Two sinusoidal waves of the same frequency travel in the same direction along a string. If `y_(m1) =2.0 cm, y_(m2)=4.0cm , phi_(1)=0, and phi_(2)=pi//2 rad`, what is the amplitude of the resultant wave?

Answer

To find the amplitude of the resultant wave when two sinusoidal waves of the same frequency travel in the same direction, we can use the principle of superposition and vector addition. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Amplitude of the first wave, \( y_{m1} = 2.0 \, \text{cm} \)
- Amplitude of the second wave, \( y_{m2} = 4.0 \, \text{cm} \)
- Phase of the first wave, \( \phi_1 = 0 \, \text{rad} \)
- Phase of the second wave, \( \phi_2 = \frac{\pi}{2} \, \text{rad} \)

2. **Represent the Waves as Vectors:**
- The first wave can be represented as a vector along the x-axis with a length of 2.0 cm (since its phase is 0).
- The second wave can be represented as a vector at a 90-degree angle (since its phase is \( \frac{\pi}{2} \)) with a length of 4.0 cm.

3. **Draw the Vectors:**
- Draw a horizontal vector (representing the first wave) of length 2.0 cm.
- Draw a vertical vector (representing the second wave) of length 4.0 cm.

4. **Use the Pythagorean Theorem to Find the Resultant Amplitude:**
- The resultant amplitude \( R \) can be calculated using the Pythagorean theorem:
\[ R = \sqrt{(y_{m1})^2 + (y_{m2})^2} \]
- Substitute the values:
\[ R = \sqrt{(2.0 \, \text{cm})^2 + (4.0 \, \text{cm})^2} \]
\[ R = \sqrt{4 + 16} = \sqrt{20} \, \text{cm} \]
\[ R = \sqrt{20} \approx 4.47 \, \text{cm} \]

5. **Final Result:**
- The amplitude of the resultant wave is approximately \( 4.47 \, \text{cm} \).