A sinusoidal wave trsvelling in the positive direction on a stretched string has amplitude `2.0 cm`, wavelength `1.0 m` and velocity `5.0 m//s`. At `x = 0` and ` t= 0` it is given that `y = 0` and `(dely)/(delt) lt 0`. Find the wave function `y (x, t)`.

Explanation

To find the wave function \( y(x, t) \) for the given sinusoidal wave traveling in the positive direction, we will follow these steps:

### Step 1: Identify the wave parameters
Given:
- Amplitude \( A = 2.0 \, \text{cm} = 0.02 \, \text{m} \)
- Wavelength \( \lambda = 1.0 \, \text{m} \)
- Velocity \( v = 5.0 \, \text{m/s} \)

### Step 2: Calculate the wave number \( k \)
The wave number \( k \) is given by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( \lambda \):
\[ k = \frac{2\pi}{1.0} = 2\pi \, \text{radians/m} \]

### Step 3: Calculate the angular frequency \( \omega \)
The angular frequency \( \omega \) can be calculated using the relationship:
\[ \omega = v \cdot k \]
Substituting the values of \( v \) and \( k \):
\[ \omega = 5.0 \cdot 2\pi = 10\pi \, \text{radians/s} \]

### Step 4: Write the general wave function
The general form of the wave function for a wave traveling in the positive direction is:
\[ y(x, t) = A \sin(kx - \omega t + \phi) \]
Substituting the values of \( A \), \( k \), and \( \omega \):
\[ y(x, t) = 0.02 \sin(2\pi x - 10\pi t + \phi) \]

### Step 5: Apply the initial conditions
At \( x = 0 \) and \( t = 0 \), we have:
\[ y(0, 0) = 0 \quad \text{and} \quad \frac{dy}{dt} < 0 \]
Substituting into the wave function:
\[ y(0, 0) = 0.02 \sin(\phi) = 0 \]
This implies:
\[ \sin(\phi) = 0 \implies \phi = n\pi \quad (n \in \mathbb{Z}) \]

### Step 6: Determine the value of \( n \)
Next, we differentiate \( y(x, t) \) with respect to \( t \):
\[ \frac{dy}{dt} = 0.02 \cdot (-10\pi) \cos(2\pi x - 10\pi t + \phi) \]
At \( x = 0 \) and \( t = 0 \):
\[ \frac{dy}{dt} = -0.02 \cdot 10\pi \cos(\phi) \]
For \( \frac{dy}{dt} < 0 \), we need \( \cos(\phi) > 0 \). This occurs when \( \phi = 0, 2\pi, 4\pi, \ldots \) (even multiples of \( \pi \)).

### Step 7: Final wave function
Choosing \( n = 0 \) (the simplest case), we have:
\[ \phi = 0 \]
Thus, the wave function becomes:
\[ y(x, t) = 0.02 \sin(2\pi x - 10\pi t) \]

### Conclusion
The wave function for the sinusoidal wave traveling in the positive direction is:
\[ \boxed{y(x, t) = 0.02 \sin(2\pi x - 10\pi t)} \]