Chapter 9.8, Problem 36E

To determine

To show that the formula can be rewritten as $y=-A\cos \frac{2\pi x}{5}$ .

Explanation of Solution

Given:

Suppose a standing wave can be modeled by the formula- $y=A\cos \left(\frac{2\pi t}{3}-\frac{2\pi x}{5}\right)+A\cos \left(\frac{2\pi t}{3}+\frac{2\pi x}{5}\right)$ .

Calculation:

Writing the formula as follows-

  $\begin{array}{l}{y}_1=\cos (A-B)\\ y_2=\cos (A+B)\end{array}$

Now, use the following formulas-

  $\begin{array}{l}\cos (A+B)=\cos A\cos B-\sin A\sin B\\ \cos (A-B)=\cos A\cos B+\sin A\sin B\end{array}$

This means

  $\begin{array}{l}y=y_1+y_2\\ y=2\cos A\cos B\end{array}$

So,

  $\begin{array}{l}y=A\cos \left(\frac{2\pi t}{3}-\frac{2\pi x}{5}\right)+A\cos \left(\frac{2\pi t}{3}+\frac{2\pi x}{5}\right)\\ y=2A\left[\cos \left(\frac{2\pi t}{3}\right)\cos \frac{2\pi x}{5}\right]\\ y=2A\left[\cos \left(\frac{2\pi }{3}\right)\cos \frac{2\pi x}{5}\right]\{\because t=1\}\\ y=-A\cos \frac{2\pi x}{5}\end{array}$