Show that the equation,`y=a sin(omegat-kx)` satisfies the wave equation`(del^(2) y)/(delt^(2))=v^(2) (del^(2)y)/(delx^(2))`. Find speed of wave and the direction in which it is travelling.

Explanation

To show that the equation \( y = a \sin(\omega t - kx) \) satisfies the wave equation

\[ \frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}, \]

we will follow these steps:

### Step 1: Differentiate \( y \) with respect to \( t \)

Given:
\[ y = a \sin(\omega t - kx) \]

We first find the first derivative of \( y \) with respect to \( t \):

\[ \frac{\partial y}{\partial t} = a \omega \cos(\omega t - kx) \]

### Step 2: Differentiate again with respect to \( t \)

Now, we differentiate \( \frac{\partial y}{\partial t} \) again with respect to \( t \):

\[ \frac{\partial^2 y}{\partial t^2} = -a \omega^2 \sin(\omega t - kx) \]

### Step 3: Differentiate \( y \) with respect to \( x \)

Next, we find the first derivative of \( y \) with respect to \( x \):

\[ \frac{\partial y}{\partial x} = -a k \cos(\omega t - kx) \]

### Step 4: Differentiate again with respect to \( x \)

Now, we differentiate \( \frac{\partial y}{\partial x} \) again with respect to \( x \):

\[ \frac{\partial^2 y}{\partial x^2} = -a k^2 \sin(\omega t - kx) \]

### Step 5: Substitute into the wave equation

Now we substitute \( \frac{\partial^2 y}{\partial t^2} \) and \( \frac{\partial^2 y}{\partial x^2} \) into the wave equation:

\[ -a \omega^2 \sin(\omega t - kx) = v^2 (-a k^2 \sin(\omega t - kx)) \]

### Step 6: Simplify the equation

We can simplify this equation by dividing both sides by \( -a \sin(\omega t - kx) \) (assuming \( a \neq 0 \) and \( \sin(\omega t - kx) \neq 0 \)):

\[ \omega^2 = v^2 k^2 \]

### Step 7: Solve for \( v \)

From the equation \( \omega^2 = v^2 k^2 \), we can solve for \( v \):

\[ v^2 = \frac{\omega^2}{k^2} \]

Taking the square root of both sides, we find:

\[ v = \frac{\omega}{k} \]

### Step 8: Determine the direction of the wave

The wave function \( y = a \sin(\omega t - kx) \) indicates that the wave is traveling in the positive x-direction because of the negative sign in front of \( kx \).

### Summary of Results

- The speed of the wave is given by \( v = \frac{\omega}{k} \).
- The wave is traveling in the positive x-direction.