Two waves are travelling in same direction along a stretched string. The waves are `90^@` out of phase. Each wave has an amplitude of 4.0 cm. Find the amplitude of the resultant wave.

Explanation

To find the amplitude of the resultant wave when two waves are traveling in the same direction and are 90 degrees out of phase, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Amplitude of Wave 1 (A1) = 4.0 cm
- Amplitude of Wave 2 (A2) = 4.0 cm
- Phase difference between the two waves = 90 degrees

2. **Understanding the Phase Difference:**
- When two waves are 90 degrees out of phase, it means that they do not interfere constructively or destructively in the traditional sense. Instead, we will use the Pythagorean theorem to find the resultant amplitude.

3. **Apply the Pythagorean Theorem:**
- Since the waves are out of phase by 90 degrees, we can treat their amplitudes as perpendicular vectors. The resultant amplitude (R) can be calculated using the formula:
\[ R = \sqrt{A_1^2 + A_2^2} \]

4. **Substituting the Values:**
- Substitute the amplitudes into the formula:
\[ R = \sqrt{(4.0 \, \text{cm})^2 + (4.0 \, \text{cm})^2} \]
\[ R = \sqrt{16 + 16} \]
\[ R = \sqrt{32} \]

5. **Simplifying the Result:**
- We can simplify \(\sqrt{32}\):
\[ R = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \, \text{cm} \]

6. **Calculating the Numerical Value:**
- Finally, we can calculate the numerical value of \(4\sqrt{2}\):
\[ R \approx 4 \times 1.414 = 5.656 \, \text{cm} \]

### Final Answer:
The amplitude of the resultant wave is approximately **5.66 cm**.

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