Chapter 7, Problem 2P

(a)

To determine

To find: The equation that outputs the combined wave and also find the nodes.

Answer to Problem 2P

The combined equation of the wave is $y=0.4\sin \left(1.047x\right)\cos \left(0.524t\right)$.

The nodes are at $x=\frac{k\pi }{1.047}$.

Explanation of Solution

Given:

A wave travelling in a rope of length 24 ft is a combination of two distinct waves $y=0.2\sin \left(1.047x-0.524t\right)\text{ and }y=0.2\sin \left(1.047x+0.524t\right)$.

Result used:

Sum-product rule $\sin \left(x+y\right)+\sin \left(x-y\right)=2\sin x\cos y$.

Calculation:

The combined wave equation is the sum of the two equations of the wave.

Therefore, obtain the combined equation as follows.

$\begin{array}{c}y=0.2\sin \left(1.047x-0.524t\right)+0.2\sin \left(1.047x+0.524t\right)\\ =0.2\left[\sin \left(1.047x-0.524t\right)+\sin \left(1.047x+0.524t\right)\right]\\ =0.2\left[2\sin \left(1.047x\right)\cos \left(0.524t\right)\right]\\ =0.4\sin \left(1.047x\right)\cos \left(0.524t\right)\end{array}$

Therefore, the combined wave equation is $y=0.4\sin \left(1.047x\right)\cos \left(0.524t\right)$.

Note that the node occurs when $\sin \left(1.047x\right)=0$.

That is $1.047x=k\pi \text{ or }x=\frac{k\pi }{1.047}$.

Therefore, nodes occur at $x=\frac{k\pi }{1.047}$.

(b)

To determine

To sketch: The shape of the wave for times $t=0,1,2,3,4,5\text{ and 6}$ and check whether the wave is standing or not.

Explanation of Solution

Given:

A wave travelling in a rope of length 24 ft is a combination of two distinct waves $y=0.2\sin \left(1.047x-0.524t\right)\text{ and }y=0.2\sin \left(1.047x+0.524t\right)$.

Calculation:

The position of any point x on the wave at time t in the canal is given by $y=0.4\sin \left(1.047x\right)\cos \left(0.524t\right)$.

Substitute $t=0$ and obtain the position of the point for time $t=0$.

$\begin{array}{l}y=0.4\sin \left(1.047x\right)\cos \left(0.524\left(0\right)\right)\\ y=0.4\sin \left(1.047x\right)\cos \left(0\right)\\ y=0.4\sin \left(1.047x\right)\end{array}$

Use online graphing calculator and draw the graph of $y=0.4\sin \left(1.047x\right)$ as shown below in Figure 1.

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

Substitute $t=1$ and obtain the position of the point for time $t=1$.

$\begin{array}{l}y=0.4\sin \left(1.047x\right)\cos \left(0.524\left(1\right)\right)\\ y=0.4\sin \left(1.047x\right)\cos \left(0.524\right)\end{array}$

Use online graphing calculator and draw the graph of $y=0.4\sin \left(1.047x\right)\cos \left(0.524\right)$ as shown below in Figure 2.

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

Substitute $t=2$ and obtain the position of the point for time $t=2$.

$\begin{array}{l}y=0.4\sin \left(1.047x\right)\cos \left(0.524\left(2\right)\right)\\ y=0.4\sin \left(1.047x\right)\cos \left(1.048\right)\end{array}$

Use online graphing calculator and draw the graph of $y=0.4\sin \left(1.047x\right)\cos \left(1.048\right)$ as shown below in Figure 3.

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

Substitute $t=3$ and obtain the position of the point for time $t=3$.

$\begin{array}{l}y=0.4\sin \left(1.047x\right)\cos \left(0.524\left(3\right)\right)\\ y=0.4\sin \left(1.047x\right)\cos \left(1.572\right)\end{array}$

Use online graphing calculator and draw the graph of $y=0.4\sin \left(1.047x\right)\cos \left(1.572\right)$ as shown below in Figure 4.

From Figure 4 it can be observed that the wave has very small amplitude so it appears to be coinciding with the x-axis.

Substitute $t=4$ and obtain the position of the point for time $t=4$.

$\begin{array}{l}y=0.4\sin \left(1.047x\right)\cos \left(0.524\left(4\right)\right)\\ y=0.4\sin \left(1.047x\right)\cos \left(2.096\right)\end{array}$

Use online graphing calculator and draw the graph of $y=0.4\sin \left(1.047x\right)\cos \left(2.096\right)$ as shown below in Figure 5.

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

Substitute $t=5$ and obtain the position of the point for time $t=5$.

$\begin{array}{l}y=0.4\sin \left(1.047x\right)\cos \left(0.524\left(5\right)\right)\\ y=0.4\sin \left(1.047x\right)\cos \left(2.620\right)\end{array}$

Use online graphing calculator and draw the graph of $y=0.4\sin \left(1.047x\right)\cos \left(2.620\right)$ as shown below in Figure 6.

From Figure 6 it can be observed that the wave is a sinusoidal wave.

Substitute $t=6$ and obtain the position of the point for time $t=6$.

$\begin{array}{l}y=0.4\sin \left(1.047x\right)\cos \left(0.524\left(6\right)\right)\\ y=0.4\sin \left(1.047x\right)\cos \left(3.144\right)\end{array}$

Use online graphing calculator and draw the graph of $y=0.4\sin \left(1.047x\right)\cos \left(3.144\right)$ as shown below in Figure 7.

From Figure 7 it can be observed that the wave is a sinusoidal wave.

From above all graphs it can be observed that the point is at same position at every instant of time.

Therefore, it can be concluded that the wave is standing.