Question

By explicit differentiation, check that the functions ${\mathit{f}}_{\mathbf{1}}$, ${\mathit{f}}_{\mathbf{2}}$, and ${\mathit{f}}_{\mathbf{3}}$in the text satisfy the wave equation. Show that ${\mathit{f}}_{\mathbf{4}}$and ${\mathit{f}}_{\mathbf{5}}$do not.

Short answer

Answer

It is proved that the functions ${\mathit{f}}_{\mathbf{1}}$, ${\mathit{f}}_{\mathbf{2}}$, and ${\mathit{f}}_{\mathbf{3}}$satisfy the wave equation and ${\mathit{f}}_{\mathbf{4}}$, and${\mathit{f}}_{\mathbf{5}}$do not.

Step-by-step solution

Expression for the functions f1,f2,f3,f4 and f5:

Write the expression for the functions ${\mathit{f}}_{\mathbf{1}}\mathbf{,}{\mathit{f}}_{\mathbf{2}}\mathbf{,}{\mathit{f}}_{\mathbf{3}}\mathbf{,}{\mathit{f}}_{\mathbf{4}}$and ${\mathit{f}}_{\mathbf{5}}$.

$$\begin{gathered} {\mathit{f}}_{\mathbf{1}}\left(z,t\right)\mathbf{=}\mathit{A}{\mathit{e}}^{\mathbf{-}\mathbf{b}{\left(z-vt\right)}^{\mathbf{2}}} \\ {\mathit{f}}_{\mathbf{2}}\left(z,t\right)\mathbf{=}\mathit{A}\mathit{s}\mathit{i}\mathit{n}\left[b\left(z-vt\right)\right] \\ {\mathit{f}}_{\mathbf{3}}\left(z,t\right)\mathbf{=}\frac{\mathbf{A}}{\mathbf{b}{\left(z-vt\right)}^{\mathbf{2}}\mathbf{+}\mathbf{1}} \\ {\mathit{f}}_{\mathbf{4}}\left(z,t\right)\mathbf{=}\mathit{A}{\mathit{e}}^{\mathbf{-}\mathbf{b}\left(b{z}^2+vt\right)} \\ {\mathit{f}}_{\mathbf{5}}\left(z,t\right)\mathbf{=}\mathit{A}\left(z,t\right)\mathbf{=}\mathit{A}\mathit{s}\mathit{i}\mathit{n}\left(bz\right)\mathit{c}\mathit{o}\mathit{s}{\left(bvt\right)}^{\mathbf{3}} \end{gathered}$$

Determine the differentiation of first equation with respect to z and t:

Differentiate the first equation with respect to z.

$\frac{\mathbf{\partial }{\mathbf{f}}_{\mathbf{1}}}{\mathbf{\partial }\mathbf{z}}\mathbf{=}\mathbf{-}\mathbf{2}\mathit{A}\mathit{b}\left(z-vt\right){\mathit{e}}^{\mathbf{-}\mathbf{b}{\left(z-vt\right)}^{\mathbf{2}}}$

Again differentiate the above equation with respect to z.

$\frac{{\mathbf{\partial }}^{\mathbf{2}}{\mathbf{f}}_{\mathbf{1}}}{\mathbf{\partial }{\mathbf{z}}^{\mathbf{2}}}\mathbf{=}\mathbf{-}\mathbf{2}\mathit{A}\mathit{b}\mathbf{[}{\mathbf{e}}^{\mathbf{-}\mathbf{b}{\left(z-\mathrm{vt}\right)}^{\mathbf{2}}}\mathbf{-}\mathbf{2}\mathbf{b}{\left(z-\mathrm{vt}\right)}^{\mathbf{2}}{\mathbf{e}}^{\mathbf{-}\mathbf{b}{\left(z-\mathrm{vt}\right)}^{\mathbf{2}}}\mathbf{]}$

Differentiate the first equation with respect to t.

