Mathematics of Waves Lumen University Physics 16.2

https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/16-2-mathematics-of-waves/

The wave function modeling a sinusoidal wave, allowing for an initial phase shift $\phi$, is

$$y(x,t)=Asin(kx\mp \omega t+\phi )$$

The value $(kx\mp \omega t+\phi )$ is known as the phase of the wave, where $\phi$ is the initial phase of the wave function. Whether the temporal term $\omega t$ is negative or positive depends on the direction of the wave.

First, consider the minus sign for a wave with an initial phase equal to zero $(\phi =0)$. The phase of the wave would be $(kx-\omega t)$. Consider following a point on a wave, such as a crest. A crest will occur when $\text{sin}(kx-\omega t)=1.00$, that is, when $kx-\omega t=n\pi +\frac{\pi }{2}$, for any integral value of $n$. For instance, one particular crest occurs at $kx-\omega t=\frac{\pi }{2}$. As the wave moves, time increases and $x$ must also increase to keep the phase equal to $\frac{\pi }{2}$. Therefore, the minus sign is for a wave moving in the positive $x$-direction. Using the plus sign, $kx+\omega t=\frac{\pi }{2}$. As time increases, $x$ must decrease to keep the phase equal to $\frac{\pi }{2}$. The plus sign is used for waves moving in the negative $x$-direction. In summary,

• $y(x,t)=A,\text{sin}(kx-\omega t+\phi )$ models a wave moving in the positive $x$-direction
• $y(x,t)=A,\text{sin}(kx+\omega t+\phi )$ models a wave moving in the negative $x$-direction

A wave function is any function such that $f(x,t)=f(x-vt)$. Later in this chapter, we will see that it is a solution to the linear wave equation. Note that $y(x,t)=A,\text{cos}(kx+\omega t+\phi \text{′})$ works equally well because it corresponds to a different phase shift $\phi \text{′}=\phi -\frac{\pi }{2}.$

Problem-Solving Strategy: Finding the Characteristics of a Sinusoidal Wave
To find the amplitude, wavelength, period, and frequency of a sinusoidal wave, write down the wave function in the form $y(x,t)=A,\text{sin}(kx-\omega t+\phi )$.

• The amplitude can be read straight from the equation and is equal to $A$.
• The period of the wave can be derived from the angular frequency $(T=\frac{2\pi }{\omega })$.
• The frequency can be found using $f=\frac{1}{T}$.
• The wavelength can be found using the wave number $(\lambda =\frac{2\pi }{k})$.

The wave speed is constant and represents the speed of the wave as it propagates through the medium, not the speed of the particles that make up the medium. The particles of the medium oscillate around an equilibrium position as the wave propagates through the medium. In the case of the transverse wave propagating in the x-direction, the particles oscillate up and down in the y-direction, perpendicular to the motion of the wave. The velocity of the particles of the medium is not constant, which means there is an acceleration. The velocity of the medium, which is perpendicular to the wave velocity in a transverse wave, can be found by taking the partial derivative of the position equation with respect to time. The partial derivative is found by taking the derivative of the function, treating all variables as constants, except for the variable in question. In the case of the partial derivative with respect to time t, the position x is treated as a constant. Although this may sound strange if you haven’t seen it before, the object of this exercise is to find the transverse velocity at a point, so in this sense, the x-position is not changing. We have

$$\begin{array}{ccc}\text{hfill}y(x,t)& =\text{hfill}& A\text{sin}(kx-\omega t+\phi )\text{hfill}\\ \text{hfill}{v}_y(x,t)& =\text{hfill}& \frac{\partial y(x,t)}{\partial t}=\frac{\partial }{\partial t}(A\text{sin}(kx-\omega t+\phi ))\text{hfill}\\ & =\text{hfill}& \text{−}A\omega \text{cos}(kx-\omega t+\phi )\text{hfill}\\ & =\text{hfill}& \text{−}{v}_{y\text{max}}\text{cos}(kx-\omega t+\phi ).\text{hfill}\end{array}$$

The magnitude of the maximum velocity of the medium is

|{v}_{{y}_{\text{max}}}|=A\omega $$. This may look familiar from the [Oscillations](https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/introduction-15/) and a mass on a spring. We can find the acceleration of the medium by taking the partial derivative of the velocity equation with respect to time,

\begin{array}{cc}\hfill {a}{y}(x,t)& =\frac{\partial {v}{y}}{\partial t}=\frac{\partial }{\partial t}(\text{−}A\omega ,\text{cos}(kx-\omega t+\varphi ))\hfill \ & =\text{−}A{\omega }^{2},\text{sin}(kx-\omega t+\varphi )\hfill \ & =\text{−}{a}_{y,\text{max}}\text{sin}(kx-\omega t+\varphi ).\hfill \end{array}

$$Themagnitudeofthemaximumaccelerationis$$

|{a}{{y}{\text{max}}}|=A{\omega }^{2}.

