# 16.2 Mathematics of waves  (Page 5/11)

 Page 5 / 11
$\frac{{\partial }^{2}y\left(x,t\right)}{\partial {x}^{2}}=\frac{1}{{v}^{2}}\phantom{\rule{0.2em}{0ex}}\frac{{\partial }^{2}y\left(x,t\right)}{\partial {t}^{2}}.$

[link] is the linear wave equation, which is one of the most important equations in physics and engineering. We derived it here for a transverse wave, but it is equally important when investigating longitudinal waves. This relationship was also derived using a sinusoidal wave, but it successfully describes any wave or pulse that has the form $y\left(x,t\right)=f\left(x\mp vt\right).$ 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 ${y}_{1}\left(x,t\right)=f\left(x\mp vt\right)$ and ${y}_{2}\left(x,t\right)=g\left(x\mp vt\right),$ which are solutions to the linear wave equations and are therefore linear wave functions. The sum of the wave functions is the wave function

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

Consider the linear wave equation:

$\begin{array}{ccc}\hfill \frac{{\partial }^{2}\left(f+g\right)}{\partial {x}^{2}}& =\hfill & \frac{1}{{v}^{2}}\phantom{\rule{0.2em}{0ex}}\frac{{\partial }^{2}\left(f+g\right)}{\partial {t}^{2}}\hfill \\ \hfill \frac{{\partial }^{2}f}{\partial {x}^{2}}+\frac{{\partial }^{2}g}{\partial {x}^{2}}& =\hfill & \frac{1}{{v}^{2}}\left[\frac{{\partial }^{2}f}{\partial {t}^{2}}+\frac{{\partial }^{2}g}{\partial {t}^{2}}\right].\hfill \end{array}$

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 ${y}_{1}\left(x,t\right)=f\left(x\mp vt\right)$ and ${y}_{2}\left(x,t\right)=g\left(x\mp vt\right)$ are solutions to the linear wave equation, then $A{y}_{1}\left(x,t\right)+B{y}_{2}\left(x,y\right),$ 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 .

## Interference of waves on a string

Consider a very long string held taut by two students, one on each end. Student A oscillates the end of the string producing a wave modeled with the wave function ${y}_{1}\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t\right)$ and student B oscillates the string producing at twice the frequency, moving in the opposite direction. Both waves move at the same speed $v=\frac{\omega }{k}.$ The two waves interfere to form a resulting wave whose wave function is ${y}_{R}\left(x,t\right)={y}_{1}\left(x,t\right)+{y}_{2}\left(x,t\right).$ Find the velocity of the resulting wave using the linear wave equation $\frac{{\partial }^{2}y\left(x,t\right)}{\partial {x}^{2}}=\frac{1}{{v}^{2}}\phantom{\rule{0.2em}{0ex}}\frac{{\partial }^{2}y\left(x,t\right)}{\partial {t}^{2}}.$

## Strategy

First, write the wave function for the wave created by the second student. Note that the angular frequency of the second wave is twice the frequency of the first wave $\left(2\omega \right)$ , and since the velocity of the two waves are the same, the wave number of the second wave is twice that of the first wave $\left(2k\right).$ Next, write the wave equation for the resulting wave function, which is the sum of the two individual wave functions. Then find the second partial derivative with respect to position and the second partial derivative with respect to time. Use the linear wave equation to find the velocity of the resulting wave.

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