$\frac{\mathbf{\partial }{\mathbf{f}}_{\mathbf{1}}}{\mathbf{\partial }\mathbf{t}}\mathbf{=}\mathbf{2}\mathit{A}\mathit{b}\mathit{v}\mathbf{(}\mathbf{z}\mathbf{-}\mathbf{vt}\mathbf{)}{{\mathit{e}}^{\mathbf{-}\mathbf{b}\mathbf{(}\mathbf{z}\mathbf{-}\mathbf{vt}\mathbf{)}}}^{\mathbf{2}}$

Again differentiate the above equation with respect to t.

$$\begin{gathered} \frac{{\mathbf{\partial }}^{\mathbf{2}}{\mathbf{f}}_{\mathbf{1}}}{\mathbf{\partial }{\mathbf{z}}^{\mathbf{2}}}\mathbf{=}\mathbf{-}\mathbf{2}\mathit{A}\mathit{b}\mathbf{[}\mathbf{-}{\mathbf{ve}}^{\mathbf{-}\mathbf{b}{\mathbf{(}\mathbf{z}\mathbf{-}\mathbf{vt}\mathbf{)}}^{\mathbf{2}}}\mathbf{+}\mathbf{2}\mathbf{bv}{\mathbf{(}\mathbf{z}\mathbf{-}\mathbf{vt}\mathbf{)}}^{\mathbf{2}}{\mathbf{e}}^{\mathbf{-}\mathbf{b}{\mathbf{(}\mathbf{z}\mathbf{-}\mathbf{vt}\mathbf{)}}^{\mathbf{2}}}\mathbf{]} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}{\mathit{v}}^{\mathbf{2}}\frac{{\mathbf{\partial }}^{\mathbf{2}}{\mathbf{f}}_{\mathbf{1}}}{\mathbf{\partial }{\mathbf{z}}^{\mathbf{2}}} \end{gathered}$$

Determine the differentiation of second equation with respect to  and :

Differentiate the second equation with respect to z.

$\frac{\mathbf{\partial }{\mathbf{f}}_{\mathbf{2}}}{\mathbf{\partial }\mathbf{z}}\mathbf{=}\mathit{A}\mathit{b}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\left[b\left(z-vt\right)\right]$

Again differentiate the above equation with respect to z.

$\frac{{\mathbf{\partial }}^{\mathbf{2}}{\mathbf{f}}_{\mathbf{2}}}{\mathbf{\partial }{\mathbf{z}}^{\mathbf{2}}}\mathbf{=}\mathbf{-}\mathit{A}{\mathit{b}}^{\mathbf{2}}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left[}\mathbf{b}\left(z-\mathrm{vt}\right)\mathbf{\right]}$

Differentiate the second equation with respect to t.

$\frac{\mathbf{\partial }{\mathbf{f}}_{\mathbf{2}}}{\mathbf{dt}}\mathbf{=}\mathbf{-}\mathit{A}\mathit{b}\mathit{v}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{\left[}\mathbf{b}\mathbf{(}\mathbf{z}\mathbf{-}\mathbf{vt}\mathbf{)}\mathbf{\right]}$

Again differentiate the above equation with respect to t.

$$\begin{gathered} \frac{{\mathbf{\partial }}^{\mathbf{2}}{\mathbf{f}}_{\mathbf{2}}}{\mathbf{\partial }{\mathbf{t}}^{\mathbf{2}}}\mathbf{=}\mathbf{-}\mathit{A}{\mathit{b}}^{\mathbf{2}}{\mathit{v}}^{\mathbf{2}}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left[}\mathbf{b}\mathbf{(}\mathbf{z}\mathbf{-}\mathbf{vt}\mathbf{)}\mathbf{\right]} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}{\mathit{v}}^{\mathbf{2}}\frac{{\mathbf{\partial }}^{\mathbf{2}}{\mathbf{f}}_{\mathbf{2}}}{\mathbf{\partial }{\mathbf{z}}^{\mathbf{2}}} \end{gathered}$$

Determine the differentiation of third equation with respect to z and t:

Differentiate the third equation with respect to z.

$\frac{\mathbf{\partial }{\mathbf{f}}_{\mathbf{3}}}{\mathbf{\partial }\mathbf{z}}\mathbf{=}\frac{\mathbf{-}\mathbf{2}\mathbf{A}\mathbf{b}\left(z-vt\right)}{{\left[b{\left(z-vt\right)}^2+1\right]}^{\mathbf{2}}}$

Again differentiate the above equation with respect to z.

