Two transverse symmetrical sawtooth waves, each having the form at \(t=0\) shown in the figure are traveling in opposite directions on a stretched string. Investigate the resulting disturbance and plot the wave pattern at several times in the time interval \(0 \leq t \leq l / c\). Does the pattern in the range \(0 \leq x \leq l\) correspond to Fig. \(1.6 .2\), which is a plot of \(\eta(x, l)\) as given by (1.6.9)?

At time t = 0, we are given that two symmetrical sawtooth waves are traveling in opposite directions on the stretched string. Let one wave move to the right and another to the left. Let the wave function of these two waves be represented by \(y_1(x, t)\) and \(y_2(x, t)\), respectively. Since they are symmetrical and have the same amplitude, we can express their forms as: \(y_1(x, t) = 2A \left(\frac{x - ct}{l}\right)\) for \(0 \leq x - ct \leq \frac{l}{2}\) \(y_1(x, t) = 2A \left(1 - \frac{x - ct}{l}\right)\) for \(\frac{l}{2} \leq x - ct \leq l\) and \(y_2(x, t) = 2A \left(\frac{x + ct}{l}\right)\) for \(0 \leq x + ct \leq \frac{l}{2}\) \(y_2(x, t) = 2A \left(1 - \frac{x + ct}{l}\right)\) for \(\frac{l}{2} \leq x + ct \leq l\)

According to the principle of superposition, the resulting disturbance can be found by adding the left-moving and right-moving waves. Therefore, the total wave function can be expressed as: \(y(x, t) = y_1(x, t) + y_2(x, t)\) Now, we need to find the resulting wave function in different regions of x and t: 1) For \(0 \leq x - ct \leq \frac{l}{2}\) and \(0 \leq x + ct \leq \frac{l}{2}\), we have: \(y(x, t) = 2A \left(\frac{x - ct}{l} + \frac{x + ct}{l}\right)\) 2) For \(\frac{l}{2} \leq x - ct \leq l\) and \(0 \leq x + ct \leq \frac{l}{2}\), we have: \(y(x, t) = 2A \left(1 - \frac{x - ct}{l} + \frac{x + ct}{l}\right)\) 3) For \(0 \leq x - ct \leq \frac{l}{2}\) and \(\frac{l}{2} \leq x + ct \leq l\), we have: \(y(x, t) = 2A \left(\frac{x - ct}{l} + 1 - \frac{x + ct}{l}\right)\) 4) For \(\frac{l}{2} \leq x - ct \leq l\) and \(\frac{l}{2} \leq x + ct \leq l\), we have: \(y(x, t) = 2A \left(1 - \frac{x - ct}{l} + 1 - \frac{x + ct}{l}\right)\)

To plot the wave pattern of the resulting disturbance function at different times within the time interval \(0 \leq t \leq \frac{l}{c}\), we will create several graphs for different values of t, by using the expressions derived in step 2.

Now that we have the resulting disturbance function and its wave pattern, we can compare it with the provided Fig. 1.6.2 and check if the range \(0 \leq x \leq l\) of the graph corresponds to the plot of \(\eta(x, l)\) as given by (1.6.9). If the resulting wave pattern matches Fig. 1.6.2, then we can conclude that the pattern corresponds to the plot of \(\eta(x, l)\) as given by (1.6.9).