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 their resultant wave when received by a detector.

### Step 1: Understand the Given Frequencies and Amplitudes
We have two sound sources with frequencies:
- \( n_1 = 400 \, \text{Hz} \)
- \( n_2 = 401 \, \text{Hz} \)

Both sources have equal amplitudes, denoted as \( A \).

### Step 2: Write the Equations for Each Wave
The equations for the waves produced by each source can be expressed using the sine function:
- For the first source:
\[ y_1 = A \sin(2\pi n_1 t) = A \sin(800\pi t) \]
- For the second source:
\[ y_2 = A \sin(2\pi n_2 t) = A \sin(802\pi t) \]

### Step 3: Apply the Principle of Superposition
According to the principle of superposition, the resultant wave \( y \) from both sources is given by:
\[ y = y_1 + y_2 = A \sin(800\pi t) + A \sin(802\pi t) \]

### Step 4: Use the Sine Addition Formula
We can simplify the expression using the sine addition formula:
\[ \sin C + \sin D = 2 \sin\left(\frac{C+D}{2}\right) \cos\left(\frac{C-D}{2}\right) \]
Here, let \( C = 800\pi t \) and \( D = 802\pi t \):
\[ y = A \left( \sin(800\pi t) + \sin(802\pi t) \right) \]
Using the formula:
\[ y = A \cdot 2 \sin\left(\frac{800\pi t + 802\pi t}{2}\right) \cos\left(\frac{800\pi t - 802\pi t}{2}\right) \]
This simplifies to:
\[ y = 2A \sin(801\pi t) \cos(\pi t) \]

### Step 5: Determine the Amplitude
From the equation \( y = 2A \sin(801\pi t) \cos(\pi t) \), we can see that the amplitude of the resultant wave varies with time and is given by:
\[ \text{Amplitude} = 2A \cos(\pi t) \]

### Step 6: Condition for Detection
The detector can only detect signals with amplitude greater than or equal to \( A \). Therefore, we need to find when:
\[ 2A \cos(\pi t) \geq A \]
Dividing both sides by \( A \) (assuming \( A > 0 \)):
\[ 2 \cos(\pi t) \geq 1 \]
This simplifies to:
\[ \cos(\pi t) \geq \frac{1}{2} \]

### Step 7: Solve for \( t \)
The cosine function is greater than or equal to \( \frac{1}{2} \) in the intervals:
\[ \pi t = 2k\pi \quad \text{or} \quad \pi t = 2k\pi + \frac{\pi}{3} \quad \text{for integers } k \]
This gives:
\[ t = 2k \quad \text{or} \quad t = 2k + \frac{1}{3} \quad \text{for integers } k \]

### Final Result
The detector will detect the sound when \( t \) is of the form \( 2k \) or \( 2k + \frac{1}{3} \) for integers \( k \).

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