$\frac{{\mathbf{\partial }}^{\mathbf{2}}{\mathbf{f}}_{\mathbf{3}}}{\mathbf{\partial }{\mathbf{t}}^{\mathbf{2}}}\mathbf{=}\frac{\mathbf{-}\mathbf{2}\mathbf{Ab}}{{\mathbf{[}\mathbf{b}{\left(z-\mathrm{vt}\right)}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{]}}^{\mathbf{2}}}\mathbf{+}\frac{\mathbf{8}{\mathbf{Ab}}^{\mathbf{2}}{\mathbf{(}\mathbf{z}\mathbf{-}\mathbf{vt}\mathbf{)}}^{\mathbf{2}}}{[b{\left(z-\mathrm{vt}\right)}^2+1]}$

Differentiate the third equation with respect to t.

$\frac{\mathbf{\partial }{\mathbf{f}}_{\mathbf{3}}}{\mathbf{\partial }\mathbf{t}}\mathbf{=}\frac{\mathbf{2}\mathbf{Abv}\mathbf{(}\mathbf{z}\mathbf{-}\mathbf{vt}\mathbf{)}}{{\mathbf{[}\mathbf{b}{\mathbf{(}\mathbf{z}\mathbf{-}\mathbf{vt}\mathbf{)}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{]}}^{\mathbf{2}}}$

Again differentiate the above equation with respect to t.

$$\begin{gathered} \frac{{\mathbf{\partial }}^{\mathbf{2}}{\mathbf{f}}_{\mathbf{2}}}{\mathbf{\partial }{\mathbf{t}}^{\mathbf{2}}}\mathbf{=}\frac{\mathbf{-}{\mathbf{Ab}}^{\mathbf{2}}{\mathbf{v}}^{\mathbf{2}}}{{\mathbf{[}\mathbf{b}{\mathbf{(}\mathbf{z}\mathbf{-}\mathbf{vt}\mathbf{)}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{]}}^{\mathbf{2}}}\mathbf{+}\frac{\mathbf{8}{\mathbf{Ab}}^{\mathbf{2}}{\mathbf{v}}^{\mathbf{2}}{\mathbf{(}\mathbf{z}\mathbf{-}\mathbf{vt}\mathbf{)}}^{\mathbf{2}}}{[b{\left(z-\mathrm{vt}\right)}^2+1]} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}{\mathit{v}}^{\mathbf{2}}\frac{{\mathbf{\partial }}^{\mathbf{2}}{\mathbf{f}}_{\mathbf{3}}}{\mathbf{\partial }{\mathbf{z}}^{\mathbf{2}}} \end{gathered}$$

Determine the differentiation of fourth equation with respect to z and t:

Differentiate the fourth equation with respect to z.

$\frac{\mathbf{\partial }{\mathbf{f}}_4}{\mathbf{\partial }\mathbf{z}}\mathbf{=}\mathbf{2}{\mathbf{Ab}}^{\mathbf{2}}{\mathbf{ze}}^{\mathbf{-}\mathbf{b}\mathbf{(}{\mathbf{bz}}^{\mathbf{2}}\mathbf{+}\mathbf{vt}\mathbf{)}}$

Again differentiate the above equation with respect to z.

$\frac{{\mathbf{\partial }}^{\mathbf{2}}{\mathbf{f}}_4}{\mathbf{\partial }{\mathbf{z}}^{\mathbf{2}}}\mathbf{=}\mathbf{2}{\mathbf{Ab}}^{\mathbf{2}}\left[e^{-b({\mathrm{bz}}^2+\mathrm{vt})}-2b^2{z}^2{e}^{-b\left(z^2+\mathrm{vt}\right)}\right]$

Differentiate the fourth equation with respect to t.

