The planes sources of sound of frequency `n_(1)=400Hz,n_(2)=401Hz` of equal amplitude of `a` each, are sounded together. A detector receives waves from the two sources simultaneously. It can detect signals of amplitude with magnitude `ge` `a` only,

Answer

To solve the problem step by step, we will analyze the situation involving two sound sources and the resultant amplitude detected by the detector.

### Step-by-Step Solution:

1. **Identify the Given Frequencies**:
- The frequencies of the two sound sources are given as:
- \( n_1 = 400 \, \text{Hz} \)
- \( n_2 = 401 \, \text{Hz} \)

2. **Write the General Expression for the Waves**:
- The displacement of the wave from the first source can be expressed as:
\[ y_1 = a \sin(2\pi n_1 t) = a \sin(800\pi t) \]
- The displacement of the wave from the second source can be expressed as:
\[ y_2 = a \sin(2\pi n_2 t) = a \sin(802\pi t) \]

3. **Apply the Principle of Superposition**:
- According to the principle of superposition, the resultant displacement \( y \) at any point in time is the sum of the displacements from both sources:
\[ y = y_1 + y_2 = a \sin(800\pi t) + a \sin(802\pi t) \]

4. **Use the Trigonometric Identity**:
- We can use the trigonometric identity for the sum of two sine functions:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
- Here, let \( A = 800\pi t \) and \( B = 802\pi t \):
\[ y = a \left(2 \sin\left(\frac{800\pi t + 802\pi t}{2}\right) \cos\left(\frac{800\pi t - 802\pi t}{2}\right)\right) \]
- Simplifying this:
\[ y = 2a \sin(801\pi t) \cos(\pi t) \]

5. **Determine the Resultant Amplitude**:
- The resultant amplitude \( A \) is given by:
\[ A = 2a \cos(\pi t) \]
- The amplitude varies with time due to the cosine term.

6. **Condition for Detection**:
- The detector can only detect signals with an amplitude greater than or equal to \( a \):
\[ |A| \geq a \]
- Therefore:
\[ |2a \cos(\pi t)| \geq a \]
- Dividing both sides by \( a \) (assuming \( a \neq 0 \)):
\[ |2 \cos(\pi t)| \geq 1 \]
- This implies:
\[ |\cos(\pi t)| \geq \frac{1}{2} \]

7. **Finding the Time Intervals**:
- The condition \( |\cos(\pi t)| \geq \frac{1}{2} \) corresponds to the angles where the cosine function is greater than or equal to \( \frac{1}{2} \).
- This occurs in the intervals:
\[ t \in \left[0, \frac{1}{3}\right] \cup \left[\frac{2}{3}, 1\right] \, \text{(and repeating every 2 seconds)} \]

### Final Answer:
The detector will successfully detect the sound waves when \( |\cos(\pi t)| \geq \frac{1}{2} \), which corresponds to specific time intervals as calculated.