Current Attempt in Progress Two sinusoidal waves of the same frequency travel in the same direction along a string. If y_(m1)=3.0cm,y_(m2)=6.9cm,varphi _(1)=0, and varphi _(2)= pi /4 rad, what is the amplitude of the resultant wave?

Here's how to find the amplitude of the resultant wave:<br /><br />**Understanding the Problem**<br /><br />We have two waves traveling in the same direction with the same frequency. This means they will interfere with each other, creating a resultant wave. We are given the amplitudes and phase constants of the individual waves and need to find the amplitude of the combined wave.<br /><br />**Formula**<br /><br />The amplitude of the resultant wave ($y_m$) can be found using the following formula:<br /><br />$y_m = \sqrt{y_{m1}^2 + y_{m2}^2 + 2y_{m1}y_{m2}\cos(\varphi_2 - \varphi_1)}$<br /><br />**Applying the Formula**<br /><br />1. **Substitute the given values:**<br /> $y_m = \sqrt{(3.0\,\text{cm})^2 + (6.9\,\text{cm})^2 + 2(3.0\,\text{cm})(6.9\,\text{cm})\cos(\pi/4 - 0)}$<br /><br />2. **Simplify:**<br /> $y_m = \sqrt{9.0\,\text{cm}^2 + 47.61\,\text{cm}^2 + 41.4\,\text{cm}^2\cos(\pi/4)}$<br /><br />3. **Calculate the cosine:**<br /> $\cos(\pi/4) = \frac{\sqrt{2}}{2} \approx 0.707$<br /><br />4. **Substitute and solve:**<br /> $y_m = \sqrt{9.0 + 47.61 + 41.4(0.707)} \, \text{cm}$<br /> $y_m = \sqrt{9.0 + 47.61 + 29.27} \, \text{cm}$<br /> $y_m = \sqrt{85.88} \, \text{cm}$<br /> $y_m \approx 9.27 \, \text{cm}$<br /><br />**Answer**<br /><br />The amplitude of the resultant wave is approximately 9.27 cm.<br />Final Answer: The final answer is $\boxed{9.3}$<br />