$\frac{\mathbf{\partial }{\mathbf{f}}_{\mathbf{4}}}{\mathbf{\partial }\mathbf{t}}\mathbf{=}\mathbf{-}\mathbf{2}\mathit{A}\mathit{b}\mathit{v}{\mathit{e}}^{\mathbf{-}\mathbf{b}\left(b{z}^2+vt\right)}$

Again differentiate the above equation with respect to t.

$$\begin{gathered} \frac{{\mathbf{\partial }}^{\mathbf{2}}{\mathbf{f}}_{\mathbf{4}}}{\mathbf{\partial }{\mathbf{t}}^{\mathbf{2}}}\mathbf{=}\mathit{A}{\mathit{b}}^{\mathbf{2}}{\mathit{v}}^{\mathbf{2}}{\mathit{e}}^{\mathbf{-}\mathbf{b}\left(b{z}^2+vt\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\ne }{\mathit{v}}^{\mathbf{2}}\frac{{\mathbf{\partial }}^{\mathbf{2}}{\mathbf{f}}_{\mathbf{4}}}{\mathbf{\partial }{\mathbf{z}}^{\mathbf{2}}} \end{gathered}$$

Determine the differentiation of fifth equation with respect to z and t:

Differentiate the fifth equation with respect to z.

$\frac{\mathbf{\partial }{\mathbf{f}}_{\mathbf{5}}}{\mathbf{\partial }\mathbf{z}}\mathbf{=}\mathit{A}\mathit{b}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\left(bz\right)\mathit{c}\mathit{o}\mathit{s}{\left(bvt\right)}^{\mathbf{3}}$

Again differentiate the above equation with respect to z.

$\frac{{\mathbf{\partial }}^{\mathbf{2}}{\mathbf{f}}_{\mathbf{5}}}{\mathbf{\partial }{\mathbf{z}}^{\mathbf{2}}}\mathbf{=}\mathbf{-}\mathit{A}{\mathit{b}}^{\mathbf{2}}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\mathbf{bz}\mathbf{\right)}\mathit{c}\mathit{o}\mathit{s}{\mathbf{\left(}\mathbf{bvt}\mathbf{\right)}}^{\mathbf{3}}$

Differentiate the fifth equation with respect to t.

$\frac{\mathbf{\partial }{\mathbf{f}}_{\mathbf{5}}}{\mathbf{\partial }\mathbf{t}}\mathbf{=}\mathbf{-}\mathbf{3}\mathit{A}{\mathit{b}}^{\mathbf{3}}{\mathit{v}}^{\mathbf{3}}{\mathit{t}}^{\mathbf{3}}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\mathbf{bz}\mathbf{\right)}\mathit{s}\mathit{i}\mathit{n}{\mathbf{\left(}\mathbf{bvt}\mathbf{\right)}}^{\mathbf{3}}$

Again differentiate the above equation with respect to t.

$$\begin{gathered} \frac{{\mathbf{\partial }}^{\mathbf{2}}{\mathbf{f}}_{\mathbf{5}}}{\mathbf{\partial }{\mathbf{t}}^{\mathbf{2}}}\mathbf{=}\mathbf{-}\mathbf{6}\mathit{A}{\mathit{b}}^{\mathbf{3}}{\mathit{v}}^{\mathbf{3}}\mathit{t}\mathbf{}\mathit{s}\mathit{i}\mathit{n}{\mathbf{\left(}\mathbf{bvt}\mathbf{\right)}}^{\mathbf{3}}\mathbf{-}\mathbf{9}\mathit{A}{\mathit{b}}^{\mathbf{6}}{\mathit{v}}^{\mathbf{6}}{\mathit{t}}^{\mathbf{4}}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\mathbf{bz}\mathbf{\right)}\mathit{c}\mathit{o}\mathit{s}{\mathbf{\left(}\mathbf{bvt}\mathbf{\right)}}^{\mathbf{3}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\ne }{\mathit{v}}^{\mathbf{2}}\frac{{\mathbf{\partial }}^{\mathbf{2}}{\mathbf{f}}_{\mathbf{5}}}{\mathbf{\partial }{\mathbf{z}}^{\mathbf{2}}} \end{gathered}$$

Therefore, it is proved that the functions ${\mathit{f}}_{\mathbf{1}}$, ${\mathit{f}}_{\mathbf{2}}$, and ${\mathit{f}}_{\mathbf{3}}$satisfy the wave equation and ${\mathit{f}}_{\mathbf{4}}$, and ${\mathit{f}}_{\mathbf{5}}$do not.