### The Linear Wave Equation We have just determined the velocity of the medium at a position _x_ by taking the partial derivative, with respect to time, of the position _y_. For a transverse wave, this velocity is perpendicular to the direction of propagation of the wave. We found the acceleration by taking the partial derivative, with respect to time, of the velocity, which is the second time derivative of the position:

{a}_{y}(x,t)=\frac{{\partial }^{2}y(x.t)}{\partial {t}^{2}}=\frac{{\partial }^{2}}{\partial {t}^{2}}(A,\text{sin}(kx-\omega t+\varphi ))=\text{−}A{\omega }^{2},\text{sin}(kx-\omega t+\varphi ).

Now consider the partial derivatives with respect to the other variable, the position _x_, holding the time constant. The first derivative is the slope of the wave at a point _x_ at a time _t_,

\text{slope}=\frac{\partial y(x,t)}{\partial x}=\frac{\partial }{\partial x}(A,\text{sin}(kx-\omega t+\varphi ))=Ak,\text{cos}(kx-\omega t+\varphi ).

$$Thesecondpartialderivativeexpresseshowtheslopeofthewavechangeswithrespecttoposition—inotherwords,thecurvatureofthewave,where$$

\text{curvature}=\frac{{\partial }^{2}y(x,t)}{\partial {x}^{2}}=\frac{{\partial }^{2}}{{\partial }^{2}x}(A,\text{sin}(kx-\omega t+\varphi ))=\text{−}A{k}^{2},\text{sin}(kx-\omega t+\varphi ).

$$Theratiooftheaccelerationandthecurvatureleadstoaveryimportantrelationshipinphysicsknownasthe\ast \ast linearwaveequation\ast \ast .Takingtheratioandusingtheequation$$

v=\omega \text{/}k

\begin{array}{cc}\hfill \frac{\frac{{\partial }^{2}y(x,t)}{\partial {t}^{2}}}{\frac{{\partial }^{2}y(x,t)}{\partial {x}^{2}}}& =\frac{\text{−}A{\omega }^{2},\text{sin}(kx-\omega t+\varphi )}{\text{−}A{k}^{2},\text{sin}(kx-\omega t+\varphi )}\hfill \ & =\frac{{\omega }^{2}}{{k}^{2}}={v}^{2},\hfill \end{array}

\frac{{\partial }^{2}y(x,t)}{\partial {x}^{2}}=\frac{1}{{v}^{2}},\frac{{\partial }^{2}y(x,t)}{\partial {t}^{2}}. $thelinearwaveequation,whichisoneofthemostimportantequationsinphysicsandengineering.Wederivedithereforatransversewave,butitisequallyimportantwheninvestigatinglongitudinalwaves.Thisrelationshipwasalsoderivedusingasinusoidalwave,butitsuccessfullydescribesanywaveorpulsethathastheform$ y(x,t)=f(x\mp vt). $$ These waves result due to a linear restoring force of the medium—thus, the name linear wave equation. Any wave function that satisfies this equation is a linear wave function.

An interesting aspect of the linear wave equation is that if two wave functions are individually solutions to the linear wave equation, then the sum of the two linear wave functions is also a solution to the wave equation. Consider two transverse waves that propagate along the x-axis, occupying the same medium. Assume that the individual waves can be modeled with the wave functions

{y}_{1}(x,t)=f(x\mp vt) $$ and $$ {y}_{2}(x,t)=g(x\mp vt), $$ which are solutions to the linear wave equations and are therefore linear wave functions. The sum of the wave functions is the wave function

{y}{1}(x,t)+{y}{2}(x,t)=f(x\mp vt)+g(x\mp vt).

$$Considerthelinearwaveequation:$$

\begin{array}{ccc}\hfill \frac{{\partial }^{2}(f+g)}{\partial {x}^{2}}& =\hfill & \frac{1}{{v}^{2}},\frac{{\partial }^{2}(f+g)}{\partial {t}^{2}}\hfill \ \hfill \frac{{\partial }^{2}f}{\partial {x}^{2}}+\frac{{\partial }^{2}g}{\partial {x}^{2}}& =\hfill & \frac{1}{{v}^{2}}[\frac{{\partial }^{2}f}{\partial {t}^{2}}+\frac{{\partial }^{2}g}{\partial {t}^{2}}].\hfill \end{array}

This has shown that if two linear wave functions are added algebraically, the resulting wave function is also linear. This wave function models the displacement of the medium of the resulting wave at each position along the _x_-axis. If two linear waves occupy the same medium, they are said to interfere. If these waves can be modeled with a linear wave function, these wave functions add to form the wave equation of the wave resulting from the interference of the individual waves. The displacement of the medium at every point of the resulting wave is the algebraic sum of the displacements due to the individual waves. Taking this analysis a step further, if wave functions

{y}{1}(x,t)=f(x\mp vt) $and$ {y}{2}(x,t)=g(x\mp vt) $aresolutionstothelinearwaveequation,then$ A{y}{1}(x,t)+B{y}{2}(x,y), $$ where A and B are constants, is also a solution to the linear wave equation. This property is known as the principle of superposition. Interference and superposition are covered in more detail in Interference of Waves.