The displacement of a string carrying a travelling sinusoidal wave is given by `y(x,t) =y_m sin(kx-omegat-phi)`
At time t=0 the point at x=0 has a velocity of 0 and a positive displacement. The phase constant `phi` is

Options

A. `90^(@)`
B. `135^(@)`
C. `180^(@)`
D. `270^(@)`

Answer

To find the phase constant \( \phi \) for the given wave equation, we start with the displacement function:

\[ y(x,t) = y_m \sin(kx - \omega t - \phi) \]

### Step 1: Determine the velocity of the wave
The velocity \( v \) of the wave can be found by differentiating the displacement \( y \) with respect to time \( t \):

\[ v(x,t) = \frac{\partial y}{\partial t} = y_m \frac{\partial}{\partial t} \left( \sin(kx - \omega t - \phi) \right) \]

Using the chain rule, we have:

\[ v(x,t) = y_m \cos(kx - \omega t - \phi) \cdot (-\omega) \]

Thus, the velocity becomes:

\[ v(x,t) = -\omega y_m \cos(kx - \omega t - \phi) \]

### Step 2: Evaluate the velocity at \( t = 0 \) and \( x = 0 \)
Now, we substitute \( t = 0 \) and \( x = 0 \) into the velocity equation:

\[ v(0,0) = -\omega y_m \cos(0 - 0 - \phi) = -\omega y_m \cos(-\phi) = -\omega y_m \cos(\phi) \]

### Step 3: Set the velocity to zero
According to the problem, at \( t = 0 \) and \( x = 0 \), the velocity is given to be zero:

\[ -\omega y_m \cos(\phi) = 0 \]

Since \( \omega \) and \( y_m \) are both non-zero constants, we can conclude that:

\[ \cos(\phi) = 0 \]

### Step 4: Solve for the phase constant \( \phi \)
The cosine function is zero at:

\[ \phi = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]

### Step 5: Determine the correct phase constant
Given that the displacement at \( t = 0 \) and \( x = 0 \) is positive, we can evaluate the sine function:

\[ y(0,0) = y_m \sin(0 - 0 - \phi) = y_m \sin(-\phi) \]

For \( \phi = \frac{\pi}{2} \):

\[ y(0,0) = y_m \sin\left(-\frac{\pi}{2}\right) = -y_m \quad (\text{not positive}) \]

For \( \phi = \frac{3\pi}{2} \):

\[ y(0,0) = y_m \sin\left(-\frac{3\pi}{2}\right) = y_m \quad (\text{positive}) \]

Thus, the phase constant \( \phi \) that satisfies both conditions (zero velocity and positive displacement) is:

\[ \phi = \frac{3\pi}{2} \quad \text{or} \quad 270^\circ \]

### Final Answer:
The phase constant \( \phi \) is \( \frac{3\pi}{2} \) or \( 270^\circ \).

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