The stationary wave produced in a stretched string is given by `Y=A Cos ((2pix)/lambda) sin ((2pit)/T)`
The corresponding progresing wave has an amplitude equal to

Options

A. A/2
B. A
C. 2A
D. `A//sqrt2`

Explanation

To solve the problem, we need to find the amplitude of the corresponding progressive wave from the given stationary wave equation:

**Given stationary wave equation:**
\[ Y = A \cos\left(\frac{2\pi x}{\lambda}\right) \sin\left(\frac{2\pi t}{T}\right) \]

### Step 1: Identify the components of the stationary wave
The stationary wave can be expressed as a product of a cosine function (which depends on position \(x\)) and a sine function (which depends on time \(t\)).

### Step 2: Relate the stationary wave to progressive waves
The stationary wave can be represented as the superposition of two progressive waves traveling in opposite directions. This can be expressed as:
\[ Y = Y_1 + Y_2 \]
where:
- \( Y_1 = A \sin\left(\frac{2\pi t}{T} + \frac{2\pi x}{\lambda}\right) \)
- \( Y_2 = A \sin\left(\frac{2\pi t}{T} - \frac{2\pi x}{\lambda}\right) \)

### Step 3: Use the sine addition formula
Using the sine addition formula:
\[ \sin(a + b) + \sin(a - b) = 2 \sin(a) \cos(b) \]
we can rewrite the stationary wave as:
\[ Y = 2 \cdot \frac{A}{2} \sin\left(\frac{2\pi t}{T}\right) \cos\left(\frac{2\pi x}{\lambda}\right) \]

### Step 4: Identify the amplitude of the progressive wave
From the above expression, we can see that the amplitude of the resulting wave (which is the amplitude of the progressive wave) is:
\[ \text{Amplitude} = \frac{A}{2} \]

### Conclusion
Thus, the amplitude of the corresponding progressive wave is:
\[ \text{Amplitude} = \frac{A}{2} \]

### Final Answer
The amplitude of the corresponding progressive wave is \( \frac{A}{2} \).

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