A wave is represented by an equation, `Y=A coskxsinomegat`, then

Options

A. it is a progresive wave with amplitude A
B. it is a progresive wave with amplitude A`coskx`
C. it is a stationary wave with amplitude A
D. it is a stationary wave with amplitude A`coskx`

Answer

To determine whether the wave represented by the equation \( Y = A \cos(kx) \sin(\omega t) \) is a standing wave or a progressive wave, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Form of the Wave Equation**:
The given wave equation is \( Y = A \cos(kx) \sin(\omega t) \). This equation can be recognized as a product of two functions: one that depends on position \( x \) (i.e., \( \cos(kx) \)) and another that depends on time \( t \) (i.e., \( \sin(\omega t) \)).

2. **Understand the Characteristics of Standing Waves**:
A standing wave is formed by the interference of two progressive waves traveling in opposite directions. The general form of a standing wave can be expressed as:
\[ Y(x, t) = f(x) g(t) \]
where \( f(x) \) is a function of position and \( g(t) \) is a function of time.

3. **Compare with Progressive Waves**:
A progressive wave typically has the form:
\[ Y(x, t) = A \sin(kx - \omega t) \quad \text{or} \quad Y(x, t) = A \sin(kx + \omega t) \]
In these forms, the wave travels in a specific direction over time.

4. **Analyze the Given Equation**:
In the equation \( Y = A \cos(kx) \sin(\omega t) \), the \( \cos(kx) \) term indicates a spatial variation, while the \( \sin(\omega t) \) term indicates a temporal variation. This structure is characteristic of standing waves, where the amplitude varies with position but oscillates in time.

5. **Determine the Amplitude of the Standing Wave**:
The amplitude of the standing wave is given by the spatial part of the equation, which is \( A \cos(kx) \). This means that the amplitude is not constant but varies with position \( x \).

6. **Conclusion**:
Since the wave equation can be expressed as a product of a function of position and a function of time, we conclude that the wave represented by the equation \( Y = A \cos(kx) \sin(\omega t) \) is indeed a standing wave. The amplitude of this standing wave is \( A \cos(kx) \).

### Final Answer:
The wave represented by the equation \( Y = A \cos(kx) \sin(\omega t) \) is a standing wave with amplitude \( A \cos(kx) \).

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