution, hence the sum is a solution.

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