Chapter 7, Problem 4P

Traveling Wave A traveling wave has period 2π/3, amplitude 5, and velocity 0.5.

1. (a) Find the equation of the wave.
2. (b) Sketch the graph for t = 0, 0.5, 1, 1.5, and 2.

To determine

To find: The equation of the travelling wave with period $\frac{2\pi }{3}$, amplitude 5, and velocity 0.5.

Answer to Problem 4P

The equation of the wave is $y\left(x,t\right)=5\sin \left(3x-1.5t\right)$.

Explanation of Solution

The general equation of the wave is of the form $y\left(x,t\right)=A\sin \left(kx-kvt\right)$ where A is amplitude, v is velocity and k is a constant which is calculated for period p by formula $k=\frac{2\pi }{p}$.

The amplitude of the wave is A = 5.

The period of the wave is $\frac{2\pi }{3}$.

Calculate the value of the k as follows.

$\begin{array}{c}k=\frac{2\pi }{\text{period}}\\ =\frac{2\pi }{\frac{2\pi }{3}}\\ =3\end{array}$

Therefore, $k=3$.

It is given that velocity is $v=0.5$.

Using all the above values the equation of the wave is $y\left(x,t\right)=5\sin \left(\left(3\right)x-\left(3\right)\left(0.5\right)t\right)$.

Simplifying further the equation of the wave is $y\left(x,t\right)=5\sin \left(3x-1.5t\right)$.

To determine

To sketch: The graph of the wave for $t=0,0.5,1,1.5\text{ and 2}$.

Explanation of Solution

The equation of the wave is $y\left(x,t\right)=5\sin \left(3x-1.5t\right)$.

Substitute $t=0$ in $y\left(x,t\right)=5\sin \left(3x-1.5t\right)$ and obtain the wave equation as $y\left(x,t\right)=5\sin \left(3x\right)$.

Use online graphing calculator and draw the graph of $y\left(x,t\right)=5\sin \left(3x\right)$ as shown below in Figure 1.

From Figure 1 it can be noted that the wave is sinusoidal.

Substitute $t=0.5$ in $y\left(x,t\right)=5\sin \left(3x-1.5t\right)$ and obtain the wave equation as $y\left(x,t\right)=5\sin \left(3x-0.75\right)$.

Use online graphing calculator and draw the graph of $y\left(x,t\right)=5\sin \left(3x-0.75\right)$ as shown below in Figure 2.

From Figure 2 it can be noted that the wave is sinusoidal.

Substitute $t=1$ in $y\left(x,t\right)=5\sin \left(3x-1.5t\right)$ and obtain the wave equation as $y\left(x,t\right)=5\sin \left(3x-1.5\right)$.

Use online graphing calculator and draw the graph of $y\left(x,t\right)=5\sin \left(3x-1.5\right)$ as shown below in Figure 3.

From Figure 3 it can be noted that the wave is sinusoidal.

Substitute $t=1.5$ in $y\left(x,t\right)=5\sin \left(3x-1.5t\right)$ and obtain the wave equation as $y\left(x,t\right)=5\sin \left(3x-2.25\right)$.

Use online graphing calculator and draw the graph of $y\left(x,t\right)=5\sin \left(3x-2.25\right)$ as shown below in Figure 4.

From Figure 4 it can be noted that the wave is sinusoidal.

Substitute $t=2$ in $y\left(x,t\right)=5\sin \left(3x-1.5t\right)$ and obtain the wave equation as $y\left(x,t\right)=5\sin \left(3x-3\right)$.

Use online graphing calculator and draw the graph of $y\left(x,t\right)=5\sin \left(3x-3\right)$ as shown below in Figure 5.

From Figure 5 it can be noted that the wave is sinusoidal.