1D : **$`\quad\large{\boldsymbol{\mathbf{\underline{U}(x,t)}}}`$** $`\;= A\cdot \underbrace{exp\,[\,i\,(\omega t - k x + \varphi}_{\color{blue}{exp(a+b) = exp(a)\times exp(b)})\,]}`$
1D : **$`\quad\large{\boldsymbol{\mathbf{\underline{U}(x,t)}}}`$** $`\;= A\cdot \underbrace{exp\,[\,i\,(\omega t - k x + \varphi}_{\color{blue}{exp(a+b) = exp(a)\times exp(b)})\,]}`$
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$`\quad\quad\quad\quad\quad\quad =\underbrace{A\;e^{\,i\,\varphi}}_{\color{blue}{\underline{A}=A\; e^{\,i\,\varphi}}\cdot exp\,[\,i\,(\omega t - kx)\,]`$
$`\quad\quad\quad\quad\quad\quad =\underbrace{A\;e^{\,i\,\varphi}}_{\color{blue}{\underline{A}=A\; e^{\,i\,\varphi}}}\cdot exp\,[\,i\,(\omega t - kx)\,]`$
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**$`\quad\quad\quad\quad\quad\quad\boldsymbol{\mathbf{\large{=\underline{A}\cdot e^{\,i\,(\omega t - kx)}}}}`$**
**$`\quad\quad\quad\quad\quad\quad\boldsymbol{\mathbf{\large{\,=\underline{A}\cdot e^{\,i\,(\omega t - kx)}}}}`$**
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3D : **$`\quad\large{\boldsymbol{\mathbf{\underline{U}(\vec{r},t)=\underline{A}\cdot e^{\,i\,(\omega t - \vec{k}\cdot\vec{r})}}}}`$**
3D : **$`\quad\large{\boldsymbol{\mathbf{\underline{U}(\vec{r},t)=\underline{A}\cdot e^{\,i\,(\omega t - \vec{k}\cdot\vec{r})}}}}`$**