Two vibrating tuning fork produce progressive waves given by `y_(1)=4 sin(500 pit)` and `y_(2)=2 sin(506 pi t)`. These tuning forks are held near the ear of a person. The person will hear `alpha` beats/s with intensity ratio between maxima and minima equal to `beta`. Find the value of `beta-alpha`.

Explanation

To solve the problem, we need to find the values of α (the number of beats per second) and β (the intensity ratio between maxima and minima) based on the given wave equations.

### Step 1: Identify the wave equations
The wave equations given are:
- \( y_1 = 4 \sin(500 \pi t) \)
- \( y_2 = 2 \sin(506 \pi t) \)

### Step 2: Determine the angular frequencies
From the wave equations, we can identify the angular frequencies:
- For \( y_1 \), \( \omega_1 = 500 \pi \)
- For \( y_2 \), \( \omega_2 = 506 \pi \)

### Step 3: Calculate the frequencies
The frequency \( f \) is related to the angular frequency \( \omega \) by the formula:
\[ f = \frac{\omega}{2\pi} \]
Calculating the frequencies:
- For \( y_1 \):
\[ f_1 = \frac{500 \pi}{2 \pi} = 250 \text{ Hz} \]
- For \( y_2 \):
\[ f_2 = \frac{506 \pi}{2 \pi} = 253 \text{ Hz} \]

### Step 4: Calculate the beat frequency (α)
The beat frequency is given by the absolute difference between the two frequencies:
\[ \alpha = |f_2 - f_1| = |253 - 250| = 3 \text{ beats/s} \]

### Step 5: Determine the amplitudes
From the wave equations:
- Amplitude of \( y_1 \) (A1) = 4
- Amplitude of \( y_2 \) (A2) = 2

### Step 6: Calculate the intensity ratio (β)
The intensity ratio between the maximum and minimum can be calculated using the formula:
\[ \text{Intensity ratio} = \frac{(A_1 + A_2)^2}{(A_1 - A_2)^2} \]
Substituting the values:
\[ \beta = \frac{(4 + 2)^2}{(4 - 2)^2} = \frac{6^2}{2^2} = \frac{36}{4} = 9 \]

### Step 7: Calculate β - α
Now, we need to find the difference:
\[ \beta - \alpha = 9 - 3 = 6 \]

### Final Answer
Thus, the value of \( \beta - \alpha \) is \( 6 \).

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