class: center, middle # Engineering Physics - Fall 2024 ## <p style="color: tomato;">(Waves)</p> --- <div style="text-align: center;"> <h3 style="color: black; font-size: 20px;"> A <strong>wave</strong> is a <a style="color: red">propagating</a>, <a style="color: blue">self-sustaining</a> disturbance of a medium. </div> <div style="color: black;text-align: center">\(\color{blue}{f}(x)=\color{blue}{f}(x\pm \color{red}{v}t)=:\Psi(x,t)\)</div> <div style="text-align: center;"> <h3>What dynamical law governs the motion of the <span style="color:tomato;">disturbance</span>? <p style="color: black;font-size: 16px;"> We seek a law in the form of an equation which yields a new position in any independent coordinate as a function of the old ones \[``="\] differential equation </p> </h3> </div> <p style="color: black;font-size: 14px;"> The <span style="color: dodgerblue;font-size:150%;">disturbance</span> is a function of 2 variables which can be varied <strong>independently</strong><br> (holding the respective other fixed): \[ \frac{\partial\Psi}{\partial x}=\frac{\partial f}{\partial x}=\frac{\partial f(x\pm vt)}{\partial(x\pm vt)}\frac{\partial(x\pm vt)}{\partial x}=\frac{\partial f}{\partial(x\pm vt)} \] and \[ \frac{\partial\Psi}{\partial t}=\frac{\partial f}{\partial t}=\frac{\partial f(x\pm vt)}{\partial(x\pm vt)}\frac{\partial(x\pm vt)}{\partial t}=\pm v\frac{\partial f}{\partial(x\pm vt)} \] \[ \Rightarrow \frac{\partial\Psi}{\partial t}=\pm v\frac{\partial\Psi}{\partial x} \] Repeat to find the 1-dimansional, fixed-profile (free) wave equation: <p style="color: tomato;font-size: 18px;"> \[ \frac{\partial^2\Psi}{\partial t^2}=v^2\frac{\partial^2\Psi}{\partial x^2} \] </p> </p> --- <h2>The <a style="color:darkorange;"> electro</a>-<a style="color:purple">magnetic</a> transversal wave</h2> <div class="bg"></div>